ARS MATHEMATICA CONTEMPORANEA Volume 6, Number 2, Fall/Winter 2013, Pages 187-433 Covered by: Mathematical Reviews Zentralblatt MATH COBISS SCOPUS Science Citation Index-Expanded (SCIE) Web of Science ISI Alerting Service Current Contents/Physical, Chemical & Earth Sciences (CC/PC & ES) The Society of Mathematicians, Physicists and Astronomers of Slovenia The Institute of Mathematics, Physics and Mechanics The University of Primorska ARS MATHEMATICA CONTEMPORANEA Features of growth Every organism should grow and progress, until its development stops and it starts the path towards its end. But for it to have a long and healthy life, its growth must be moderate. Growing too fast can be very dangerous; uncontrolled growth may cause damage, and even death — for individuals and businesses, and also for journals. We started this journal in 2008, publishing 20 papers in that first year. The number of papers grew to 35 in 2013 (a 75 percent increase). More than half of the papers for the year 2014 are already on-line, and we have a growing back-log of papers accepted but not yet published. The quality of the research published in these papers and the inclusion of the journal in various databases have made it popular among mathematicians worldwide. The number of submissions to our journal is also growing, with over 15 in each of September and October this year, and correspondingly, both the number and percentage of papers that we have to reject or redirect to other journals are increasing as well. We would like to shape this journal into a self-consistent form that will attract the best possible papers from a rich and wide range of fields of mathematics, while retaining an expectation that their content combines at least two branches of a discrete nature. To pursue this goal, however, we must carefully control the growth of our journal, with respect to its size and maturity. That explains why we are taking some novel approaches to the journal's production. For reasons of business viability, we changed the main publisher from a learned society to a university. We decided to apply for support being offered by the Republic of Slovenia to scientific journals, and a visible consequence is the translation of abstracts into the Slovenian language. Next, because we are committed to preserving our policy that neither readers nor authors should pay for access to the journal's papers over the internet, from 2014 we are introducing a 'Creative Commons Copyright' model for our journal. We hereby announce that Ars Mathematica Contemporanea will publish four issues per year, from 2015. If you wish to support our journal and help with the long-term preservation of its contents, please subscribe to Ars Mathematica Contemporanea, and ask your library to subscribe to the printed edition. Dragan Marušic and Tomaž Pisanski Editors-in-Chief Contents The excluded minor structure theorem with planarly embedded wall Bojan Mohar.................................187 Maximum independent sets of the 120-cell and other regular polytopes Sean Debroni, Erin Delisle, Wendy Myrvold, Amit Sethi, Joseph Whitney, Jennifer Woodcock, Benoit de La Vaissière .................197 Wiener index of iterated line graphs of trees homeomorphic to the claw Ki,3 Martin Knor, Primož Potocnik, Riste Škrekovski ..............211 Orienting and separating distance-transitive graphs Italo J. Dejter.................................221 A parallel algorithm for computing the critical independence number and related sets Ermelinda DeLaVina, Craig E. Larson....................237 A note on homomorphisms of matrix semigroup Matjaž Omladic, Bojan Kuzma........................247 On geometric trilateral-free (n3) configurations Michael W. Raney...............................253 Interlacing-extremal graphs Irene Sciriha, Mark Debono, Martha Borg..................261 Unordered multiplicity lists of wide double paths Aleksandra Eric, C. M. da Fonseca......................279 GCD-Graphs and NEPS of Complete Graphs Walter Klotz, Torsten Sander.........................289 The bipartite graphs of abelian dessins d'enfants Rubén A. Hidalgo...............................301 On stratifications for planar tensegrities with a small number of vertices Oleg Karpenkov, Jan Schepers, Brigitte Servatius..............305 Relations between graphs Jan Hubicka, Jürgen Jost, Yangjing Long, Peter F. Stadler..........323 CI-groups with respect to ternary relational structures: new examples Edward Dobson, Pablo Spiga.........................351 On the rank two geometries of the groups PSL(2, q): part II Francis Buekenhout, Julie De Saedeleer, Dimitri Leemans ......... 365 A note on a conjecture on consistent cycles Štefko Miklavic................................389 Sectional split extensions arising from lifts of groups Rok Požar...................................393 Augmented down-up algebras and uniform posets Paul Terwilliger, Chalermpong Worawannotai................409 Simplicial arrangements revisited Branko Grünbaum ..............................419 Volume 6, Number 2, Fall/Winter 2013, Pages 187-433 d MFA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 187-196 The excluded minor structure theorem with planarly embedded wall Bojan Mohar * Department of Mathematics, Simon Fraser University, Burnaby, B.C. V5A 1S6, Canada Received 25 September 2009, accepted 29 May 2012, published online 11 July 2012 Abstract A graph is "nearly embedded" in a surface if it consists of graph G0 that is embedded in the surface, together with a bounded number of vortices having no large transactions. It is shown that every large wall (or grid minor) in a nearly embedded graph, many rows of which intersect the embedded subgraph G0 of the near-embedding, contains a large subwall that is planarly embedded within G0. This result provides some hidden details needed for a strong version of the Robertson and Seymour's excluded minor theorem as presented in [1]. Keywords: Graph, graph minor, surface, near-embedding, grid minor, excluded minor. Math. Subj. Class.: 05C10, 05C82 1 Introduction A graph is a minor of another graph if the first can be obtained from a subgraph of the second by contracting edges. One of the highlights of the graph minors theory developed by Robertson and Seymour is the Excluded Minor Theorem (EMT) that describes a rough structure of graphs that do not contain a fixed graph H as a minor. Two versions of EMT appear in [7, 8]; see also [3] and [4]. In [1] and [2] the authors used a strong version of EMT in which it is concluded that every graph without a fixed minor and whose tree-width is large has a tree-like structure, whose pieces are subgraphs that are almost embedded in some surface, and in which one of the pieces contains a large grid minor that is (essentially) embedded in a disk on the surface. Although not explicitly mentioned, this version of EMT follows from the published results of Robertson and Seymour [8] by applying standard techniques of routings on surfaces. * Supported in part by ARRS, Research Program P1-0297, by an NSERC Discovery Grant and by the Canada Research Chair program. On leave from Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia. E-mail address: mohar@sfu.ca (Bojan Mohar) Experts in this area are familiar with these techniques (that are also present in Robertson and Seymour's work [6]). However, they may be harder to digest for newcomers in the area, and thus deserve to be presented in the written form. The purpose of this note is to provide a proof of an extended version of EMT as stated in [1, Theorem 4.2]. It may be worth mentioning that the proof in [1] does not really need the extended version of the EMT, but the proof in [2] does. Thus, this note may also be viewed as a support for the main proof in [2]. We assume that the reader is familiar with the basic notions of graph theory and in particular with the basic notions related to graph minors; we refer to [3] for all terms and results not explained here. 2 Walls in near-embeddings In this section, we present our main lemma, which shows that for every large wall (to be defined in the sequel) in a "nearly embedded" graph, a large subwall must be contained in the embedded subgraph of the near-embedding. Let us first introduce the notion of the wall and some of its elementary properties. Figure 1: The cylindrical 6-wall Q6 For an integer r > 3, we define a cylindrical r-wall as a graph that is isomorphic to a subdivision of the graph Qr defined as follows. We start with vertex set V = {(i,j) | 1 < i < r, 1 < j < 2r}, and make two vertices (i, j) and (i',j') adjacent if and only if one of the following possibilities holds: (1) i' = i and j' G {j - 1, j + 1}, where the values j - 1 and j + 1 are considered modulo 2r. (2) j' = j and i' = i + (-1)i+j. Less formally, Qr consists of r disjoint cycles Ci,... ,Cr of length 2 r (where V (Cj) = {(i, j) | 1 < j < 2r}), called the meridian cycles of Qr. Any two consecutive cycles Ci and Ci+i are joined by r edges so that the edges joining Ci and Ci_i interlace on Ci with those joining Ci and Ci+1 for 1 < i < r. Figure 1 shows the cylindrical 6-wall Q6. By deleting the edges joining vertices (i, 1) and (i, 2r) for i = 1,..., r, we obtain a subgraph of Qr. Any graph isomorphic to a subdivision of this graph is called an r-wall. To relate walls and cylindrical walls to (r x r)-grid minors, we state the following easy correspondence: (a) Every (4r + 2)-wall contains a cylindrical r-wall as a subgraph. (b) Every cylindrical r-wall contains an (r x r)-grid as a minor. (c) Every (r x r)-grid minor contains an |_r__1 J -wall as a subgraph. Lemma 2.1. Suppose that 1 < i < j < r and let t = j - i - 1. Let Si C Ci and Sj C Cj be paths of length at least 2t - 1 in the meridian cycles Ci, Cj of Qr. Then Qr contains t disjoint paths linking Ci and Cj. Moreover, for each of these paths and for every cycle Ck, i < k < j, the intersection of the path with Ck is a connected segment of Ck. Proof. The lemma is easy to prove and the idea is illustrated in Figure 2, in which the edges on the left are assumed to be identified with the corresponding edges on the right. The paths are shown by thick lines and the segments Si and Sj are shown by thick broken lines. □ A surface is a compact connected 2-manifold (with or without boundary). The components of the boundary are called the cuffs. If a surface S has Euler characteristic c, then the non-negative number g = 2 - c is called the Euler genus of S. Note that a surface of Euler genus g contains at most g cuffs. Disjoint cycles C, C' in a graph embedded in a surface S are homotopic if there is a cylinder in S whose boundary components are the cycles C and C'. The cylinder bounded by homotopic cycles C, C is denoted by int(C, C'). Disjoint paths P, Q whose initial vertices lie in the same cuff C and whose terminal vertices lie in the same cuff C in S (possibly C' = C) are homotopic if P and Q together with a segment in C and a segment in C' form a contractible closed curve A in S. The disk bounded by A will be denoted by int(P, Q). The following basic fact about homotopic curves on a surface will be used throughout (cf., e.g., [5, Propositions 4.2.6 and 4.2.7]). Lemma 2.2. Let S be a surface of Euler genus g. Then every collection of more than 3g disjoint non-contractible cycles contains two cycles that are homotopic. Similarly, every collection of more than 3g disjoint paths, whose ends are on the same (pair of) cuffs in S, contains two paths that are homotopic. Let G be a graph and let W = {w0,..., wn}, n = |W| - 1, be a linearly ordered subset of its vertices such that wi precedes wj in the linear order if and only if i < j. The pair (G, W) is called a vortex of length n, W is the society of the vortex and all vertices in W are called society vertices. When an explicit reference to the society is not needed, we will as well say that G is a vortex. A collection of disjoint paths R1, ...,Rk in G is called a transaction of order k in the vortex (G, W) if there exist i, j (0 < i < j < n) such that all paths have their initial vertices in {wj, wi+i,..., wj} and their endvertices in W \{wj,wi+i,...,wj }. Let G be a graph that can be expressed as G — Go U Gi U • • • U Gv, where Go is embedded in a surface S of Euler genus g with v cuffs Oi,..., Qv, and G i (i = 1,..., v) are pairwise disjoint vortices, whose society is equal to their intersection with G0 and is contained in the cuff Qi, with the order of the society being inherited from the circular order around the cuff. Then we say that G is near-embedded in the surface S with vortices Gi,..., Gv .A subgraph H of a graph G that is near-embedded in S is said to be planarly embedded in S if H is contained in the embedded subgraph G0, and there exists a cycle C C G0 that is contractible in S and H is contained in the disk on S that is bounded by C. Our main result is the following. Theorem 2.3. For every non-negative integers g, v, a there exists a positive integer s = s (g, v, a) such that the following holds. Suppose that a graph G is near-embedded in the surface S with vortices Gi,..., Gv, such that the maximum order of transactions of the vortices is at most a. Let Q be a cylindrical r-wall contained in G, such that at least r0 > 3s of its meridian cycles have at least one edge contained in G0. Then Q n G0 contains a cylindrical r'-wall that is planarly embedded in S and has r' > r0/s. Proof. Let Cpi, Cp2,..., Cp (pi < p2 < • • • < pr0 ) be meridian cycles of Q having an edge in G0. For i = 1,..., r0, let Li be a maximal segment of CPi containing an edge in E(CPi ) n E(G0) and such that none of its vertices except possibly the first and the last vertex are on a cuff. It may be that Li = CPi if CPi contains at most one vertex on a cuff; if not, then Li starts on some cuff and ends on (another or the same) cuff. (We think of the meridian cycles to have the orientation as given by the meridians in the wall.) At least r0/(v2 + 1) of the segments Li either start and end up on the same cuffs Qx and Qy (possibly x = y), or are all cycles. In each case, we consider their homotopies. By Lemma 2.2, these segments contain a subset of q > r0/((3g + 1)(v2 + 1)) homotopic segments (or cycles). Since we will only be interested in these homotopic segments or cycles, we will assume henceforth that Li,..., Lq are homotopic. Let us first look at the case when Li,..., Lq are cycles. Since s = s(g, v, a) can be chosen to be arbitrarily large (as long as it only depends on the parameters), we may assume that q is as large as needed in the sequel. If the cycles L are pairwise homotopic and non-contractible, then it is easy to see that two of them bound a cylinder in S containing many of these cycles. This cylinder also contains the paths connecting these cycles; thus it contains a large planarly embedded wall and hence also a large planarly embedded cylindrical wall. So, we may assume that the cycles Ll,...,Lq are contractible. By Lemma 2.1, Q contains t paths linking any two of these cycles that are t apart in Q, say C = Lj and C' = Li+t+1. (Here we take t large enough that the subsequent arguments will work.) Again, many of these paths either reach C' without intersecting any of the cuffs, or many reach the same cuff Q. A large subset of them is homotopic. In the former case, the paths linking C' with C'' = Lj+2t+2 can be chosen so that their initial vertices interlace on C' with the end-vertices of the homotopic paths coming from C. This implies that C or C'' lies in the disk bounded by C' (cf. Figure 3). By repeating the argument, we obtain a sequence of nested cycles and interlaced linkages between them. This clearly gives a large subwall, which contains a large cylindrical subwall that is planarly embedded. In the latter case, when the paths from C to C' go through the same cuff Qj, we get a contradiction since the vortex on Qj does not admit a transaction of large order, and thus too many homotopic paths cannot reach C ''. Figure 4: Many homotopic segments joining two cuffs We get a similar contradiction as in the last case above, when too many homotopic segments Lj start and end up on the same cuffs Qx and Qy. We shall give details for the case when x = y, but the same approach works also if x = y. (In the case when x = y and the homotopic segments Li are contractible, the proof is similar to the part of the proof given above.) Let us consider the "extreme" segments Li, L j, whose disk int(Li, L j ) contains many homotopic segments (cf. Figure 4). Let us enumerate these segments as L' = Li, L'2,..., Lm = L j in the order as they appear inside int(Li,Lj ). Let C't (for 1 < t < m) be the meridian cycle containing the segment L't. Since vortices admit no transactions of order more than a, at most 4a of the cycles C' (1 < t < m) can leave int(Li, Lj). By adjusting m, we may thus assume that none of them does. In particular, each L't has another homotopic segment in int(Li, Lj). Since there are no transactions of order more than a, there is a large subset of the cycles C' that follow each other in int(Li, L j ) as shown by the thick cycles in Figure 4. Consider four of these meridian cycles A, B,C, D that are pairwise far apart in the wall Q and appear in the order A, B, C, D within int(Li, Lj). Then A and C are linked in Q by a large collection of disjoint paths by Lemma 2.1. At most 8a of these paths can escape intersecting two fixed segments L'u and L'v of B or two such segments of D by passing through a vortex. All other paths linking A and C intersect either two segments of B or two segments of D. However, this is a contradiction since the paths linking A and C can be chosen in Q so that each of them intersects each meridian cycle in a connected segment (Lemma 2.1). This completes the proof. □ 3 The excluded minor structure In this section, we define some of the structures found in Robertson-Seymour's Excluded Minor Theorem [7] which describes the structure of graphs that do no contain a given graph as a minor. Robertson and Seymour proved a strengthened version of that theorem that gives a more elaborate description of the structure in [8]. Our terminology follows that introduced in [1]. Let G0 be a graph. Suppose that (G'1,G'2) is a separation of G0 of order t < 3, i.e., G0 = G'1 U G', where Gl n G'2 = {v1 ,...,vt} C V (G0), 1 < t < 3, V (G'2) \ V (G') = 0. Let us replace G0 by the graph G', which is obtained from G' by adding all edges vivj (1 < i < j < t) if they are not already contained in G'. We say that G' has been obtained from G0 by an elementary reduction. If t = 3, then the 3-cycle T = v'v2v3 in G' is called the reduction triangle. Every graph G" that can be obtained from G0 by a sequence of elementary reductions is a reduction of G0. We say that a graph G0 can be embedded in a surface S up to 3-separations if there is a reduction G" of G0 such that G" has an embedding in S in which every reduction triangle bounds a face of length 3 in S. Let H be an r-wall in the graph G0 and let G" be a reduction of G0. We say that the reduction G" preserves H if for every elementary reduction used in obtaining G" from G0, at most one vertex of degree 3 in H is deleted. (With the above notation, G' \ G' contains at most one vertex of degree 3 in H.) Lemma 3.1. Suppose that G" is a reduction of the graph G0 and that G" preserves an r-wall H in G0. Then G" contains an |_(r + 1)/3_|-wall, all of whose edges are contained in the union of H and all edges added to G" when performing elementary reductions. Proof. Let H' be the subgraph of the r-wall H obtained by taking every third row and every third "column". See Figure 5 in which H' is drawn with thick edges. It is easy to Figure 5: Smaller wall contained in a bigger wall see that for every elementary reduction we can keep a subgraph homeomorphic to H' by replacing the edges of H' which may have been deleted by adding some of the edges vivj involved in the reduction. The only problem would occur when we lose a vertex of degree 3 and when all vertices vi, v2, v3 involved in the elementary reduction would be of degree 3 in H'. However, this is not possible since G" preserves H. □ Suppose that for i = 0,... ,n, there exist vertex sets, called parts, Xi C V (G), with the following properties: (V1) Xi n W = {wi, wi+1} for i = 0,... ,n, where wn+1 = wn, (V2) Uc 2, admits a linked vortex decomposition; just take Xi = (V(G) \ W) U {wi, wi+1}. The (linked) adhesion of the vortex is the minimum adhesion taken over all (linked) decompositions of the vortex. Let us observe that in a linked decomposition of adhesion q, there are q disjoint paths linking Z1 with Zn in G - W. For us it is important to note that a vortex with adhesion less than k does not admit a transaction of order more than k. Let G be a graph, H an r-wall in G, S a surface, and a > 0 an integer. We say that G can be a-nearly embedded in S if there is a set of at most a cuffs C1,..., Cb (b < a) in S, and there is a set A of at most a vertices of G such that G - A can be written as G0 U Gì U • • • U Gb where G0, Gì,..., Gb are edge-disjoint subgraphs of G and the following conditions hold: (N1) G0 can be embedded in S up to 3-separations with G" being the corresponding reduction of G0. (N2) If 1 < i < j < b, then V (G,) n V (Gš ) = 0. (N3) Wi = V (G0) n V (G,) = V (G") n C, for every i = 1,..., b. (N4) For every i = 1,..., b, the pair (G,, W,) is a vortex of adhesion less than a, where the ordering of W, is consistent with the (cyclic) order of these vertices on C,. The vertices in A are called the apex vertices of the a-near embedding. The subgraph G0 of G is said to be the embedded subgraph with respect to the a-near embedding and the decomposition G0, G1,..., Gb. The pairs (G,, W,), i = 1,..., b, are the vortices of the a-near embedding. The vortex (G,, W,) is said to be attached to the cuff C, of S containing W,. If G is a-near-embedded in S, let G0, G1,..., Gb be as above and let G" be the reduction of G0 that is embedded in S. If H is an r-wall in G, we say that H is captured in the embedded subgraph G0 of the a-near-embedding if H is preserved in the reduction G" and for every separation G = K U L of order less than r, where G0 C K, at least § of the degree-3 vertices of H lie in K. We shall use the following theorem which is a simplified version of one of the cornerstones of Robertson and Seymour's theory of graph minors, the Excluded Minor Theorem, as stated in [8]. For a detailed explanation of how the version in this paper can be derived from the version in [8], see the appendix of [1]. Theorem 3.2 (Excluded Minor Theorem). For every graph R, there is a constant a such that for every positive integer w, there exists a positive integer r = r (R, a, w), which tends to infinity with w for any fixed R and a, such that every graph G that does not contain an R-minor either has tree-width at most w or contains an r-wall H such that G has an a-near embedding in some surface S in which R cannot be embedded, and H is captured in the embedded subgraph of the near-embedding. We can add the following assumptions about the r-wall in Theorem 3.2. Theorem 3.3. It may be assumed that the r-wall H in Theorem 3.2 has the following properties: (a) H is contained in the reduction G" of the embedded subgraph G0. (b) H is planarly embedded in S, i.e., every cycle in H is contractible in S and the outer cycle of H bounds a disk in S that contains H. (c) We may prespecify any constant p and ask that the face-width of G" be at least p. (d) G" is 3-connected. Proof. The starting point is Theorem 3.2. By making additional elementary reductions if necessary, we can achieve (d). The property (c) is attained as follows. If the face-width is too small, then there is a set of less than p vertices whose removal reduces the genus of the embedding of G". We can add these vertices in the apex set and repeat the procedure as long as the face-width is still smaller than p. The only subtlety here is that the constant a in Theorem 3.2 now depends not only on R but also on p. See also [4]. After removing the apex set A, we are left with an (r - a)-wall in G - A. By applying Lemma 3.1, we may assume that H is contained in the reduced graph G" U G\ U • • • U Gb. The wall H contains a large cylindrical wall Q. Since the vortices have bounded adhesion, they do not have large transactions. Since the wall is captured in G", edges of many meridians of Q lie in G". Therefore, we can apply Theorem 2.3 for the near embedding of the reduced graph together with the vortices. This shows that a large cylindrical subwall of Q is planarly embedded in the surface. The size r' of this smaller wall still satisfies the condition that r' = r'(R, a,w) ^ to as w increases. □ It is worth mentioning that there are other ways to show that a graph with large enough tree-width that does not contain a fixed graph R as a minor contains a subgraph that is a-near-embedded in some surface S in which R cannot be embedded, and moreover, there is an r-wall planarly embedded in S (after reductions taking care of at most 3-separations). Let us describe two of them: (A) Large face-width argument: One can use property (c) in Theorem 3.3 that the face-width p can be made as large as we want if a = a(R, w, p) is large enough. Once we have that, it follows from [6] that there is a planarly embedded r-wall, where r = r(R, p) ^ to as p ^ to. While this easy argument is sufficient for most applications, it appears to be slightly weaker than Theorem 3.3 since the quantifiers change. The difference is that the number of apex vertices is no longer bounded as a function of a = a(R) but rather as a function depending on R and r, where the upper bound has linear dependence on r, i.e. it is of the form ß(R)r. However, other parameters of the near-embedding keep being only dependent on R. (B) Irrelevant vertex: The third way of establishing the same result is to go through the proof of Robertson and Seymour that there is an irrelevant vertex, i.e. a vertex v such that G has an R-minor if and only if G - v has. (Compared to the later, more abstract parts of the graph minors series of papers, this part is very clean and well understood; it could (and should) be explained in a(ny) serious graduate course on graph minors.) In that proof, one starts with an arbitrary wall W that is large enough. A large wall exists since the tree-width is large. Then one compares the W-bridges attached to W. They may give rise to < 3-separations, to jumps (paths in bridges whose addition to W yields a nonplanar graph), crosses (pairs of disjoint paths attached to the same planar face of W whose addition to W yields a nonplanar graph). If there are many disjoint jumps or crosses on distinct faces of W, one can find an R-minor. If there are just a few, there is a large planar wall. If there are many of them on the same face, we get a structure of a vortex with bounded transactions (or else an R-minor can be discovered). The proof then discusses ways for many jumps and crosses but no large subset of them being disjoint. One way is to have a small set of vertices whose removal destroys most of these jumps and crosses. This gives rise to the apex vertices. The final conclusion is that the jumps and crosses can affect only a bounded part of the wall, so after the removal of the apex vertices and after elementary reductions which eliminate < 3-separations, there is a large subwall W0 such that no jumps or crosses are involved in it. The "middle" vertex in W0 is then shown to be irrelevant. For our reference, only this planar wall is needed. By being planar, we mean that the rest of the graph is attached only to the outer face of this wall. Then we define the tangle corresponding to this wall and the proof of the EMT preserves this tangle while making the modifications yielding to an a-near-embedding. References [1] T. Böhme, K. Kawarabayashi, J. Maharry and B. Mohar, Linear connectivity forces large complete bipartite minors, J. Combin. Theory, Ser. B 99 (2009), 557-582. [2] T. Bohme, K. Kawarabayashi, J. Maharry and B. Mohar, K3>k -minors in large 7-connected graphs, submitted to J. Combin. Theory, Ser. B, May 2008. [3] R. Diestel, Graph Theory, 3rd Edition, Springer, 2005. [4] K. Kawarabayashi and B. Mohar, Some recent progress and applications in graph minor theory, Graphs Combin. 23 (2007), 1-46. [5] B. Mohar and C. Thomassen, Graphs on Surfaces, Johns Hopkins University Press, Baltimore, MD, 2001. [6] N. Robertson and P. D. Seymour, Graph minors. VII. Disjoint paths on a surface, J. Combin. Theory Ser. B 45 (1988), 212-254. [7] N. Robertson and P. D. Seymour, Graph minors. XVI. Excuding a non-planar graph, J. Combin. Theory Ser. B 89 (2003), 43-76. [8] N. Robertson and P. D. Seymour, Graph minors. XVII. Taming a vortex, J. Combin. Theory Ser. B 77 (1999), 162-210. d MFA Also available at http://amc.imfm.si ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 197-210 Maximum independent sets of the 120-cell and other regular polytopes SeanDebroni, ErinDelisle, Wendy Myrvold *, Amit Sethi Joseph Whitney, Jennifer Woodcock Department of Computer Science, University of Victoria P. O. Box 3055, Stn CSC, Victoria, BC Canada V8W 3P6 Patrick W. Fowler t, Benoit de La Vaissiere Department of Chemistry, University of Sheffield, Sheffield S3 7HF, UK Michel Deza CNRS and LIGA, Ecole Normale Supérieure, 45 rue d'Ulm, 75230 Paris, France Received 4 August 2010, accepted 20 March 2012, published online 26 July 2012 Abstract A d-code in a graph is a set of vertices such that all pairwise distances are at least d. As part of a study of d-codes of three-and four-dimensional regular polytopes, the maximum independent set order of the 120-cell is calculated. A linear program based on counting arguments leads to an upper bound of 221. An independent set of order 110 in the antipodal collapse of the 120-cell (also known as the hemi-120-cell) gives a lower bound of 220 for the 120-cell itself. The gap is closed by the computation described here, with the result that the maximum independent set order of the 120-cell is 220. All maximum d-code orders of the icosahedron, dodecahedron, 24-cell, 600-cell and 120-cell are reported. Keywords: Graphs, independent sets, polyhedra, polytopes. Math. Subj. Class.: 05C69, 05C85, 05C10 * Supported by NSERC. t Supported by Royal Society/Wolfson Research Merit Award, 2004-2009. E-mail addresses: wendym@cs.uvic.ca (Wendy Myrvold), P.W.Fowler@sheffield.ac.uk (Patrick W. Fowler), Michel.Deza@ens.fr (Michel Deza) 1 Introduction An independent set in a graph is a set of pairwise non-adjacent vertices such that all pairs are at distance two or more. A clique (a subset of the vertices that are pairwise adjacent) in a graph corresponds to an independent set in the complement of the graph. Hence algorithms for maximum clique can be applied to find maximum independent sets. The problem of finding a maximum independent set in a graph is NP-Hard [12]. The DIMACS Clique Challenge arose from the need to find practical algorithms for the maximum clique problem, and the proceedings volume is an excellent place to start looking for information about practical algorithms for clique finding [14]. The challenge also included a database of difficult clique problems. Bomze, Budinich, Pardalos, and Pelillo [1] provide a comprehensive survey on the maximum clique problem. Ostergard focusses on solving maximum clique problems on graphs arising from various combinatorial problems. Both surveys cite the problem of finding a maximum clique of the Keller graph of dimension 7 as a open problem. This problem has subsequently been solved [5]. In general, a d-code is a set of vertices such that pairwise distances are all at least d. The concept of distance between vertices a and b may be cast in terms of graphs (number of edges in the shortest path between the vertices), coding theory (the Hamming distance between (0,1) sequences representing coordinates of vertices), or sets (the cardinality of the symmetric difference between the two subsets that represent the vertices). A d-code in a graph G corresponds to an independent set in the graph H which has the same vertex set as G and the property that two vertices u and v are adjacent in H if and only if the distance between them in G is at most d -1. The interest for applications is usually to find maximum d-codes, and one standard problem in the theory of error-correcting codes [16, 2] is to find the largest d-code in the n-cube. Here we consider the problem of finding largest d-codes in the graphs corresponding to other regular polytopes. For polycyclic and polyhedral graphs in two and three dimensions, the construction of d-codes has applications to chemistry [15, 4, 7, 8]. For example, sets of codes with increasing d may be seen as templates for addition to an underlying molecular framework by addends of increasing steric demand. Codes have been presented for chemically relevant regular and semi-regular polyhedra [15] and arguments based on d-codes, coupled with spectral information, give useful rationalisations of the extent and symmetry of addition in fullerene chemistry, for example [9, 10, 11]. Although not invoked in chemistry so far, d-anticodes, defined by the requirement that pairwise distances should not exceed d, would model the opposite regime of attachment to a framework where the added groups cluster under strong inter-addend attraction. Extension of the existing lists [15] to d-codes in the graphs of all regular polytopes is a finite task, as there are only the following convex regular polytopes [3]: in dimension n the n-simplex (an), the n-cross-polytope (ßn), the n-cube (Y„), and additionally in dimension 2 the regular polygons, and in dimensions 3 and 4, five sporadic polytopes. In dimension 3, a3 is the tetrahedron, ß3 the octahedron and 73 the cube, and there is a dual pair of sporadic polyhedra: the icosahedron and the dodecahedron. In dimension 4, the analogues of the icosahedron and dodecahedron are the 600-cell (all of its independent sets have been enumerated previously [6]) and the 120-cell (again a dual pair), and there is also the self-dual 24-cell, without analogue in higher or lower dimensions [3]. Codes for the polytopes common to all dimensions (an, ßn and jn) are well studied. The 1-skeleton of an is the complete graph Kn, of diameter 1, and the 1-skeleton of ßn is the complete multipartite graph (the Cocktail-Party graph) Kn(2) = Cp(n) = K2,2,...,2, of diameter 2, so coding problems are trivial for both. Codes for y„ are the subject of classical coding theory [16]. As a bipartite graph with equal partite sets, y„ has independence number 2n-1. The coding problem is also trivial in two dimensions, where the order of the d-code is [n/d\ for the cycle of length n. It only remains to study the five exceptional regular polytopes in dimensions 3 and 4. Most of these problems are easy (see the summary in Table 3, §3). The difficult case is that of the independence number of the 600-vertex 120-cell, which does not appear to be computable in a reasonable amount of time by use of standard algorithms. The solution to this problem is described in the following. 2 Maximum Independent Sets of the 120-cell The 120-cell is the largest regular polytope in 4 dimensions. Its properties are described in Coxeter's book on Regular Polytopes [3] and Stillwell's survey paper [20], for example. The 1-skeleton of the 120-cell is a 4-regular graph with 600 vertices, 720 pentagonal faces and 120 dodecahedral cells. Models exhibiting three-dimensional projections of the complete object have been constructed; photographs of Donchian's models are shown in [3]. A partial model of the 120-cell is given as an example of a construction using Zome Models [13, Ch. 21]; this 330-vertex subgraph has 45 of the 120 dodecahedra, arranged in concentric shells of 1, 12 and 32 face-sharing cells. In the following subsections, the steps leading to the solution of the problem of finding the maximum independent set order of the 120-cell are described. First, a description is given of how the vertices of the graph are numbered and how its automorphism group is computed (§2.1). Then (§2.2) a lower bound of 220 for the maximum independent set order is derived from an independent set of order 110 in the antipodal collapse of the 120-cell. An upper bound of 221 is established by use of a linear program (§2.3), and the information from the solution to the integer program is then exploited (§2.4) to infer structural information about a putative independent set of order 221. This information is subsequently used in the computational search described in the remaining subsections, which establishes that the maximum independent set order of the 120-cell is 220. 2.1 Numbering the Graph and its Automorphism Group A special breadth-first search was used for numbering the 120-cell and finding a permutation representation of its automorphism group on the 600 vertices. The search in question is performed as follows: Clockwise BFS Labelling Algorithm: Input: an adjacency list for the 120-cell or its collapse. Output: a canonical labelling for the graph and its automorphisms expressed in terms of permutations of the vertices of the canonically labelled graph. To obtain the initial canonical labelling, select one vertex to be labelled as vertex 0 and then choose one way to label its four neighbours as 1, 2, 3 and 4. The remaining vertices are labelled using a breadth-first search starting at vertex 0, and visiting its neighbours 1, 2, 3, and 4 in order. In order to make the breath-first search labelling deterministic, the neighbours of a vertex v are visited in an order which is decided as follows. Each unlabelled neighbour u of vertex v is in one pentagon with vertex v and the breadth-first search parent p of vertex v. Let the other two vertices of the pentagon be x and y so that the vertices of rithm. this pentagon in cyclic order are u, v, p, x, y. Since x is a neighbour of p (and p is the breadth-first search parent of v), x has already been labelled. The order of the neighbours of vertex v is selected so that the labels of the vertices indicated by x in their pentagons are sorted in increasing order. Figure 1 shows a portion of the 120-cell labelled this way. The 120-cell has an automorphism group of order 14,400. Obtaining the permutations of the automorphism group is easy using the clockwise BFS as described above, as they correspond to choosing a start vertex for the BFS in each of the 600 possible ways, and for each start, considering each of the 4! permutations of its neighbours (14,400 = 600 x 4!). 2.2 A Lower Bound from the Antipodal Collapse Given a vertex v of a graph, its antipodal vertices are those at maximum distance from v. The 120-cell has the property that each of its vertices has a unique antipodal vertex. The antipodal collapse of the 120-cell is obtained by identifying each vertex of the 120-cell with its antipodal vertex. If {u, u'} and {v, v'} are two sets of antipodal pairs of vertices of a graph G, then in the collapse, there is one edge between {u, u'} and {v, v'} corresponding to each edge of the form (u, v), (u, v'), (u', v), or (u', v') of the original graph G. Since multiple edges are inconsequential for the independent-set problem, each multiple edge is replaced by a single edge. The result is a 4-regular graph on 300 vertices which has the same local structure as the 120-cell. This graph is the 1-skeleton of the hemi-120-cell, one of the projective regular polytopes of rank 4 in projective 3-space [17, Section 6C]. The automorphism group order of the collapse is 300 x 4! = 7200, and the Clockwise BFS Labelling Algorithm from Section 2.1 is first used to label the vertices and find the automorphisms. An independent set of order k in the antipodal collapse can be lifted to one of order 2k in the 120-cell (if a vertex is in the independent set in the collapse, then include the two corresponding vertices of the 120-cell). A non-exhaustive computer search indicated that the antipodal collapse has at least 60 independent sets of order 110 up to isomorphism. This shows that the 120-cell has an independent set of order 220. The most symmetrical of the sets that we found in the antipodal collapse has stabiliser group of order 8. This set is lifted (Table 1) to give an independent set of order 220 in the 120-cell, with 16 automorphisms. 2.3 A Linear Programming Upper Bound An upper bound of 221 is not difficult to prove by solving a linear programming problem which sets up necessary constraints for a maximum independent set of the 120-cell. The nine variables for this linear program are as follows: R = the number of red (independent set) vertices Bi for i = 0,1, 2, 3,4 = the number of blue (not in the independent set) vertices having i red neighbours. Pi for i = 0,1, 2 = the number of pentagons with i red vertices. Each of the nine variables is constrained to be non-negative. The LP has six further constraints (five equalities and one inequality). These are introduced after proving some theorems required to justify the sixth constraint. The other constraints are all trivial conditions. A blue pentagon is a pentagon whose vertices are all blue (i.e., none of them are in the independent set). An isolated blue pentagon is defined to be a blue pentagon such that all of its ten incident vertices (i.e., the ten vertices that are adjacent to a vertex of the pentagon but are not themselves in the pentagon) are red. A blue pentagon with at least one incident blue vertex is called a non-isolated blue pentagon. A blue vertex with one red and three blue neighbours is called a key. A blue vertex with four red neighbours is called an isolated blue vertex. Remark 2.1. The independent set of order 220 listed in Table 1 has no isolated blue vertices and no keys. Theorem 2.2. For any maximum independent set, the number of non-isolated blue pentagons is at most Bi (the number of keys). Proof. Note that for a maximum independent set, it is not possible to have a blue vertex v with four blue neighbours since otherwise the independent set order could be increased by colouring v red. Therefore, if there is a blue pentagon which is a non-isolated blue 0 5 6 9 10 13 14 21 22 23 24 29 32 37 39 42 46 47 55 58 60 61 64 68 69 71 74 76 78 81 83 85 89 90 91 93 95 98 100 102 105 108 109 113 114 116 119 122 129 132 133 136 138 142 148 150 154 155 162 167 171 172 173 178 182 185 186 190 193 194 195 196 197 199 202 210 211 216 217 220 222 227 228 229 230 232 236 242 243 248 249 253 259 260 263 265 267 274 277 280 281 282 283 284 286 289 290 292 293 297 300 304 309 311 312 316 317 318 319 322 324 326 328 329 333 334 335 336 343 346 348 350 357 366 370 373 374 375 379 380 381 385 390 391 398 400 406 410 411 414 417 419 421 423 426 427 428 431 433 435 436 437 440 441 442 443 455 458 462 465 466 470 475 476 480 482 489 493 495 497 500 505 507 508 509 510 511 514 517 518 520 524 525 528 529 533 535 537 540 542 544 545 551 556 557 560 563 566 570 571 574 579 581 584 586 588 589 592 594 599 Table 1: An independent set of order 220 in the 120-cell generated from a set of order 110 in the antipodal collapse. pentagon, there must be at least one blue vertex on that pentagon which has one red and three blue neighbours (a key). The number of vertices like this is B1. This does not complete the proof however because a key can be on 0,1, 2, or 3 blue pentagons. To finish the proof, start by assigning a weight to each vertex v of the graph which is a key: assign a weight of one if v is contained in at least one non-isolated blue pentagon and zero otherwise. The sum of the weights of the keys is at most B1. Next, assign fractional weights to the non-isolated blue pentagons. If a key v is on r blue pentagons, this key contributes a weight of 1/r to each of its blue pentagons. The sum of the weights of the non-isolated blue pentagons is equal to the sum of the weights of the keys. To finish the proof, we argue that for each of the non-isolated blue pentagons, the sum of the contributions from its keys is at least one, meaning that the number of non-isolated blue pentagons is at most B1. This argument is broken down into three cases according to the types of keys on each non-isolated blue pentagon P. Case 1: Pentagon P contains a key v which is only in one non-isolated blue pentagon. In this case, the weight that v contributes to P is one and so P has weight at least one. Case 2: Pentagon P contains a key v which is in two non-isolated blue pentagons. Let A and B the the two non-isolated blue pentagons containing v and let (v, x) be the edge common to A and B. Vertex x is also a key. If it is a key which is in exactly two non-isolated blue pentagons then the weight of P is at least one, since each of v and x contributes 1/2 to the weight of P. If x is in three blue pentagons, then consider Case 3 instead of Case 2. Case 3: Pentagon P contains a key v which is in three non-isolated blue pentagons. Let the three blue neighbours of v be x, y and z where x and y are the vertices which are on P. Since v is on three non-isolated blue pentagons, x and y are either on two or three nonisolated blue pentagons and hence they contribute at least 1/3 to each pentagon they are on. Since P has contributions of at least 1/3 from v, x, and y, the sum of its contributions is at least one, as required. □ Corollary 2.3. For any maximum independent set of the 120-cell, the number of isolated blue pentagons is at least P0 — B1. Theorem 2.4. For a maximum independent set of the 120-cell, if I is the number of isolated blue pentagons, then 2I < Pi. Proof. In a dodecahedron, an isolated blue pentagon P appears as a blue pentagon with five incident red vertices. This means that the only possibility for another blue pentagon in the dodecahedron is the pentagon Q antipodal to P (all other pentagons contain at least one of the five reds). But the vertices in the dodecahedron incident to Q cannot be red (they have neighbours which are red) and therefore, a dodecahedron contains at most one isolated blue pentagon. Any edge (u, v) of the pentagon Q antipodal to P with both endpoints blue is in a pentagon P' with exactly one red vertex in the dodecahedron which contains P and Q. Since the pentagon Q has at least one edge with both endpoints blue, there is at least one pentagon P' with exactly one red vertex in the dodecahedron with P and Q. Each pentagon of the 120-cell falls into exactly two dodecahedra. To finish the proof, we argue that a pentagon P' with exactly one red vertex occurs as one which must be present as described above because of an isolated blue pentagon in at most one of its two dodecahedra. Suppose that P' corresponds to isolated blue pentagons in both of its two dodecahedra. Then the picture must be as in Figure 2 where the isolated blue pentagons are A and B, and P' is the pentagon with the bold edges. Vertex x is incident to A and vertex y is incident to B so both x and y must be red. This is a contradiction since x and y are adjacent to each other in the 120-cell. Figure 2: Two isolated blue pentagons A and B sharing a pentagon P' that has one red vertex (P' outlined in bold). The conclusion is that each isolated blue pentagon maps to at least one pentagon with exactly one red in each of its two dodecahedra. Further, such pentagons with one red correspond to at most one isolated blue pentagon of the graph. This implies that 21 < Pi. □ We now have the necessary theory to justify an integer programming problem which provides necessary constraints on a maximum independent set of the 120-cell. The conditions for the integer programming problem are: 1. Bi + B2 + B3 + B4 + R = 600 2. Po + Pi + P2 = 720 3. 4R = 1Bi + 2B2 + 3B3 + 4B4 4. 6R = 0P0 + 1Pi + 2P2 5. 5Po + 2Pi = 3Bi + 1B2 6. Pi > 2(Po - Bi) The justifications for these constraints are: 1. The 120-cell has 600 vertices and for a maximum independent set B0 = 0 as noted earlier. 2. The 120-cell has 720 pentagons. 3. Each red vertex is incident to four blue vertices. Hence, four times the number of red vertices is equal to the number of times a blue vertex is adjacent to a red one. 4. Each vertex is in six pentagons. Hence, six times the number of red vertices is equal to the number of times a red vertex occurs in a pentagon. 5. A blue 2-path is a path on 3 vertices (and hence two edges) whose vertices are all blue. Since each 2-path of the graph is in a unique pentagon, the number of blue 2-paths is 5P0 + 2Pi. The number of blue 2-paths is also equal to 3B1 + 1B2 since a blue vertex with three blue neighbours is the centre of three blue 2-paths, a blue vertex with two blue neighbours is the centre of one blue 2-path, and a blue vertex with zero or one blue neighbours is not the central vertex of any blue 2-path. 6. This constraint comes from combining Corollary 2.3 with Theorem 2.4. To get an upper bound for the maximum independent set order, the objective function is to maximize the value of R. Solving the linear programming relaxation gives an upper bound of 2880/13 = 221.538... which gives an upper bound of 221 on the integer programming problem. (The optimum solution is attained for the vector P0 = 360/13, P1 = 720/13, P2 = 8280/13, B1 = 0, B2 = 3240/13, B3 = 1680/13, B4 = 0, and the polytope thus defined is three-dimensional and has nine-vertices.) Applying the same tactic to the antipodal collapse gives an upper bound of 110 for the collapse, implying that the independent set of order 110 found in §2.2 is a maximum independent set of the antipodal collapse. 2.4 Exploiting the Integer Program Information The example in §2.2 gives a lower bound of 220 for the order of a maximum independent set of the 120-cell. On the other hand, §2.3 proves an upper bound of 221. This implies that if the independent set from §2.2 is not optimal, then there is an independent set of the 120-cell of order 221. The next step is to examine the solutions of the integer programming problem from §2.3 which have the number R of red vertices equal to 221 to gain structural information as to what a solution of order 221 must look like. Table 2 shows all the integer solutions that could result in an independent set of order 221. Correctness of the LP code is not an issue since it is not hard to check the final solutions by hand. Scanning the table of solutions, we observe that P0 - B1 is always at least 25. From Corollary 2.3, the implication is that any independent set of order 221 has at least 25 isolated blue pentagons. Observe also that all cases satisfy the constraint that the number B1 of keys plus two times the number B4 of isolated blues is at most seven. The existence of an independent set of order 221 requires that there is some way to add at least 25 isolated blue pentagons to the 120-cell without creating too many keys or isolated blue vertices (B1 + 2B4 < 7). The next two sections explain how we first tried planting a smaller number of isolated blue pentagons in part of the graph in all ways up to isomorphism and give an account of how the search for the 25 isolated blue pentagons was completed. P0 P1 P2 B1 B2 Bs B4 25 64 631 0 253 126 0 26 62 632 0 254 124 1 27 60 633 0 255 122 2 28 58 634 0 256 120 3 26 62 632 1 251 127 0 27 60 633 1 252 125 1 28 58 634 1 253 123 2 29 56 635 1 254 121 3 27 60 633 2 249 128 0 28 58 634 2 250 126 1 29 56 635 2 251 124 2 28 58 634 3 247 129 0 29 56 635 3 248 127 1 30 54 636 3 249 125 2 29 56 635 4 245 130 0 30 54 636 4 246 128 1 30 54 636 5 243 131 0 31 52 637 5 244 129 1 31 52 637 6 241 132 0 32 50 638 7 239 133 0 Table 2: Solutions to the Linear Program that would correspond to an independent set of order 221. 2.5 Planting Blue Pentagons A typical approach to trying to plant 25 isolated blue pentagons into the 120-cell that covers all possibilities up to isomorphism is to choose some smaller number of pentagons (for example, seven) that are placed in all ways up to isomorphism and then add the rest without concern for duplication since at some point, isomorphism screening is too costly for the amount of duplication prevented. This approach was taken first and it resulted in too many cases for a practical solution. The next strategy applied was to consider only a subgraph of the 120-cell for which it is possible to prove that at least some number k of isolated pentagons must be present in order to reach an independent set of order 221, and then to place the k pentagons in this region in all ways up to isomorphism. It is assumed that vertices of the 120-cell are labelled by the Clockwise BFS Labelling Algorithm from Section 2.1 The restricted region for consideration is defined by first sorting the pentagons. Before sorting, a list of five integers is created for each pentagon which contains the labels of its vertices in reverse sorted order (which is not necessarily the same as a cyclic ordering). Then these lists are compared lexicographically while sorting the pentagons. This gives a sorted order of pentagons P0,P1,... ,P719. The first six pentagons are the ones pictured in Figure 1. The sequences used to sort them are: P0 : 8, 5, 2,1, 0 P1 : 11, 6, 3,1, 0 P2 : 12, 9, 3, 2, 0 P3 : 14, 7,4,1, 0 P4 : 15,10,4,2,0 and P5 : 16,13,4, 3,0. The last two pentagons (illustrating how lexicographic order is used to break ties) are: P718 : 599, 598, 596, 592, 591 and P7ig : 599, 598, 597, 594, 593. This (slightly unnatural) ordering was selected so that the maximum vertex number occurring in the pentagons numbered P0, P1,..., Pk is minimized given a chosen value of k. Intuitively, this helps to compress the first k pentagons into a small subgraph of the 120-cell. After some experimentation, it was decided that planting seven pentagons in all ways up to isomorphism was the best compromise between the number of cases created and the difficulty for finishing the cases. The restricted region for planting these pentagons is shown to consist of the first 173 pentagons (P0, P1,..., P172) in the following lemma. Lemma 2.5. If the 120-cell has an independent set of order 221 then it is possible to find an independent set of order 221 such that there are at least seven isolated blue pentagons in the first 173 pentagons (P0, P1,..., P172). Proof. We already know from the results in §2.3 that the entire graph contains at least 25 isolated blue pentagons if there is an independent set of order 221. The idea for this proof is to count the number of isolated blue pentagons in the graph by considering the sets of pentagons numbered P0, P1,..., P172 for each of the automorphisms of the graph. If the average count over each of these sets P0, P1,..., P172 is greater than six, then we can conclude that there is at least one automorphism of the graph such that the count for P0, P^ ..., P172 is at least seven. Owing to the structure of the automorphism group of the graph, taking into consideration the sets of pentagons labelled P0, P1,..., P172 over all automorphisms accounts for each pentagon the same number of times; each is included 14,400 x 173/720 times. If the graph has 25 or more isolated blue pentagons, then the sum of the number of isolated blue pentagons over each choice for P0, P1,..., P172 is equal to at least 25 x 14400 x 173/720. Hence, the average term is equal to at least 25 x 173/720. But 25 x 173/720 > 6 and therefore, since the average is greater than six, at least one case must be greater than six. □ The total number of ways to plant seven isolated blue pentagons in the set P0,..., P172 is equal to 8,211,380. It is a little more difficult than usual to define a canonical form for these, because some of the automorphisms of the graph can map a choice of seven pentagons selected from P0, P^ ..., P172 to another choice of seven pentagons which is lexicographically smaller, but is no longer a selection from the pentagons P0, P1,..., P172 because the new set contains a pentagon numbered 173 or higher. To accommodate this difficulty, the canonical form is selected so that it is the lexicographically minimum set of seven pentagons with the additional property that the pentagon with the largest number in the set corresponds to some Pk for k < 172. The algorithms used for this phase were very simple. A nested set of seven loops was used to plant all possible choices for seven isolated blue pentagons. For each isolated blue pentagon selected, the ten incident vertices are coloured red. Vertices adjacent to a red vertex are coloured blue. To determine if an additional choice for an isolated blue pentagon is compatible with a previously chosen set, it suffices to check if its ten incident vertices can all legally be coloured red (that is, they are either uncoloured or red already, but cannot be blue). Then the 8,211,380 ways to place the isolated blue pentagons were run through a screen which kept only the canonical cases. For this step, the automorphism group of the graph was precomputed as described in §2.1. As a check on the computation, for each of the 1,379,646 cases retained, we determined the number of valid images it had (that is, the number of ways to renumber it with an automorphism such that the largest label on a pentagon is 172). The sum of these was equal to 8,211,380 (the number of cases possible without removing duplicates), a necessary condition for correctness. As an additional check of correctness, the number of cases to consider up to isomorphism matches that from a computation done earlier with a different approach. 2.6 Finishing the Search by a Backtrack For each of the 1,379,646 non-isomorphic ways of planting seven blue pentagons in the pentagons P0, P1, ...Pm (described in §2.5), the next step is to determine if it is possible to extend the configuration so that it contains at least 25 isolated blue pentagons. The possibilities for an independent set of order 221 outlined in §2.4 indicate that a solution of order 221 does not have many keys or isolated blues, more specifically, that B + 2B4 < 7. A backtracking routine was used to try to extend each of the cases with the seven isolated blue pentagons to 25 isolated blue pentagons without creating too many isolated blues or keys in the process. Some tricks were used to make this computation finish in a reasonable amount of time. The backtracking algorithm at level k considers two cases: one where the pentagon Pk is not included as an isolated blue pentagon, and if feasible, a second case where the pentagon Pk is included as an isolated blue pentagon (which means that its ten incident vertices are coloured red). The colour of a vertex is recorded as an integer which is 0 if a vertex is not coloured. The colour is decremented each time a vertex is coloured red, or incremented each time a vertex is coloured blue. This permits the algorithm to colour vertices then backtrack by uncolouring the vertices without using a data structure such as a stack to indicate vertices with a status change. Only blue or white vertices can legally be coloured blue. Only red or white vertices can legally be coloured red. If a vertex is coloured red, then its neighbours are immediately coloured blue. When the colour of a vertex returns to zero, it returns to the uncoloured status. As the algorithm progresses, certain vertices can safely be coloured red. These are characterized in the next theorem. Theorem 2.6. Suppose a 120-cell has an independent set of vertices coloured red, the neighbours of these are coloured blue, and the remaining vertices are uncoloured. If there is an uncoloured vertex v with three blue neighbours and one uncoloured neighbour w, then if there is a maximum independent set of the 120-cell which is consistent with the entire colouring, there is a maximum independent set with v coloured red. Proof. If v is red in the maximum independent set then there is nothing to prove. If v is not red, then w is red because if w is blue instead, v is a blue vertex with four blue neighbours, contradicting the maximality of the independent set (colouring v red increases the independent set order). An independent set of the same order can then be found by changing the colouring so that v is red and w is blue. □ The algorithm first inserts the initial seven isolated blue pentagons, waiting until they are all included before applying Theorem 2.6. The delay is needed because applying the theorem earlier can result in a colouring inconsistent with the initial pentagons. If there are two uncoloured vertices u and v which are adjacent to each other and also each is adjacent to three blues, there are two choices for how to apply Theorem 2.6, and it is possible that only one of these is consistent with the initial selection of the seven isolated blue pentagons. During the course of the backtrack, each time an isolated blue pentagon is added to the current configuration, a queue is used to record vertices which evolve to being white with three blue neighbours. As the goal is to try to add 25 isolated blue pentagons, 25 queues suffice. After addition of the ten incident reds of the isolated blue pentagon, the algorithm traverses the queue, and each vertex in the queue which is not blue is coloured red (as noted in the last paragraph, applying Theorem 2.6 at a vertex may prevent its subsequent use at another vertex). This process can trigger the addition of further vertices to the queue. When the isolated blue pentagon is removed (when the computation backtracks), the queue is first traversed in the reverse order to undo these changes. New isolated blue vertices are recorded at the point when the fourth neighbour of the isolated blue is initially coloured red. The number is decremented when this fourth neighbour becomes uncoloured again. Keys arise either when an uncoloured vertex with three blue neighbours is coloured blue or when a third neighbour of a blue vertex is initially coloured blue. To facilitate the detection of isolated blue vertices and keys, respectively, the number of red neighbours and the number of blue neighbours of each vertex are maintained. The algorithm takes exponential time to run, which is not surprising as the problem is hard. A careful selection of the data structures results in an approach such that the work it does to maintain the data structures is at most a constant times the number of times a vertex is coloured red. The algorithm also used some precomputed upper bounds. We determined the maximum number of isolated blue pentagons possible if the pentagons are chosen from Pk, Pk+i,..., P7i9 such that the penalty (equal to Bi + 2B4) is at most seven (as required for an independent set of order 221). There is no point in continuing this computation past the point where 18 isolated blue pentagons are possible: since we start with seven, only 18 more are required. Theorem 2.6 was not used for computing these upper bounds, owing to its interference with what we were trying to compute. At a given level of the backtrack for placing the 25 blue pentagons, if the number of isolated blue pentagons included so far plus the bound for the level is less than 25, the algorithm backs up, since it is necessary to have at least 25 isolated blue pentagons for an independent set of order 221. It is possible in the course of the algorithm that an isolated blue pentagon which has yet not been considered ends up with all ten of its incident vertices red. This however does not preclude the algorithm from adding it: the incident vertices just get coloured red more than once. The algorithm for this last backtrack was coded independently twice to ensure correctness. The 1,379,646 cases were split into 64 slices, and run in parallel on a 64-processor array, with the run of the C program for a typical slice taking 16 -18 hours. Both programs concluded that it is not possible to include 25 blue pentagons in the 120-cell with a penalty of seven or less after applications of Theorem 2.6 are taken into account. Because this must be possible for an independent set of order 221 to exist, the maximum independent set order of the 120-cell is 220. Polytope n m f r g D |Cd| Icosahedron 12 30 20 5 3 3 3,2 Dodecahedron 20 30 12 3 5 5 8, 4, 2, 2 24-cell 24 96 96 8 3 3 8,2 600-cell 120 720 1200 12 3 5 24, 8, 3, 2 120-cell 600 1200 720 4 5 15 220, 120, 48, 28, 24, 10, 8, 5, 5, 3, 2, 2, 2, 2 Table 3: Exceptional polytopes in dimensions three and four. The columns n, m, and f give the numbers of vertices, edges and two-dimensional faces of the polytope; r, g and D are the vertex degree, girth and diameter of the graph. The entries |Cd| are the maximum orders of d-codes for d =2,3, ...D — 1. 3 Other Results Table 3 lists the orders of the maximum d-codes for all five exceptional polytopes. Apart from the 2-code of the 120-cell, the only case requiring special tactics is the 4-code of the 120-cell, which was solved as described in [18]. All five polytopes have antipodal pairs as their d-codes for d = D, the diameter of the graph. When the codes are considered in terms of their 'contact polytopes' [15], some interesting 'Russian Doll'-like interconnections are seen. In the sense used in previous work [15], the contact polytope of a d-code has the same vertices as the independent set, and two vertices of the contact polytope are joined by an edge if the independent-set vertices are at distance d in the parent graph. Sim-plices of dimensions two, three and four appear: the triangle (a2 ) is the contact polygon of the 3-code of the icosahedron, the 4-code of the 600-cell and the 11-code of the 120-cell; the tetrahedron (a3) is the contact polyhedron of the 3-code of the dodecahedron; the four-dimensional simplex (a4) is the contact polytope of the 9-code of the 120-cell. The cube appears (73) appears as the contact polytope of the 2-code of the dodecahedron. The hyperoctahedron (ß4) appears as the contact polytope of 2-code of the 24-cell, 3-code of the 600-cell and the 8-code of the 120-cell. The 24-cell itself is the contact polytope of the 2-code of the 600-cell. The 3-code of the 120-cell is a perfect code [10] in the sense that each vertex of the code is at the centre of a ball of radius 1 containing one vertex of the 120-cell and its four nearest neighbours; the 120-cell is then partitioned exactly into 120 such balls, with centres on the vertex set of a 600-cell whose edges are paths of length 3 in the 120-cell. These observations are closely related to the fact, pointed out by Coxeter [3], that the vertices of the 120-cell embedded as an equilateral object in four-dimensional space include the vertices of all fifteen of the other regular polytopes in four dimensions, a property that has no analogy in three dimensions, where the dodecahedron contains the vertices of the cube and tetrahedron, but not those of the octahedron or icosahedron. References [1] I. M. Bomze, M. Budinich, P. M. Pardalos and M. Pelillo, The maximum clique problem, in: Handbook of combinatorial optimization, Supplement Vol. A, Kluwer Acad. Publ., Dordrecht, 1999, 1-74. [2] J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, Springer-Verlag, New York, 1988. [3] H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover, New York, 1973. [4] J. D. Crane, Maximal Non-Adjacent Addition to fullerene-70: Computation of All the Closed Shell Isomers of C70X26, Fullerene Sci. Technol. 2 (1994), 427-435. [5] J. Debroni, J. D. Eblen, M. A. Langston, W. Myrvold, P. Shor and D. Weerapurage, A complete resolution of the Keller maximum clique problem, Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 11), (2011), 129-135. [6] M. Dutour Sikiric and W. Myrvold. The special cuts of the 600-cell. Beiträge Algebra Geom. 49 (2008), 269-275. [7] S. Fajtlowicz and C. E. Larson, Graph-theoretic independence as a predictor of fullerene stability, Chem. Phys. Lett. 377 (2003), 485-490. [8] P. W. Fowler, S. Daugherty and W. Myrvold, Independence number and fullerene stability, Chem. Phys. Lett. 448 (2007), 75-82. [9] P. W. Fowler, P. Hansen, K.M. Rogers and S. Fajtlowicz, C60Br24: a chemical illustration of graph theoretical independence, J. Chem. Soc., Perkin Trans. 2 (1998), 1531-1533. [10] P. W. Fowler, B. de La Vaissiere and M. Deza, Addition patterns, codes and contact graphs for fullerene derivatives, J. Mol. Graphics Mod., 19 (2001), 199-204. [11] P. W. Fowler, K. M. Rogers, K. R. Somers and A. Troisi, Independent sets and the prediction of addition patterns for higher fullerenes, J. Chem. Soc., Perkin Trans. 2 (1999), 2023-2027. [12] M. Garey and D. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman and Co., San Fransico, 1979. [13] G. W. Hart and H. Picciotto, Zome Geometry: Hands-on Learning with Zome Models, Key Curriculum Press, Emeryville, CA, 2000. [14] D. S. Johnson and M. A. Trick, Cliques, Coloring, and Satisfiability, Second DIMACS Implementation Challenge, Oct. 11-13, 1993, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 26, AMS, Providence, RI, 1996. [15] B. de La Vaissiere, P. W. Fowler and M. Deza, Codes in Platonic, Archimedean, Catalan and related polyhedra: a model for maximum addition patterns in chemical cages, J. Chem. Inf. Comp. Sci. 41 (2001), 376-386. [16] F. J. MacWilliams and N. J. A. Sloane, Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1978. [17] P. McMullen and E. Schulte, Abstract Regular Polytopes, Cambridge University Press, Cambridge, 2002. [18] W. Myrvold and P. W. Fowler, Fast enumeration of all independent sets up to isomorphism, Preprint. [19] P. R. J. Östergard, Constructing combinatorial objects via cliques, Surveys in combinatorics 2005, London Math. Soc. Lecture Note Ser., 327 (2005), 57-82, Cambridge Univ. Press, Cambridge. [20] J. Stillwell, The story of the 120-cell, Notices of the AMS 48 (2001), 17-24. d MFA Also available at http://amc.imfm.si ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 211-219 Wiener index of iterated line graphs of trees homeomorphic to the claw Kx,3 Martin Knor Department of Mathematics, Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 813 68, Bratislava, Slovakia Primož PotoCnik, Riste Skrekovski Faculty of Mathematics and Physics, University of Ljubljana, and Institute of Mathematics, Physics and Mechanics, Jadranska 21, 1111 Ljubljana, Slovenia Received 24 October 2011, accepted 20 March 2012, published online 26 July 2012 Abstract Let G be a graph. Denote by Ll(G) its i-iterated line graph and denote by W(G) its Wiener index. Dobrynin, Entringer and Gutman stated the following problem: Does there exist a non-trivial tree T and i > 3 such that W(Li(T)) = W(T)? In a series of five papers we solve this problem. In a previous paper we proved that W(Li(T)) > W(T) for every tree T that is not homeomorphic to a path, claw Ki,3 and to the graph of "letter H", where i > 3. Here we prove that W(Li(T)) > W(T) for every tree T homeomorphic to the claw, T = Ki,3 and i > 4. Keywords: Wiener index, iterated line graph, tree, claw. Math. Subj. Class.: 05C12 1 Introduction Let G be a graph. For any two of its vertices, say u and v, denote by dG (u, v) (or by d(u, v) if no confusion is likely) the distance from u to v in G. The Wiener index of G, W(G), is defined as W(G) = £ d(u, v), u=v where the sum is taken through all unordered pairs of vertices of G. Wiener index was introduced by Wiener in [12]. It is related to boiling point, heat of evaporation, heat of E-mail addresses: knor@math.sk (Martin Knor)primož.potocnik@fmf.uni-lj.si (Primož Potocnik), skrekovski@gmail.com (Riste Skrekovski), formation, chromatographic retention times, surface tension, vapour pressure, partition coefficients, total electron energy of polymers, ultrasonic sound velocity, internal energy, etc., see [8]. For this reason Wiener index is widely studied by chemists. The interest of mathematicians was attracted in 1970's. It was reintroduced as the distance and transmission, see [5] and [11], respectively. Recently, there are whole special issues of journals devoted to (mathematical properties) of Wiener index, see [6] and [7], as well as several surveys, see e.g. [3] and [4]. By definition, if G has a unique vertex, then W(G) = 0. In this case, we say that the graph G is trivial. We set W(G) = 0 also when the set of vertices (and hence also the set of edges) of G is empty. The line graph of G, L(G), has vertex set identical with the set of edges of G. Two vertices of L(G) are adjacent if and only if the corresponding edges are adjacent in G. Iterated line graphs are defined inductively as follows: In [1] we have the following statement. Theorem 1.1. Let T be a tree on n vertices. Then W(L(T)) = W(T) - (n). Since (n) > 0 if n > 2, there is no nontrivial tree for which W(L(T)) = W(T). However, there are trees T satisfying W(L2(T)) = W(T), see e.g. [2]. In [3], the following problem was posed: Problem 1.2. Is there any tree T satisfying the equality W(Ll(T)) = W(T) for some i > 3? As observed above, if T is a trivial tree then W(Ll(T)) = W(T) for every i > 1, although here the graph L® (T) is empty. Denote by H the tree on six vertices out of which two have degree 3 and four have degree 1. Since H can be drawn to resemble the letter H, it is often called the H-graph. Graphs Gi and G2 are homeomorphic if and only if the graphs obtained from Gi and G2, respectively, by substituting the vertices of degree two together with the two incident edges with a single edge, are isomorphic. In [10] we proved the following: Theorem 1.3. Let T be a tree, not homeomorphic to a path, claw K13 and the graph H. Then W(L®(T)) > W(T) for all i > 3. Since the case when T is a path is trivial, it remains to consider graphs homeomorphic to the claw K13 and those homeomorphic to H. In this paper we concentrate on graphs homeomorphic to the claw K13. The remaining two cases, namely the trees homeomorphic to H for i > 3 and trees homeomorphic to K13 for i = 3, are dealt with in a forthcoming paper. First, consider the case of the claw K1j3 itself. Then L® (K1j3) is a cycle of length 3 for every i > 1. Since W(K1j3) = 9 and the Wiener index of the cycle of length 3 is 3, we have W(L®(Kl ,3)) < W(K13) for every i > 1. For other trees homeomorphic to K13, we prove the opposite inequality, provided that i > 4: Theorem 1.4. Let T be a tree homeomorphic to K13, such that T = K13. Then it holds that W(L®(T)) > W(T) for every i > 4. In [9] we proved the following statement: Theorem 1.5. Let G be a connected graph. Then fG(i) = W(Ll(G)) is a convex function in variable i. Hence, to prove Theorem 1.4 it suffices to prove: Theorem 1.6. Let T be a tree homeomorphic to K1j3, such that T = K1j3. Then it holds W(L4(T)) > W(T). 2 Proofs Let a, b, c > 1. Denote by Ca,b,c a tree that has three paths of lengths a, b and c, starting at a common vertex of degree 3. Obviously, Ca,b,c is homeomorphic to K13 and Ci 11 = Kij3. By symmetry, we may assume a > b > c, see Figure 1 for C5i4i3. Our aim is to prove ACa,b,c > 0 if a > 2. We start with the case a < 3. This case will serve as the base of induction in the proof of Theorem 1.6. Lemma 2.1. Let 3 > a > b > c > 1 and a =1. Then ACa,b,c > 0. Proof. Since 3 > a > b > c > 1 and a =1, there are 9 cases to consider. In Table 1 we present ACa b c for each of these cases. The results were found by a computer, though it is rather easy to find W(Ca,b,c) by hand, and W(L4(Ca,b,c)) can be found by applying Proposition 2.3 to L2(Ca b c). □ Figure 1: The graph C5j4j3. Denote ACa,b,c = W(L4(Ca,b,c)) - W(Ca,b,c). (a ^ c) W (Ca,b,c) W (L4(Ca,b,c)) ACa,b,c (3, 3, 3) 138 642 504 (3, 3, 2) 102 533 431 (3, 3,1) 75 257 182 (3, 2, 2) 72 435 363 (3, 2,1) 50 192 142 (3,1,1) 32 65 33 (2, 2, 2) 48 348 300 (2, 2,1) 31 138 107 (2,1,1) 18 38 20 Table 1: ACa,b,c for a < 3. In what follows we assume that a > 4. Denote So(a,b,c) = W(Ca,b,c) - W(Ca-1,b,c) S4(a,b,c) = W(L4(Ca,b,c)) - W(L4(Ca-1,b,c)). Then A Ca,b,c - ACa-i,b,c = Ma, b, c) - So (a, b, c), (2.1) so if we prove S4 (a, b, c) - S0(a, b, c) > 0, we obtain ACa,b,c > ACa-1,b,c. We distinguish 4 vertices in Ca b c. Denote by y the vertex of degree 3, and denote by x1, x2 and x3 the pendant vertices so that d(x1, y) = a, d(x2, y) = b and d(x3, y) = c, see Figure 1. As is the custom, by V (G) we denote the vertex set of G. Lemma 2.2. Let a,b,c > 1. Then w fa + b +1\ fa + c +1\ fa + 1\ So(a,b,c)=^ 2 JH 2 2 ). Proof. Since Ca-1,b,c is a subgraph of Ca,b,c with V(Ca,b,c) - V(Ca-1,b,c) = {x1}, we have So (a, b, c) = W(Ca,b,c) - W(Ca-1,b,c) = £ d(u, x{), u where the sum goes through all u G V(Ca,b,c) \ {x1}. For vertices u of the x1 - x2 path, the sum of all d(u, x1) is 1 + 2 +-----+ (a+b) = (a+b+^. For vertices of the x1 - x3 path which do not lay on x1 - x2 path, the sum of d(u, x1) is (a+1) + (a+2) + • • • + (a+c) = (a+2+1) - , see Figure 1. Since the paths x1 - x2 and x1 - x3 contain all vertices of Ca,b,c, weha^e So(a,b,c) = t^1) + (a+cc+1e - (a+1e. □ For two subgraphs S1 and S2 of G, by d(S1, S2) we denote the shortest distance in G between a vertex of S1 and a vertex of S2. If S1 and S2 share an edge then we set d(S1,S2) = -1. Analogously as a vertex of L(G) corresponds to an edge of G, a vertex of L2(G) corresponds to a path of length two in G. For x G V (L2 (G)) we denote by B2(x) the corresponding path in G. Let x and y be two distinct vertices of L2 (G). It was proved in [9] that d_L2(g)(x, y) = dG(B2(x),B2(y)) + 2. Let u,v G V (G), u = v. Denote by ßi(u, v) the number of pairs x, y G V (L2(G)), with u being the center of B2(x) and v being the center of B2(y), such that d(B2(x), B2(y)) = d(u,v) — 2 + i. Since d(u,v) — 2 < d(B2(x), B2(y)) < d(u,v), we have ßi(u, v) = 0 for all i G {0,1, 2}. Denote by deg(w) the degree of w in G. In [9] we have the following statement: Proposition 2.3. Let G be a connected graph. Then fdeg(u)\ fdeg(v) W(L2(G)) = ]T u=v + E 22 d(u,v) + ßi(u,v) + 2ß2(u,v) 3^deg(uA +^deg(u) (2.2) where the first sum goes through unordered pairs u,v G V (G) and the second one goes through u G V (G). We apply Proposition 2.3 to L2(Ca,b,c) and L2(Ca_i,b,c). This enables us to calculate S4(a, b, c) using degrees and distances of the second iterated line graph. Figure 2: The graph L2(C5,4,3). Denote by w1 the pendant vertex of L2(Ca,b,c) corresponding to the path of length 2 terminating at x1. Since a > 4, the unique neighbour of w1 has degree 2. Denote by w this neighbour, see Figure 2. For every vertex u G V(L2(Ca,b,c)) \ {w, w1}, denote by n(u) the number of neighbours of u, whose distance to w is at least d(u, w). We have: Lemma 2.4. Let a > 4 and b,c > 1. Then where the sum goes through all vertices of V(L2(Ca,b,c)) \ {w, w1}. Proof. Observe that L2(Ca-1,b,c) is a subgraph of L2(Ca,b,c) and V(L2(Ca,b,c)) \ V(L2(Ca-1,b,c)) = {w1}. Since deg(w1) = 1, the vertex w1 cannot be the center of a path of length 2, implying that ßi(u,w1) = 0 for every u and i. Since (des2W1^ = 0, all summands of (2.2) containing w1 contribute 0 to W(L4(Ca,b,c)). The vertices of L2 (Ca-1,b,c), except w, have the same degree in L2(Ca,b,c) as in L2 (Ca-1,b,c). The degree of w is 1 in L2(Ca_i,b,c), and it is 2 in L2(Ca,b,c). Therefore £u[3(deg3(u)C + 6(deg4(u)C] has the same value in L2(Ca,b,c) as in L2(Ca-1,b,c), so these sums will cancel out. Thus, we have 64(0,, b, c) = W(L2(L2(C\b,c))) - W(L2(L2(Ca_iAc))) 'deg(u)\ /2^ E 2 d(u, w) + ßi (u, w) + 2ß2 (u, w) where the sum goes through u G V(L2(Ca-1,b,c)) \ {w}. Let u G V(L2(Ca-1,b,c)) \ {w}. Since deg(w1) = 1 and deg(w) =2 in L2(Ca,b,c), the unique path of length 2 centered at w contains an endvertex closer to u than w. Hence, ß2(u,w) = 0. Consequently, ß1(u,w) equals the number of paths of length 2 centered at u, both endvertices of which have distance to w at least d(u,w). Hence, ß1 (u, w) = (n(2u)), which completes the proof. □ 2 Using Lemma 2.4 we prove the induction step. Lemma 2.5. Let 0 > b > c > 1 and 0 > 4. Then 64(o, b, c) > 60(o, b, c). Proof. We distinguish 8 more vertices in L2(Ca,b,c). Denote by w2 and w3 pendant vertices corresponding to the paths of length 2 containing x2 and x3, respectively, see Figure 1 and 2. Denote by z1, z2 and z3 the vertices corresponding to the paths of length 2, whose endvertex is y ; and denote by z4, z5 and z6 the vertices corresponding to the paths of length 2 centered at y. Of course, if b < 2 or c < 2, then some of these vertices are not defined. For u G V(L2(Ca-1,b,c)) \ {w}, denote h(u)= (deg2(u))d(u,w)+ (n(2u)). By Lemma 2.4, we have 64(o, b, c) = J2u h(u), where the sum goes through all vertices of V(L2(Ca,b,c)) \ {w, w1}. If u G {w2, w3} then deg(u) = 1 and n(u) = 0, so h(u) = 0. Thus, vertices of degree 1 contribute 0 to J2u h(u). Denote Si = ^2 h(u), u where the sum is taken over all interior vertices u of the wi — zi path, u = w and 1 < i < 3. Then 64(0, b, c) = ^3=1 Si + £6=1 h(zi). Regarding the values of 0, b and c, we distinguish 4 cases: Case 1. 0 > 4 and b,c > 3. If u is a vertex of degree 2, then n(u) = 1 and (deg(u)) = 1. Hence h(u) = d(u, w). Thus, 51 = 1 + 2 + • • • + (0—4) = (0 — 3C 52 = 0 + (o+1) + ••• + (o+b—4) = (0 + b — 3C — (0C 53 = 0 + (0+1) + ••• + (o+c—4) = (o + c— 3C — (0 C. If u G {z1,z2,z3}, then deg(u) = 3 and n(u) = 2. Thus h(u) = 3d(u,w) + 1. If u G {z4, z5 }, then deg(u) = 4 and n(u) = 3, so h(u) = 6d(u, w) + 3. Finally, if u = z6, then deg(u) = 4 and n(u) = 2, so h(u) = 6d(u, w) + 1. This gives h(z1) = 3(a-3) + 1 h(z4) = h(z5) = 6(a-2) + 3 h(z2) = h(z3) = 3(a-1) + 1 h(z6) = 6(a-1) + 1. Since J4(a, b, c) = ^3=1 Sj + ^6=1 h(zj), we have ^(a,b,c) = (a - 3) + (a + b - 3) + (a + C - 3) - 2(a) +(3a-8) + 2(3a-2) + 2(6a-9) + (6a-5). Denote P = J4(a, b, c) - J0(a, b, c). By Lemma 2.2 we have J0(a, b, c) = (a+2+1) + (a+2+1) - . Expanding the terms we get P = 17a - 4b - 4c - 17. Since a > b and a > c, we have P > 9a - 17. Finally, since a > 4, we have P = &4(a, b, c) - ^o(a, b, c) > 0. Case 2. a > 4, b > 3 and c < 2. We calculate first J4(a, b, 1). In L2(Caibj1) we have S3 = 0; note that z3 is not defined here and that deg(z5) = deg(z6) = 3 (see Figure 2). Analogously as in Case 1 we get: 51 = (°-3) h(z2) = 3(a-1) + 1 52 = (a+26-3) - (2) h(z4) = 6(a-2) + 3 53 = 0 h(z5 ) = 3(a-2) + 1 h(z1) = 3(a-3) + 1 h(z6) = 3(a-1) since n(z5) = 2 and n(z6) = 1. Thus, Ma,b, 1) = (a - 3) + (a + b - 3) - (a) + (3a-8) +(3a-2) + (6a-9) + (3a-5) + (3a-3). Denote P = ^(a, b, 1) - Jo(a, b, 2). By Lemma 2.2 we have Jo(a, b, 2) = (a+2+1) + (a+3) - . Expanding the terms we get P = 9a - 4b - 18. Since a > b, we have P > 5a - 18, and as a > 4, we have P > 0. Since ó4(a, b, 2) > J4(a, b, 1) and J0(a, b, 2) > J0(a, b, 1), we conclude J4(a, b, i) - J0(a, b, i) > P > 0 for i G {1, 2}. Case 3. a > 4, b = 2 and c < 2. We find J4(a, 2,1). In L2(Cai2l1) we have S2 = S3 = 0. Again, the vertex z3 is not defined here, deg(z2) = 2 and deg(z5 ) = deg(z6) = 3 (see Figure 2). Analogously as in the previous cases we get: S1 = (°-3) h(z4) = 6(a-2) + 3 h(z1) = 3(a-3) + 1 h(z5) = 3(a-2) + 1 h(z2) = (a-1) h(z6) = 3(a-1) since n(z2) = 1, n(z5) = 2 and n(z6) = 1. Thus, 64(a, 2,1) = (a - 3) + (3a-8) + (a-1) + (6a-9) + (3a-5) + (3a-3). Denote P = 64(a, 2,1) -60(a, 2, 2). By Lemma 2.2 we have ö0(a, 2,2) = 2(a+3) - ("+1). Expanding the terms we get P = 8a - 26. Since a > 4, we have P > 0. Since S4(a, 2, 2) > 64(a, 2,1) and S0(a, 2,2) > S0(a, 2,1), we conclude 64(a, 2, i) - ö0(a, 2, i) > P > 0 for i e {1,2}. Case 4. a > 4 and b = c =1. In L?(Caii) we have S2 = S3 = 0. Note that the vertices z2 and z3 are not defined, while deg(z4) = deg(z5) = 3 and deg(z6) = 2 (see Figure 2). Analogously as in the previous cases we get: Si = (a-3) h(z4) = h(z5) = 3(a-2) + 1 h(z1) = 3(a-3) + 1 h(z6) = (a-1) since n(z4) = n(z5) = 2 and n(z6) = 0. Thus, 64(a, 1, 1) = (a - 3) + (3a-8) + 2(3a-5) + (a-1). Denote P = ^(a, 1,1)-^(a, 1,1). By Lemma 2.2 we have 60 (a, 1,1) = 2(a+2) - (a+1). Expanding the terms we get P = 4a - 15. Since a > 4, we have P > 0, and hence 64(a, 1,1) - 60(a, 1,1) > P > 0. □ Proof of Theorem 1.6. Let T be the tree Ca,b,c with a > b > c > 1, such that a = 1. If a < 3, then ACaAc = W(L4(Ca,b,c)) - W(Ca,6,c) > 0, by Lemma 2.1. Now suppose that a > 4. Consider lexicographical ordering of triples (a', b', c') with a' > b' > c' > 1 and a' > 2. Further, assume that ACa/jb/,c/ > 0 for every triple (a', b', c'), such that a' > b' > c' > 1 and a' > 2, which is in the lexicographical ordering smaller than (a, b, c). Let (a*, b*, c*) be ordering of triple (a-1, b, c), such that the multisets {a*, b*, c*} and {a-1, b, c} are the same and a* > b* > c*. Then Ca_1jbjc and Ca»jb»,c» are isomorphic graphs. Moreover, since a > 4, we have a* > 3. By (2.1) and Lemma 2.5 we see that ACa,b,c - ACa* ,b* ,c* = ACa,b,c - ACa- 1,b,c = 64(a, b, c) - 60(a, b, c) > 0. Since (a*, b*, c*) is in the lexicographical ordering smaller than (a, b, c) and a* > 2, by the induction hypothesis we have ACa* b* c* > 0. Hence, ACa b c = W(L4(Ca b c)) - W(Ca , b, c) > 0. □ Acknowledgements The first author acknowledges partial support by Slovak research grants VEGA 1/0871/11 and APVV-0223-10. All authors acknowledge the support of bilateral Slovak-Slovenian grant. References [1] F. Buckley, Mean distance in line graphs, Congr. Numer. 32 (1981), 153-162. [2] A. A. Dobrynin, Distance of iterated line graphs, Graph Theory Notes of New York 37 (1999), 50-54. [3] A. A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: Theory and applications, Acta Appl. Math. 66 (2001), 211-249. [4] A. A. Dobrynin, I. Gutman, S. Klavžar, P. Žigert, Wiener index of hexagonal systems, Acta Appl. Math. 72 (2002), 247-294. [5] R. C. Entringer, D. E. Jackson, D. A. Snyder, Distance in graphs, Czechoslovak Math. J. 26 (1976), 283-296. [6] I. Gutman, S. Klavžar, B. Mohar (eds), Fifty years of the Wiener index, MATCH Commun Math. Comput. Chem. 35 (1997), 1-259. [7] I. Gutman, S. Klavžar, B. Mohar (eds), Fiftieth Anniversary of the Wiener index, Discrete Appl. Math. 80 (1997), 1-113. [8] I. Gutman, I. G. Ženkevich, Wiener index and vibrational energy, Z. Naturforsch. 57 (2002), 824-828. [9] M. Knor, P. Potocnik, R. Skrekovski, Wiener index in iterated line graphs, submitted, (see also IMFM preprint series 48 (2010), 1128, http://www.imfm.si/preprinti/index. php?langlD=1). [10] M. Knor, P. Potocnik, R. Skrekovski, On a conjecture about Wiener index in iterated line graphs of trees, Discrete Math. 312 (2012), 1094-1105. [11] J. Plesnfk, On the sum of all distances in a graph or digraph, J. Graph Theory 8 (1984), 1-21. [12] H. Wiener, Structural determination of paraffin boiling points, J. Amer. Chem. Soc. 69 (1947), 17-20. d MFA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 221-236 Orienting and separating distance-transitive graphs Italo J. Dejter University of Puerto Rico, Rio Piedras, PR 00936-8377, Puerto Rico Received 3 October 2011, accepted 7 March 2012, published online 6 September 2012 Abstract It is shown that exactly 7 distance-transitive cubic graphs among the existing 12 possess a particular ultrahomogeneous property with respect to oriented cycles realizing the girth that allows the construction of a related Cayley digraph with similar ultrahomogeneous properties in which those oriented cycles appear minimally "pulled apart", or "separated" and whose description is truly beautiful and insightful. This work is proposed as the initiation of a study of similar ultrahomogeneous properties for distance-transitive graphs in general with the aim of generalizing to constructions of similar related "separator" Cayley digraphs. Keywords: Distance-transitive graph, ultrahomogeneous graph, Cayley graph. Math. Subj. Class.: 05C62, 05B30, 05C20, 05C38 1 Introduction A graph is said to be distance-transitive if its automorphism group acts transitively on ordered pairs of vertices at distance i, for each i > 0 [3, 10, 15]. In this paper we deal mainly with finite cubic distance-transitive graphs. While these graphs are classified and very well-understood since there are only twelve examples, for this very restricted class of graphs we investigate a property called ultrahomogeneity that plays a very important role in logic, see for example [7, 18]. For ultrahomogeneous graphs (resp. digraphs), we refer the reader to [5, 9, 11, 17, 19] (resp. [6, 8, 16]). Distance-transitive graphs and ultrahomogeneous graphs are very important and worthwhile to investigate. However, to start with, the following question is answered in the affirmative for 7 of the 12 existing cubic distance-transitive graphs G and negatively for the remaining 5: Question 1.1. If k is the largest I such that G is ^-arc-transitive, is it possible to orient all shortest cycles of G so that each two oppositely oriented (k - 1)-arcs of G are just in two corresponding oriented shortest cycles? E-mail address: ijdejter@uprrp.edu (Italo J. Dejter) The answer (below) to Question 1 leads to 7 connected digraphs S (G) in which all oriented shortest cycles of G are minimally "pulled apart" or "separated". Specifically, it is shown that all cubic distance-transitive graphs are {Cg}Pk-ultrahomogeneous, where g = girth, but only the 7 cited G are {Cg}pfc-ultrahomogeneous digraphs and in each of these 7 digraphs G, the corresponding "separator" digraph S (G) is: (a) vertex-transitive digraph of indegree = outdegree = 2, underlying cubic graph and automorphism group as that of G; (b) {Cg, C2}-ultrahomogeneous digraph, where Cg = induced oriented g-cycle, with each vertex taken as the intersection of exactly one such Cg and one C2; (c) a Cayley digraph. The structure and surface-embedding topology [2, 12, 20] of these S (G) are studied as well. We remark that the description of these S (G) is truly beautiful and insightful. It remains to see how Question 1 can be generalized and treated for distance-transitive graphs of degree larger than 3 and what separator Cayley graphs could appear via such a generalization. 2 Preliminaries We may consider a graph G as a digraph by taking each edge e of G as a pair of oppositely oriented (or O-O) arcs e and (e)-1 inducing an oriented 2-cycle C2. Then, fastening e and (e)-1 allows to obtain precisely the edge e in the graph G. Is it possible to orient all shortest cycles in a distance-transitive graph G so that each two O-O (k — 1)-arcs of G are in just two oriented shortest cycles, where k = largest l such that G is l-arc transitive? It is shown below that this is so just for 7 of the 12 cubic distance-transitive graphs G, leading to 7 corresponding minimum connected digraphs S (G) in which all oriented shortest cycles of G are "pulled apart" by means of a graph-theoretical operation explained in Section 4 below. Given a collection C of (di)graphs closed under isomorphisms, a (di)graph G is said to be C-ultrahomogeneous (or C-UH) if every isomorphism between two induced members of C in G extends to an automorphism of G. If C is the isomorphism class of a (di)graph H, we say that such a G is {H}-UH or H-UH. In [14], C-UH graphs are defined and studied when C is the collection of either the complete graphs, or the disjoint unions of complete graphs, or the complements of those unions. Let M be an induced subgraph of a graph H and let G be both an M-UH and an H-UH graph. We say that G is an {H}M-UH graph if, for each induced copy H0 of H in G and for each induced copy M0 of M in H0, there exists exactly one induced copy H1 = H0 of H in G with V(H0) n V(H1) = V(M0) and E(H0) n E(H1) = E(M0). The vertex and edge conditions above can be condensed as H0 n H1 = M0. We say that such a G is tightly fastened. This is generalized by saying that an {H }M -UH graph G is an l-fastened { H}M -UH graph if given an induced copy H0 of H in G and an induced copy M0 of M in H0, then there exist exactly l induced copies H = H0 of H in G such that H n H0 D M0, for each i = 1,2,..., l, with at least H1 n H0 = M0. Let M be an induced subdigraph of a digraph H and let the graph G be both an M-UH and an H-UH digraph. We say that G is an {H} m -UH digraph if for each induced copy H0 of H in G and for each induced copy M0 of M in H0 there exists exactly one induced copy Hi = Ho of H in G with V (Ho) n V (H ) = V (M0) and A(Ho ) n A(Hi) = A(M0), where A(Hi) is formed by those arcs (e)-1 whose orientations are reversed with respect to the orientations of the arcs e of A(Hi). Again, we say that such a G is tightly fastened. This case is used in the constructions of Section 4. Given a finite graph H and a subgraph M of H with |V(H)| > 3, we say that a graph G is (strongly fastened) SF {H}M-UH if there is a descending sequence of connected subgraphs M = Mi,M2..., Mt = K such that: (a) Mm is obtained from Mi by the deletion of a vertex, for i = 1,... ,t -1 and (b) G is a (2i -1)-fastened {H }Mi -UH graph, for i = 1,... , t. This paper deals with the above defined C-UH concepts applied to cubic distance-transitive (CDT) graphs [3]. A list of them and their main parameters follows: CDT graph G n d g k n a b h K Tetrahedral graph K4 4 1 3 2 4 24 0 1 1 Thomsen graph K3,3 6 2 4 3 9 72 1 1 2 3-cube graph Q3 8 3 4 2 6 48 1 1 1 Petersen graph 10 2 5 3 12 120 0 0 0 Heawood graph 14 3 6 4 28 336 1 1 0 Pappus graph 18 4 6 3 18 216 1 1 0 Dodecahedral graph 20 5 5 2 12 120 0 1 1 Desargues graph 20 5 6 3 20 240 1 1 3 Coxeter graph 28 4 7 3 24 336 0 0 3 Tutte 8-cage 30 4 8 5 90 1440 1 1 2 Foster graph 90 8 10 5 216 4320 1 1 0 Biggs-Smith graph 102 7 9 4 136 2448 0 1 0 where n = order; d = diameter; g = girth; k = AT or arc-transitivity (= largest i such that G is i-arc transitive); n = number of g-cycles; a = number of automorphisms; b (resp. h) = 1 if G is bipartite (resp. hamiltonian) and = 0 otherwise; and k is defined as follows: let Pk and Pk be respectively a (k - 1)-path and a directed (k - 1)-path (of length k - 1); let Cg and Cg be respectively a cycle and a directed cycle of length g; then (see Theorem 3 below): k = 0, if G is not (Cg; Pk)-UH; k = 1, if G is planar; k = 2, if G is {Cg}Pfc-UH with g = 2(k - 1); k = 3, if G is {Cg}p -UH with g > 2(k - 1). In Section 3 below, Theorem 2 proves that every CDT graph is an SF {Cg}Pfc-UH graph, while Theorem 3 establishes exactly which CDT graphs are not {Cg}pfc-UH digraphs; in fact 5 of them. Section 4 shows that each of the remaining 7 CDT graphs G yields a digraph S (G) whose vertices are the (k - 1)-arcs of G, an arc in S (G) between each two vertices representing corresponding (k - 1)-arcs in a common oriented g-cycle of G and sharing just one (k - 2)-arc; additional arcs of S (G) appearing in O-O pairs associated with the reversals of (k - 1)-arcs of G. Moreover, Theorem 4 asserts that each S (G) is as claimed and itemized at end of the Introduction above. 3 (Cg, Pk)-UH properties of CDT graphs Theorem 3.1. Let G be a CDT graph of girth = g, AT = k and order = n. Then, G is an SF {Cg}Pk-UH graph. In particular, G has exactly 2k-23ng-i g-cycles. Proof. We have to see that each CDT graph G with girth = g and AT = k is a (2i+1 - Unfastened {Cg}pk-i-UH graph, for i = 0,1,..., k - 2. In fact, each (k - i - 1)-path P = Pk-i of any such G is shared by exactly 2i+1 g-cycles of G, for i = 0,1,..., k - 2. For example if k = 4, then any edge (resp. 2-path, resp. 3-path) of G is shared by 8 (resp. 4, resp. 2) g-cycles of G. This means that a g-cycle Cg of G shares a P2 (resp. P3, resp. P4) with exactly other 7 (resp. 3, resp. 1) g-cycles. Thus G is an SF {Cg}Pi+2-UH graph, for i = 0,1,..., k - 2. The rest of the proof depends on the particular cases analyzed in the proof of Theorem 3 below and on some simple counting arguments for the pertaining numbers of g-cycles. □ Given a CDT graph G, there are just two g-cycles shared by each (k - 1)-path. If in addition G is a {Cg } pfc -UH graph, then there exists an assignment of an orientation for each g-cycle of G, so that the two g-cycles shared by each (k - 1)-path receive opposite orientations. We say that such an assignment is a {Cg}pfc-O-O assignment (or {Cg}pfc-OOA). The collection of n oriented g-cycles corresponding» to the n g-cycles of G, for a particular {Cg}pfc -OOA will be called an {nCg}pfc -OOC. Each such g-cycle will be expressed with its successive composing vertices expressed between parentheses but without separating commas, (as is the case for arcs uv and 2-arcs uvw), where as usual the vertex that succeeds the last vertex of the cycle is its first vertex. Theorem 3.2. The CDT graphs G of girth = g and AT = k that are not {Cg} ^ -UH digraphs are the graphs of Petersen, Heawood, Pappus, Foster and Biggs-Smith. Thke remaining 7 CDT graphs are {Cg }pfc -UH digraphs. Proof. Let us consider the case of each CDT graph sequentially. The graph K4 on vertex set {1,2,3,0} admits the {4 C3}p2-OOC {(123), (210), (301), (032)}. The graph K3,3 obtained from K6 (with vertex set {1,2,3,4, 5,0}) by deleting the edges of the triangles (1,3, 5) and (2,4,0) admits the {9 C4}p3-OOC {(1234), (3210), (4325), (1430), (2145), (0125), (5230), (0345), (5410)}. The graph Q3 with vertex set {0,..., 7} and edge set {01, 23, 45, 67, 02,13, 46, 57, 04, 15, 26, 37} admits the {6 C4}p2-OOC {(0132), (1045), (3157), (2376), (0264), (4675)}. 2 The Petersen graph Pet is obtained from the disjoint union of the 5-cycles = (m0m1m2 w3w4) and v= (v0v2v4v1v3) by the addition of the edges (ux, vx), for x G Z5. Apart from the two 5-cycles given above, the other 10 5-cycles of Pet can be denoted by mx = (ux-1«xUx+1Vx+1Vx-1) and vx = (vx-2VxVx+2Ux+2Ux-2), for each x G Z5. Then, the following sequence of alternating 5-cycles and 2-arcs starts and ends up with opposite orientations: 2 00 1 0 2 W3W2U1 W0U1W2 M— U2V2V0 v_ V3W3W2 M+, where the subindexes ± indicate either a forward or backward selection of orientation and each 2-path is presented with the orientation of the previously cited 5-cycle but must be present in the next 5-cycle with its orientation reversed. Thus Pet cannot be a {C5}p3 -UH digraph. Another way to see this is via the auxiliary table indicated below, that presents the form in which the 5-cycles above share the vertex sets of 2-arcs, either O-O or not. The table details, for each one of the 5-cycles £ = m0, v0, (expressed as £ = (£0,..., £4) in the shown vertex notation), each 5-cycle n in i = to, 0,..., 4}\{£} that intersects £ in the succeeding 2-paths £ì£ì+1£ì+2, for i = 0,..., 4, with additions involving i taken mod 5. Each such n in the auxiliary table has either a preceding minus sign, if the corresponding 2-arcs in £ and n are O-O, or a plus sign, otherwise. Each —nj (resp. nj) shown in the table has the subindex j indicating the equality of initial vertices nj = £«+2 (resp. nj = £0 of those 2-arcs, for i = 0, . . . , 4: ,+m°,+mo,+m°), v~:(+v°° ,+vJJ ,+v1 ,+v0 ,+v°° ), M° :(+Mr. + v33,-vf,-vi, + v0), v0 :(+v4-=,-m°,+mi,+mo _MJ). This partial auxiliary table is extended to the whole auxiliary table by adding x e Z4 uniformly mod 5 to all superindexes = to, reconfirming that Pet is not {C5}p -UH. For each positive integer n, let In stand for the n-vertex cycle (0,1,... ,n — 1). The Heawood graph Hea is obtained from I14 by adding the edges (2x, 5 + 2x), where x e [1,..., 7} and operations are in Zi4. The 28 6-cycles of Hea include the following 7 6-cycles: Yx = (2x, 2x+1, 2x+2, 2x+3, 2x+4, 2x+ 5), Sx = (2x ,2x+5, 2x+6, 2x+7, 2x+8, 2x+13), ex = (2x, 2x+5, 2x+4, 2x+9, 2x+8, 2x+13), Zx = (2x+12, 2x+3, 2x+4, 2x+5, 2x ,2x+13), where x e Z7. Now, the following sequence of alternating 6-cycles and 3-arcs starts and ends with opposite orientations for 70 : Y+ 2345 7- 7654 y+ 6789 7- ba98 y+ abcd 7- 10dc 7+ 0123 7-, (where tridecimal notation is used, up to d = 13). Thus Hea cannot be a {C7}p■ -UH digraph. Another way to see this is via an auxiliary table for Hea obtained in a fashion similar to that of the one for Pet above from: 70:(+726, + Ä5,+7o1,+C16,-e2,-C40); e°:(+e5,-74°, + e0,+C54, + Ä0,-C26); ,W8,-ZÌ,WJ,-W0), U 0:( Y39 ,-U16,Z1T,-Wg ,-W00,Z09,-Uf ,Y00,Sg), Y0:( U? ,-T» ,-Tff, U* ,-Yg2, Vg ,W% ,V£ ,-YJ), V0:(-Z|,-v64,y52,-x3ì,-x»,Y70,-V1d,-Z°,Tg), Z0:( U8,-Z8,-V04,-S8,-S®,-V70,-Z0, Ug,-X0), This table is extended by adding x e Zi7 uniformly mod 17 to all superindexes, confirming that B-S is not {C9}p -UH.. □ 4 Separator digraphs of 7 CDT graphs For each of the 7 CDT graphs G that are {Cg}p-UH digraphs according to Theorem 3, the following construction yields a corresponding digraph S (G) of outdegree and indegree two and having underlying cubic graph structure and the same automorphism group of G. The vertices of S (G) are defined as the (k - 1)-arcs of G. We set an arc in S (G) from each vertex a1a2... ak-l into another vertex a2... ak-lak whenever there is an oriented g-cycle (a^a2... ak-lak ...) in the {n has order 36 and acts regularly on the vertices of S(K3,3). For example, (0, 2)(1, 5) stabilizes the edge (145,541), and the permutation (0,5,4,1)(2, 3) permutes (clockwise) the black oriented 4-cycle (541,410,105,054). Thus S(K3,3) is a Cayley digraph. Also, observe the oriented 9-cycles in S(K3,3) obtained by traversing alternatively 2-arcs in the oriented 4-cycles and transposition edges; there are 6 such oriented 9-cycles. (E) The collection of oriented cycles of S (Des) corresponding to the {nCg} pfc -OOC of Des is formed by the following oriented 6-cycles, where x G Z5: (X0 X1 X2 , Xi X2 X3 , X2 X3 X0 , £3X0X3 , X0X3Xo, X3X0X1), (X2X1XQ, Xo X3 x2 , Xo xl1, xjjxlx^ , X^X^Xo, X^XoXi), respectively for the 6-cycles AX, BX, CX, DX, where xj stands for (x + j)ž; each of the participant vertices here is an end of a transposition edge. Figure 3 represents a subgraph S (M3) of S (Des) associated with the matching M3 of Des indicated in its representation "inside" the left-upper "eye" of the figure, where vigesimal integer notation is used (up to j = 19); in the figure, additional intermittent edges were added that form 12 square pyramids, 4 such edges departing from a corresponding extra vertex; so, 12 extra vertices appear that can be seen as the vertices of a cuboctahedron whose edges are 3-paths with inner edge in S(M3) and intermittent outer edges. There is a total of 5 matchings, like M3, that we denote MX, where x = 3,7, b, f, j. In fact, S (Des) is obtained as the union U{S (MX); x = 3, 7, b, f, j }. Observe that the components of the subgraph induced by the matching MX in Des are at mutual distance 2 and that MX can be divided into three pairs of edges with the ends of each pair at minimum distance 4, facts that can be used to establish the properties of S (Des). Figure 3: M3 and the subgraph of S (Des) associated to it In Figure 3 there are: 12 oriented 6-cycles (dark-gray interiors); 6 alternate 8-cycles (thick-black edges); and 8 9-cycles with alternate 2-arcs and transposition edges (light-gray interiors). The 6-cycles are denoted by means of the associated oriented 6-cycles of Des. Each 9-cycle has its vertices sharing the notation of a vertex of V(Des) and this is used to denote it. Each edge e in M3 has associated a closed walk in Des containing every 3-path with central edge e; this walk can be used to determine a unique alternate 8-cycle in S(M3), and viceversa. Each 6-cycle has two opposite (black) vertices of degree two in S (M3). In all, S (Des) contains 120 vertices; 360 arcs amounting to 120 arcs in oriented 6-cycles and 120 transposition edges; 20 dark-gray 6-cycles; 30 alternate 8-cycles; and 20 light-gray 9-cycles. By filling the 6-cycles and 8-cycles here with 2-dimensional faces, then the 120 vertices, 180 edges (of the underlying cubic graph) and resulting 20 + 30 = 50 faces yield a surface of Euler characteristic 120 - 180 + 50 = -10, so this surface genus is 6. The automorphism group of Des is G = S5 x Z2. Now, G contains three subgroups of index 2: two isomorphic to to S5 and one isomorphic to A5 x Z2. One of the two subgroups of G isomorphic to S5 (the diagonal copy) acts regularly on the vertices of S (Des) and hence S (Des) is a Cayley digraph. (F) The collection of oriented cycles of S (Cox) corresponding to the {n(pg} pfc -OOC of Cox is formed by oriented 7-cycles, such as: 01 = (m1m2w3, U2«3U4, W3W4W5, W4W5W6, W5W6U0, m6w0w1, m0w1w2), and so on for the remaining oriented 7-cycles xy with x G {0,..., 7} and y G {1, 2,3}, based on the corresponding table in the proof of Theorem 3. Moreover, each vertex of S (Cox) is adjacent via a transposition edge to its reversal vertex. Thus S (Cox) has: underlying cubic graph; indegree = outdegree = 2; 168 vertices; 168 arcs in 24 oriented 7-cycles; 84 transposition edges; and 42 alternate 8-cycles. Its underlying cubic graph has 252 edges. From this information, by filling the 7- and 8-cycles mentioned above with 2-dimensional faces, we obtain a surface with Euler characteristic 168 - 252 + (24 + 42) = -18, so its genus is 10. On the other hand, S (Cox) is the Cayley digraph of the automorphism group of the Fano plane, namely PSL(2,7) = GL(3, 2) [4], of order 168, with a generating set of two elements, of order 2 and 7, representable by the 3 x 3-matrices (100,001,010)T and (001,101,010)T over the field F2, where T stands for transpose. Figure 4 depicts a subgraph of S (Cox) containing in its center a (twisted) alternate 8-cycle that we denote (in gray) z1u1, and, around it, four oriented 7-cycles adjacent to it, (namely 11, 7^, 2^, 6^, denoted by their corresponding oriented 7-cycles in Cox, also in gray), plus four additional oriented 7-cycles (namely 0°, 3^, 41, 5^), related to four 9-cycles mentioned below. Black edges represent arcs, and the orientation of these 8 7-cycles is taken clockwise, with only gray edges representing transposition edges of S (Cox). Each edge of Cox determines an alternate 8-cycle of S (Cox). In fact, Figure 4 contains not only the alternate 8-cycle corresponding to the edge mentioned above, but also those corresponding to the edges ulu2, vizi, tizi and m0m1 . These 8-cycles and the 7-cycles in the figure show that alternate 8-cycles C and C adjacent to a particular alternate 8-cycle C '' in S (Cox) on opposite edges e and e' of C '' have the same opposite edge e" both to e and e' in C and C', respectively. There are two instances of this property in Figure 4, where the two edges taking the place of e'' are the large central diagonal gray ones, with C'' corresponding to uizl. As in (E) above, the fact that each edge e of Cox determines Figure 4: A subdigraph of S (Cox) associated with an edge of Cox an alternate 8-cycle of S (Cox) is related with the closed walk that covers all the 3-paths having e as central edge, and the digraph S (Cox) contains 9-cycles that alternate 2-arcs in the oriented 7-cycles with transposition edges. In the case of Figure 4, these 9-cycles are, in terms of the orientation of the 7-cycles: A convenient description of alternate 8-cycles, as those denoted in gray in Figure 4 by the edges zi«i, zivi, u2ui, uo«i, ziti of Cox, is given by indicating the successive passages through arcs of the oriented 7-cycles, with indications by means of successive subindexes in the order of presentation of their composing vertices, which for those 5 alternate 8-cycles looks like: In a similar fashion, the four bi-alternate 9-cycles displayed just above can be presented by means of the shorter expressions: By the same token, there are twenty four tri-alternate 28-cycles, one of which is expressible as: (U2U1 Z1, U1Z1V1, Z1V1V3,V3V1Z1,V1Z1t1, Z11115 , t5t1Z1, t1Z1U1, Z1U1U2 ), (i6o,756,26o,656), (512,323,745,io1), (001,i56,66o,456), (256,76o,0o6,46o), (312,523,645,231). (i61,513,646), (71o,05o,i1o), (261,3!3,716), (65o,45o,23o ). (01 61 32 23 53 42 1 ° ) (0o3,6o3,34o,236,536,462,i4o ). At this point, we observe that S (K4), S (Q3) and S (A) have alternate 6-cycles, while S (K3,3), S (Des) and S (Cox) have alternate 8-cycles. (G) The collection of oriented cycles of S (Tut) corresponding to the {nC} pfc -OOC of Tut is formed by oriented 8-cycles, such as: A0 = (45O0O1O2O3,00O1O2O3O4,01O2O3O4O5,02O3O4O5I0,03O4O5I045,04O5I045O0,05I045O0O1,1045O0O1O2) and so on for the remaining oriented 8-cycles Xy with X G {A,..., R} and y G Z5 based on the corresponding table in the proof of Theorem 3. Moreover, each vertex of S (Tut) is adjacent via a transposition edge to its reversal vertex. Thus S (Tut) has: underlying cubic graph; indegree = outdegree = 2; 720 vertices; 720 arcs in 90 oriented 8-cycles; 360 transposition edges; and 180 alternate 8-cycles, (36 of which are displayed below); its underlying cubic graph has 1080 edges. From this information, by filling the 900 8-cycles above with 2-dimensional faces, a surface with Euler characteristic 720 - 1080 + 240 = -120 is obtained, so genus = 61. On the other hand, the automorphism group of Tut is the projective semilinear group G = PrL(2, 9) [13], namely the group of collineations of the projective line PG(1,9). The group G contains exactly three subgroups of index 2 (and so of order 720), one of which (namely M10, the Mathieu group of order 10, acts regularly on the vertices of S (Tut). Thus S (Tut) is a Cayley digraph. A fifth of the 180 alternate 8-cycles of S (Tut) can be described by presenting in each case the successive pairs of vertices in each oriented 8-cycle Xy as follows, each such pair denoted by means of the notation XU(U+1) , where u stands for the 4-arc in position u in Xy , with 0 indicating the first position: (A°1, M°4, B34 ' K°02) (A12, P1 P01 ' 7° 770 ' M2°3) (A23' H34 ' B1° ' p7o) 7° L 3 L70, M^' hj3 ) El4) (A§5' E7„0 ' P 0 7 0 ) J6°7) C45' C34 ' F1 ' "y' (A°7, g45, (A70, K°°3, 72 G34) (B°1 ' Q23' P67) (B°2, D34' D34' N°16' C23) (B°7' M|15' Q67' R23) D23) (B4°5' D§7' r73°' Koi) O°°6' DÌ) H45' c4 C67 ' (coi ' G70' H70' M°°1) (C06' (C70 , n47' F 3 Fi35' 701 ' ) D3^2) L 0 L67' 112 ' 72 ' O^ ' Q312) 7536) (cr (D°5' H56' O2, O67, O4 ' 7O144'' 7 45 ' L°6) O45) (DJ°- (E03 ' 13 M7°°, P 4 H?' r6>7 ) NI3) (eL N°41' F°°1' q45, P 0 ) pi4) F 67 ) (F1' N 2 N4' J56 ' F3' P56 ' N70) q34) (GL N445, R#5' 743253) (F°6' Q|6' m256' q7°) (G°1' 7435' O43' KL' H°°7) qo1' 72, J12 ' 4) (G23, L 2 723' R12, (H?2' 7 3 N!47) (1°°3, 71 7 23 ' K17' OŠ1) ( J34 ' R56' P4 ' P2 3 ' K56) (K°o' do R°1' 7 0 734' O1°) Again, as in the previously treated cases, we may consider the oriented paths that alternatively traverse two arcs in an oriented 8-cycle and then a transposition edge, repeating this operation until a closed path is formed. It happens that all such bi-alternate cycles are 12-cycles. For example with a notation akin to the one in the last table, we display the first row of the corresponding table of 12-cycles: I (AO2, P^, R°0, K°°2) I (A°3, hO5, C°°0, M03) I (A24, J701, Q13, P6O) I As in the case of the alternate 8-cycles above, which are 180, there are 180 bi-alternate 12-cycles in S (Tut). On the other hand, an example of a tri-alternate 32-cycle in S (Tut) is given by: MO TT° r>4 r>2 r° r>1 r>1 \ (A03, H 36 ' O36' D50' 761' D14' O50 ' K72) There is a total of 90 such 32-cycles. Finally, an example of a tetra-alternate 15-cycle in S (Tut) is given by (A04, J03, K02), and there is a total of 240 such 15-cycles. More can be said about the relative structure of all these types of cycles in S (Tut). The automorphism groups of the graphs S (G) in items (A)-(G) above coincide with those of the corresponding graphs G because the construction of S (G) depends solely on the structure of G as analyzed in Section 3 above. Salient properties of the graphs S (G) are contained in the following statement. Theorem 4.1. For each CDT graph that is a {Cg } p■ -UH digraph, S (G) is: (a) a vertex-transitive digraph with indegree = outdegree = 2, underlying cubic graph and the automorphism group of G; (b) a {Cg, C2}-ultrahomogeneous digraph, where Cg stands for oriented g-cycle coincident with its induced subdigraph and each vertex is the intersection of exactly one such Cg and one C2 ; (c) a Cayley digraph. Moreover, the following additional properties hold, where s(G) = subjacent undirected graph of S(G): (A) S (K4) = Cay (A4, {(123), (12)(34)}), s(K4) = truncated octahedron; (B) S(Q3) = Cay(S4, {(1234), (12)}), s(Q3) = truncated octahedron; (C) S (A) = Cay(A5, {(12345), (23)(45)}), s(A) = truncated icosahedron; (D) S(K3,3) is the Cayley digraph of the subgroup of S6 on the vertex set {0,1,2, 3,4,5} generated by (0,5,4,1)(2,3) and (0, 2)(1,5) and has a toroidal embedding whose faces are delimited by 9 oriented 4-cycles and 9 alternate 8-cycles; (E) S (Des) is the Cayley digraph of a diagonal copy of S5 in the automorphism group S5 X Z2 of Des and has a 6-toroidal embedding whose faces are delimited by 20 oriented 6-cycles and 30 alternate 8-cycles; (F) S (Cox) = Cay(GL(3, 2), {(100,001,010)T, (001,101,010)T }), has a 10-toroidal embedding whose faces are delimited by 24 oriented 7-cycles and 42 alternate 8-cycles; (G) S (Tut) is the Cayley digraph of a subgroup M10 of order 2 in the automorphism group PrL(2,9) of Tut and has a 61-toroidal embedding whose faces are delimited by 90 oriented 8-cycles and 180 alternate 8-cycles. Corollary 4.2. The bi-alternate cycles in the graphs S (G) above are 9-cycles unless either G = Q3 or G = A, in which cases they are respectively 12-cycles and 15-cycles. Acknowledgement The author is grateful to the referee of a previous version of this paper for indications and corrections that lead to the present manuscript and for encouragement with respect to the value of the construction of the 7 directed graphs S (G) in Section 4. References [1] N. L. Biggs, Algebraic Graph Theory (2nd ed.), Cambridge University Press, 1993. [2] B. Mohar and C. Thomassen, Graphs on surfaces, Johns Hopkins Univ. Press, 2001. [3] A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance-Regular Graphs, Springer-Verlag, New York 1989. [4] E. Brown and N. Loehr, Why is PSL(2, 7) = GL(3, 2)?, Amer. Math. Mo. 116-8 (2009), 727-732. [5] P. J. Cameron, 6-transitive graphs, J. Combin. Theory Ser. B 28 (1980), 168-179. [6] G. L. Cherlin, The Classification of Countable Homogeneous Directed Graphs and Countable Homogeneous n-tournaments, Memoirs Amer. Math. Soc., vol. 131, number 612, Providence RI, January 1988. [7] A. Devillers, Classification of some homogeneous and ultrahomogeneous structures, Dr. Sc. Thesis, Universite Libre de Bruxelles, 2001-2002. [8] R. Fraisse, Sur l'extension aux relations de quelques proprietes des ordres, Ann. Sci. École Norm. Sup. 71 (1954), 363-388. [9] A. Gardiner, Homogeneous graphs, J. Combin. Theory Ser. B 20 (1976), 94-102. [10] C. Godsil and G. Royle, Algebraic Graph Theory, Springer-Verlag, 2001. [11] Ja. Ju. Gol'fand and M. H. Klin, On k-homogeneous graphs, Algorithmic studies in combinatorics 186 (1978), 76-85, in Russian. [12] J. L. Gross and T. W. Tucker, Topological Graph Theory, Wiley Interscience, 1987. [13] J. W. P. Hirschfeld, Projective Geometries over Finite Fields, Oxford Science Publications, 2nd ed., Clarendom Press, Oxford 1998. [14] D. C. Isaksen, C. Jankowski and S. Proctor, On K„-ultrahomogeneous graphs, Ars Combinatoria 82 (2007), 83-96. [15] I. A. Faradzev, A. A. Ivanov, M. Klin et al., The Algebraic Theory of Combinatorial Objects, Math. Appl. (Soviet Series), 84, Dordrecht, Kluwer, 1992. [16] A. H. Lachlan and R. Woodrow, Countable ultrahomogeneous undirected graphs, Trans. Amer. Math. Soc. 262 (1980), 51-94. [17] C. Ronse, On homogeneous graphs, J. London Math. Soc. 17 (1978), 375-379. [18] B. I. Rose and R. E. Woodrow, Ultrahomogeneous structures, Mathematical Logic Quarterly 27 (1981), 23-30. [19] J. Sheehan, Smoothly embeddable subgraphs, J. London Math. Soc. 9 (1974), 212-218. [20] A. T. White, Graphs of groups on surfaces, North-Holland, Mathematics Studies 188, 2001. d MFA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 237-245 A parallel algorithm for computing the critical independence number and related sets Ermelinda DeLaVifta * Department of Computer and Mathematical Sciences University of Houston—Downtown, Houston, TX 77002 Craig E. Larson t Department of Mathematics and Applied Mathematics Virginia Commonwealth University, Richmond, VA 23284 Received 9 July 2010, accepted 10 May 2012, published online 28 October 2012 Abstract An independent set Ic is a critical independent set if \Ic\ — \N(Ic)\ > | J| - |N(J)|, for any independent set J. The critical independence number of a graph is the cardinality of a maximum critical independent set. This number is a lower bound for the independence number and can be computed in polynomial-time. The existing algorithm runs in O(n2-5^f m/ log n) time for a graph G with n = \V(G)\ vertices and m edges. It is demonstrated here that there is a parallel algorithm using n processors that runs in O(n1-^y/m/ log n) time. The new algorithm returns the union of all maximum critical independent sets. The graph induced on this set is a Konig-Egervary graph whose components are either isolated vertices or which have perfect matchings. Keywords: Critical independent set, critical independence number, independence number, matching number, König-Egerväry graph. Math. Subj. Class.: 05C69 1 Introduction A new faster parallel algorithm is given for finding maximum critical independent sets and calculating the critical independence number of an arbitrary graph. *Work supported in part by the United States Department of Defense and resources of the Extreme Scale Systems at Oak Ridge National Laboratories. t Corresponding author. E-mail addresses: delavinae@uhd.edu (Ermelinda DeLaVina), clarson@vcu.edu (Craig E. Larson) Copyright © 2013 DMFA Slovenije The following notation is used throughout: the vertex set of a graph G is V (G), the order of G is n = n(G) = |V(G)|, the set of neighbors of a vertex v is NG(v) (or simply N (v) if there is no possibility of ambiguity), the set of neighbors of a set S C V (G) in G is Ng(S ) = U„eS N (u) (or simply N (S ) if there is no possibility of ambiguity), the set N [S] = N (S) U S, and the graph induced on S is G[S ]. All graphs are assumed to be finite and simple. A set I C V (G) of vertices is an independent set if no pair of vertices in the set are adjacent. The independence number a = a(G) is cardinality of a maximum independent set (MIS) of vertices in G. An independent set of vertices Ic is a critical independent set if |Ic|-|N (Ic)| > |J | — |N ( J) |, for any independent set J. A maximum critical independent set (MCIS) is a critical independent set of maximum cardinality. The critical independence number of a graph G, denoted a' = a'(G), is the cardinality of a maximum critical independent set. If Ic is a maximum critical independent set, and so a'(G) = |Ic|, then clearly a' < a. The critical difference d is max{|Ic| — |N(Ic)| : Ic is an independent set}. Critical independent sets are of interest for both practical and theoretical reasons. By a theorem of Butenko and Trukhanov, any critical independent set can be extended to a maximum independent set [4]. Zhang and Ageev gave polynomial-time algorithms for finding critical independent sets [17, 1]. Thus, finding a critical independent set is a polynomial-time technique for reducing the search for the well-known widely-studied NP-hard problem of finding a maximum independent set in a graph [7]. Maximum critical independent sets are central in the investigation of the structure of maximum independent sets, a connection via the Independence Decomposition Theorem, recounted in the next section. _ The existing algorithm for finding aMCIS and calculating a' runs in O(n2-5^/m/log n) time [11]. It is demonstrated here that there is a parallel algorithm using n processors that runs in O^n1-5^m/ log n) time. The new algorithm finds the set H of vertices which are in some maximum critical independent set, that is, the union of all MCISs. The graph induced on this set is a Konig-Egervary graph whose non-trivial components each have a perfect matching. 2 The set H of vertices in some MCIS The correctness of the algorithm presented in the next section depends on the following decomposition theorem, a corollary of, and equivalent to, the Independence Decomposition Theorem in [12]. A matching in a graph is a set of pairwise non-incident (or independent) edges. The matching number p of a graph is the cardinality of a maximum matching. A Konig-Egervary graph is one where a + p = n. Theorem 2.1. (Larson, [12]) For any graph G, there is a unique set X C V (G) such that 1. a(G) = a(G[X]) + a(G[Xc]), 2. G[X] is a König-Egerväry graph, 3. for every non-empty independent set I in G[Xc], |N(I)| > |I|, and 4. for every maximum critical independent set Jc of G, X = Jc U N (Jc). According to the theorem there is a unique set X C V (G) such that, for any maximum critical independent set Ic, Ic U N(Ic) = X. For any graph G let X = X(G) be the set guaranteed by Theorem 2.1. Call G[X] the distinguished Konig-Egervary subgraph of G. Konig-Egervary graphs were first characterized by Deming [6] and Sterboul [16] in the 1970s. The first author's Graffiti.pc program conjectured (number 329 in [5]) a characterization in terms of the critical independence number: a graph G is a Konig-Egervary graph if, and only if, a(G) = a'(G). The conjecture was first proven by Larson in [12]. In [14] Levit & Mandrescu extended the statement of this result as follows. Theorem 2.2. (Levit & Mandrescu, [14]) The following are equivalent: 1. G is a Konig-Egervary graph, 2. there is a maximum independent set of G that is a MCIS, 3. every maximum independent set of G is a MCIS, Figure 1: The vertices Ic = {a, b} form a (maximum cardinality) critical independent set; this set of vertices can be extended to a maximum independent set of the graph. According to Theorem 2.1 the sets X = Ic U N(Ic) = {a, b, c, d} and Xc = V\ X = {e,f,g} induce a decomposition of the graph into a Konig-Egervary subgraph G[X] and one, G[Xc], where every non-empty independent set of vertices I has more than |I| neighbors. It will now be shown that the graph induced on the set H (the union of all MCISs) is Konig-Egervary. This fact will be used in the proof of correctness of the parallel algorithm. While the class of Konig-Egervary graphs contains all the bipartite graphs (by the Konig-Egervary Theorem, [15]) and subgraphs of bipartite graphs are bipartite, it is not true in general that subgraphs of Konig-Egervary graphs are Konig-Egervary. So it is worth noting that G[X] is Konig-Egervary, G[H] is a subgraph of G[X], and G[H] is Konig-Egervary. The following results are required in the proof of Theorem 2.5. Lemma 2.3. (The Matching Lemma, [11]) If I is a critical independent set, then there is a matching from N (I ) to I. Let Q = Q(G) be the set of maximum independent sets in G. The core of a graph G, denoted core(G), is defined to be n{S : S e Q}, namely, the set of vertices which are in every maximum independent set; and £ = £(G) = |core(G) |. This notation follows [3]. Theorem 2.4. (Levit & Mandrescu, [13]) If G is a König-Egerväry graph, then G has a perfect matching if, and only if, | n {S : S e Q(G)}| = | n {V(G) - S : S e Q}|. Theorem 2.5. If Ic is a maximum critical independent set of a graph G, X = Ic U N (Ic), and H is the union ofall maximum critical independent sets, then 1. H U N (H ) = X, 2. G[H] is a Konig-Egervàry graph, 3. I is a maximum independent set of G[H] if, and only if, I is a MCIS of G and a'(G) = a(G[H ]), 4. The non-trivial components of G[H] have a perfect matching, 5. If Io are the isolated vertices in G[H] then a(G[H]) = |I0| + 2 |H \ I0|. Proof. Let Ic be a MCIS of a graph G and X = Ic U N(Ic). By Theorem 2.1 it follows that, for any MCIS Jc of G, Jc U N( Jc) = X. Let Oc be the set of MCISs of G. Then H U N (H ) = U{Jc : Jc G Oc} U N (U{Jc : Jc G Oc}) = [U{Jc : Jc G Oc}] U [U{N (Jc) : Jc G Oc}] = U{ Jc U N( Jc) : Jc G Oc} = X, proving 1. Now, Ic C H. Let H' = H \ Ic. So n(G[H]) = |Ic| + |H'|. Furthermore, a(G[H]) > |Ic|. By construction H' C N(Ic). By the Matching Lemma there is a matching from N(Ic) into Ic in G. Thus there is amatching from H' into Ic in G[H] and ^(G[H]) > |H'|. So a(G[H]) + m(G[H]) > |IC| + |H'| = n(H) and, for any graph, a + ^ < n, it follows that a(G[H]) + ^(G[H]) = n(G[H]), that is, G[H] is Konig-Egervary, proving 2. It now follows easily that a ( G [H]) = | Ic | and thus that Ic is a maximum independent set of G [H], proving one direction of 3. Now let I be a maximum independent set of G[H]. So I is an independent set in G[X], |I| > |Ic|, and a(G[X]) > |I|. It will now be argued that a(G[X]) = |IC| and, hence, |Ic| = |I|. Theorem 2.1 implies that G[X] is Konig-Egervary. Then it is not difficult to see that the Matching Lemma implies that ^(G[X]) = |N(Ic)|. Finally, we have n(G[X]) = a(G[X]) + m(g[x]) > |Ic| + |N(Ic)| = |X| = n(G[X]). The claim them follows. Then, since I and Ic are maximum independent sets of G[H], I U N (I ) C H U N (H) = X. N (I ) C X \ I and N (Ic) C X\Ic. It is worth noting here that, N (I ) is the set of neighbors of I in graph G (as opposed to graph G[H]). No use is made in this proof of neighbors of a set of vertices in graph G[H] and no subscripts are ever required for clarity. To continue, it follows that |N(I)| < |X \ 11 and |N(Ic)| = |X \ Ic|. Since |X \ 11 = |X \ Ic|, it follows that |N(I)| < |N(Ic)| and, thus, that |I| - |N(I)| > |I| - |N(Ic)|. But, if |I| - |N(I)| > |Ic| - |N(Ic)|, Ic is not a critical independent set, contradicting our assumption. Thus |I| - |N(I)| = |Ic| - |N(Ic)|, and I is a critical independent set of G, proving the other direction of 3. Let I0 be the set of isolated vertices in G[H]. These are contained in any maximum independent set of G[H]. Let I' = Ic \ I0 and H' = H \ I0. It is then claimed that G[H'] has a perfect matching. Let v G H'. Suppose v G core(G[H']), that is, v is in every maximum independent set of G[H']. Then v is in every maximum independent set of G[H] and, thus, in every maximum critical independent set of G. But H is the set of vertices in some maximum critical independent set of G. So no vertex in N(v) is in any maximum independent set of G[H], or in any maximum critical independent set of G, which is a contradiction. Thus | n {S : S G O(G[H'])}| = 0. By similar reasoning it can be shown that | n {V(G[H']) - S : S G O(G[H'])}| = 0. Thus | n {S : S G O(G[H'])}| = | n {V(G[H']) - S : S G O(G[H'])}|. Theorem 2.4 then implies that G[H'] has aperfect matching, proving 4. It is clear that, since G[H] is Konig-Egervary, and G[H'] has a perfect matching, G[H'] is also Konig-Egervary; that is, a(G[H']) + ^(G[H']) = n(G[H']). Since n(G[H']) = 2^(G[H']), it follows that a(G[H']) = 2n(G[H']) = 2|h \ I0|. Finally a(G[H]) = |I0| + a(G[H']) = |I0| + 2|H \ I0|, prov2ng 5. □ 3 A parallel MCIS algorithm The criterion given for testing whether a vertex belongs to a critical independent set begins by passing to a certain bipartite graph. The computational speed of the following algorithm is due to the fact that the independence number of a bipartite graph can be computed in polynomial time. Definition 3.1. For a graph G, the bi-double graph B(G) has vertex set VU V', where V' is a copy of V. If V = {vi, v2,..., v„|, let V' = {vi ,v2 ,...,< }. Then, (x,y') e E(B(G)) if, and only if, (x, y) e E (G). The bi-double graph B(G) of G can also be described as K2DG, the cartesian product of K2 and G. Corollary 3.2. (Larson, [11]) A graph G contains a non-empty critical independent set if, and only if, there is a vertex v e V (G) such that a(B(G)) = a(B(G) — {v, v'} — N ({v,v'}) + 2. In fact, the proof of this corollary actually shows that a vertex v satisfying the specified condition is in some critical set. It is also shown in [11] that any critical independent set can be extended to a MCIS. These results are now combined in a form directly useful in the current context. Theorem 3.3. (MCIS Criterion) A vertex v in a graph G is in some MCIS if, and only if, a(B(G)) = a(B(G) — {v, v'} — N({v, v'}) + 2. The following algorithm results in the set of all vertices which are in some maximum critical independent set. Step 4 requires n iterations—but, due to the MCIS Criterion, these n tests can be run independently on n processors. This is where the parallelization takes place. MCIS subgraph algorithm 1. Construct B(G). 2. a := a(B(G)). 3. H := 0. 4. For each vertex v in V(G), (a) t := a(B(G) — {v, v'} — N({v, v'}) + 2. (b) If t = a, H := H U {v}. According to Theorem 3.3 these steps will result in the construction of a set H consisting of all vertices which are in some MCIS. This can be extended in various ways to find the following invariants or sets. 1. Find a'. In order to calculate a', the remaining step is to identify the trivial and nontrivial components of H. Let I0 be the isolated vertices in H. Then, by Theorem 2.5, a'(G) = |Io| + i(|H \ Io|). 2. Find X. In order to calculate the decomposition guaranteed by the Independence Decomposition Theorem, it remains to find the neighbors of the vertices in H. Again by Theorem 2.5, H U N (H ) = X. Let Xc = V (G) \ X. Then G[Xc] will have the property that, for every non-empty independent set J, |N (J )| > | J |. 3. Find a MCIS Ic of G. In order to find a MCIS, Theorem 2.5 implies that it suffices to find a maximum independent set Ic in H. Then Ic is a MCIS in G. Since B(G) is a bipartite graph on 2n vertices, calculating a maximum matching of B(G) and, hence, calculating a(B(G)) requires O(n1-5y/m/ logn) operations, using the algorithm of Alt, et al. [2]. This algorithm will be run once and then a second time independently on each of n processors. So the total running time is still O(n1-5y/m/ log n). If M is a matching in a graph G and w is a vertex incident to an edge in M, let w' be the vertex matched to w under M. The new algorithm can now be stated. The parallelism occurs in step 1. The parallel MCIS algorithm 1. Find H. 2. Find the set I0 of isolated vertices in G[H]. H0 := I0. 3. If H \ Ho = 0, I := Io. STOP. 4. Find a maximum matching M of G[H]. 5. Let w G H \ Ho. 6. N1 := N(I0 U {w}), Mi := {v' : v G Ni}, I1 := I0 U M1, H1 := I1 U N1 (=H0 U N1 U M1). 7. i := 1. 8. While H \ Hi = 0: (a) i. If Hi = Hi-1: Ni+i := N(Ii). ii. Else if Hi = Hi-1: A. Let w G H \ Hi. B. Ni+1 := N(Ii U {w}), (b) Mi+1 := {v' : v G Ni+1}, Ii+1 := Ii U Mm. Hm := Im U Ni+1, i := i + 1. 9. I := Ii. Theorem 3.4. If G is a graph then the set I produced by the Parallel MCIS algorithm is a maximum critical independent set of G. Proof. Let G be a graph, H be the set of vertices in some maximum critical independent set of G, and M be the maximum matching of G[H] produced by the Parallel MCIS algorithm. Theorem 2.5 implies that a'(G) = a(G[H]). Thus it is enough to show that the set I produced by this algorithm is a maximum independent set of G[H]. Let I0 be the set of isolated vertices in G[H] and H' = H \ I0. It was shown that G[H] is a Konig-Egervary graph whose non-trivial components have perfect matchings. G[H'] is the union of the non-trivial components. So M is a perfect matching of G[H']. The algorithm will first be described for the convenience of the reader. The first step is to identify the isolated vertices. These can be extended to a maximum independent set of G[H]. Then choose any vertex w among the remaining vertices. By the definition of the set H, there is a MCIS and, by Theorem 2.5, this is a maximum independent set of G[H]. So there is maximum independent set I of G[H] which contains w. The neighbors of this vertex cannot be in I but each of these vertices is incident to some edge in the perfect matching M of G[H] and, since one vertex from every edge of M must be in I, it follows that the vertices matched to N(w) under M must be in I. In general, having constructed an independent set J, the neighbors of J cannot be in the maximum independent set but, since one vertex from every edge in M must be in the maximum independent set, the vertices matched to N ( J) under M must be in the set. If at some point there are no new vertices in N ( J), but the vertices in the graph have not been used up, an arbitrary vertex can be selected from the remaining vertices, added to the independent set, and the process continued. The set I0 is an independent set, H0 = Io, and there is a maximum independent set of G[H] containing I0. Assume that after the (k — 1)th iteration of the while loop, Ik is an independent set and there is a maximum independent set of G[H] containing Ik. It will be shown that Ik+1 is an independent set and there is a maximum independent set of G[H] which contains Ik+1. If H \ Hk = 0 after the (k — 1)th iteration of the while loop, then H = Hk and I is a maximum independent set of G[H]. Assume then that H \ Hk = 0 after the (k — 1)th iteration of the while loop. Hk is formed by (possibly) adding vertices to Hk-1, namely, N(Ik-1) \ Hk-1 together with the vertices matched to these under M. Either Hk = Hk-1 or Hk = Hk-1. Note that, in either case, by construction, Nk C Nk+1, Mk C Mk+1, Ifc C Ik+1, and Hk C Hk+1. In the first case, Hk \ Hk-1 = 0. Hk is formed by adding the vertices Nk \ Nk-1 and their neighbors Mk \ Mk-1 = Ik \ Ik-1 under the matching M. The vertices Mk \ Mk-1 may or may not have neighbors outside of Hk. Nk+1 = N(Ik), Mk+1 = {v' : v G Nk+1}, Ik+1 = Ik U Mk+1, and Hi+1 = Ii+1 U Ni+1. By assumption Ik is an independent set and there is a maximum independent set of G[H] containing Ik. The vertices in Ik+1 are the vertices in Ik together with the vertices matched to the neighbors of Ik under M. Let I be a maximum independent set of G[H] containing Ik. It cannot contain any neighbor of Ik. Since any maximum independent set I of G[H] must contain one vertex from each edge of M, I must contain the vertices matched to N(Ik ) under M. Thus Ik+1 is an independent set and it can be extended to a maximum independent set of G[H]. In the case where Hk = Hk-1, the kth step in the while loop of the algorithm (step 8) works as follows: A vertex w is selected from from H \ Hk. Since Ik is independent and w G N(w), Ik+1 = Ik U {w} is an independent set. By assumption there is a MCIS containing w and, following Theorem 2.5, there is a maximum independent set I containing w. Each edge in M must be incident to some vertex in I. Let I ' = (I \ Hk ) U Ik .It remains to be shown that I' is a maximum independent set of G[H]. Since Hk = Ik U N(Ik ) it follows that I' is an independent set. It is now enough to show that, for every edge xy in M, either x or y is in I '. Either x or y is in I. Assume x G I .If x G I ' then x G Hk. But then by the construction of Hk, y is matched to x under M and y is also in Hk. But Ik is a maximum independent set in G[Hk]. So either x or y must be in Ik and, thus I'. So Ik+1 is contained in a maximum independent set. □ The MCIS Subgraph algorithm, which produces H, requires O(n1-5y/m/ log n) operations. Finding a maximum matching of G[H] requires the same order or of operations. The remaining steps only require finding the neighbors of sets of vertices. So the total running time of Parallel MCIS Algorithm is O(n1'5 ^Jm/ log n). 4 Acknowledgment and future research We would like to thank an anonymous referee who pointed us to state-of-the-art references on maximum matching algorithms (including [8, 9, 10]). The referee notes that algorithm performance depends on edge density: some algorithms perform better for sparse graphs while others perform better for dense graphs. The referee also suggests that future investigation of the time complexity of the algorithm presented above would be of interest in the case that there are n2 or n3 processors. The presented algorithm requires use of n processors; the problem naturally breaks into n cases following Theorem 3.3. We do not know of a way to break the problem down further. References [1] A. Ageev, On finding critical independent and vertex sets, SIAM J. Discrete Mathematics 7 (1994), 293-295. [2] H. Alt, N. Blum, K. Melhorn and M. Paul, Computing a maximum cardinality matching in a bipartite graph in time O(n1'6 m/ log n), Information Processing Letters 37 (1991), 237240. [3] E. Boros, M. Golumbic and V. Levit, On the Number of Vertices belonging to all Maximum Stable Sets of a Graph, Discrete Applied Mathematics 124 (2002), 17-25. [4] S. Butenko and S. Trukhanov, Using Critical Sets to Solve the Maximum Independent Set Problem, Operations Research Letters 35 (2007), 519-524. [5] E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc, http://cms.uhd.edu/ faculty/delavinae/research/wowII/index.htm. [6] R. W. Deming, Independence Numbers of Graphs—an Extension of the Koenig-Egervary Theorem, Discrete Mathematics 27 (1979), 23-33. [7] M. Garey and D. Johnson, Computers and Intractability, W. H. Freeman and Company, New York, 1979. [8] N. J. A. Harvey, Matchings, Matroids and Submodular Functions, Massachusetts Institute of Technology Ph.D. thesis, 2008. [9] A. Jindal, G. Kochar and M. Pal, Maximum Matchings via Glauber Dynamics, Arxiv preprint arXiv:1107.2482, 2011. [10] K. Kaya, J. Langguth, F. Manne and B. Uc, Experiments on Push-Relabel-based Maximum Cardinality Matching Algorithms for Bipartite Graphs, Technical Report TR/PA/11/33, CER-FACS, Toulouse, France, 2011. [11] C. E. Larson, A Note on Critical Independence Reductions, Bulletin of the Institute of Combinatorics and its Applications 51 (2007), 34-46. [12] C. E. Larson, Critical Independent Sets and an Independence Decomposition Theorem, European Journal of Combinatorics 32 (2011), 294-300. [13] V. E. Levit and E. Mandrescu, On a+-stable Konig-Egervary graphs, Discrete Mathematics 263 (2003) 179-190. [14] V. E. Levit and E. Mandrescu, Critical Independent Sets and König-Egerväry Graphs, Graphs and Combinatorics 28 (2012), 243-250. [15] L. Lovasz, M. D. Plummer, Matching Theory, North Holland, Amsterdam, 1986. [16] F. Sterboul, A characterization of the graphs in which the transversal number equals the matching number, Journal of Combinatorial Theory. Series B 27 (1979), 228-229. [17] C.-Q. Zhang, Finding critical independent sets and critical vertex subsets are polynomial problems, SIAM J. Discrete Mathematics 3 (1990), 431-438. d MFA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 247-252 A note on homomorphisms of matrix semigroup Matjaž OmladiC Faculty of Mathematics and Physics, University of Ljubljana Jadranska 19, SI-1000, Ljubljana Slovenia Bojan Kuzma University of Primorska, FAMNIT, Glagoljaška 8, SI-6000 Koper, Slovenia and Institute of Mathematics, Physics and Mechanics, Department of Mathematics, Jadranska 19, SI-1000 Ljubljana, Slovenia Received 12 August 2011, accepted 5 June 2012, published online 28 October 2012 Abstract Let F be a field. We classify multiplicative maps from Mn(F) to M(n)(F) which annihilate a zero matrix and map rank-k matrix into a rank-one matrix. Keywords: Matrix semigroup, Homomorphism, Representation. Math. Subj. Class.: 20M15, 15A33, 20G05 1 Introduction and preliminaries Let Mn(F) denote the semigroup of all n-by-n matrices with coefficients in a field F, let Eij be its matrix units, and let Id = Idn := J2 Eii be its identity. In [5], Jodeit and Lam classified nondegenerate semigroup homomorphisms n : Mn(F) ^ Mn(F), that is, maps which are (i) multiplicative n( AB) = n(A)n(B) and (ii) their restriction on singular matrices is nonconstant. It was shown that the semigroup of such maps is generated by three simple types: (i) a similarity, (ii) a fixed field homomorphism applied entry-wise on a matrix, and (iii) the map which sends A to a matrix of its cofactors. We refer below for more precise definitions. The complete classification of degenerate maps on Mn(F) is more involved. They are all of the type A ^ n1(A) © Idn-m for some integer m G {0,..., n} and some degenerate multiplicative ni : Mn(F) ^ Mm(F) with ni(0) =0 [5]. When m = 1, Đokovic [2, Theorem 1] proved the following. E-mail addresses: matjaz.omladic@fmf.uni-lj.si (Matjaž Omladic), bojan.kuzma@famnit.upr.si (Bojan Kuzma) Lemma 1.1. Let F be a field, and n > 2. If n : Mn(F) ^ F is multiplicative, then there exists multiplicative ^ : F ^ F so that n(X) — ^(det X). EH When m < n and the characteristic of F differs from 2, ĐokoviC [2, Theorem 2] also showed n1 factors through determinant so that n1 — f o det for some multiplicative f : F ^ Mm(F). The classification of those seems to be difficult, and as far as we know they are known only in case F — C is the filed of complex numbers, by the work of Omladic, Radjavi, and Šemrl [8]. Later, Guralnick, Li, and Rodman [4], extended the result of Đokovic to include also the case n — m. Semigroup homomorphisms mapping into higher dimensional algebras are less known. Kokol-Bukovšek [6, 7] classified them in case they are nondegenerate and map 2-by-2 matrices into 3-by-3 or into 4-by-4. Under additional assumption that a degenerate homomorphism is a polynomial in matrix entries, the classification is well-known, see a book by Weyl [9] (see also Fulton and Harris [3] for holomorphic homomorphisms over a field of complex numbers). It is our aim to show that all homomorphisms from n-by-n matrices to (n) -by-(n) matrices which map a rank-k matrix into a rank-one come from exterior product. Both assumptions on the dimension of the target space as well as on the rank of the matrices are essential; otherwise there are many more maps as we show in Remark 1.4 below. We remark that the main idea, that rank-k idempotents are mapped into rank-1 idempotents, is essentially due to Jodeit and Lam [5]. To be self-contained, we briefly repeat the basics about exterior products. Let ei,..., en be the standard basis of column vectors in Fn. Given a linear operator X on Fn, denote by /\k(X) the k-th exterior product of X, acting on /\k(Fn), i.e., a k-th exterior product of Fn. Recall [3] that, as a vector space, Ak (Fn) has a basis consisting of (k) elements {ejj A • • • A eik ; 1 < i1 < i2 < • • • < ik < n}, where x Ay — —y Ax and x A x — 0 is the alternating tensor. Then by definition, /\k (X) : e^ A • • • A eik ^ (Xeil ) A • • • A (Xeik ). It follows easily that Ak(AB) — Ak(A) Ak(B). Also, in lexicographic order of a basis (eil A • • • A eik) \ 2 be an integer, and let m — ("") for some integer k — 1,..., n. If n : Mn(F) ^ Mm(F), n(0) — 0, is a multiplicative map such that rk(n(Ao)) — 1 for some matrix A0 of rank-k, then n(X )— S^(Ak (X ))S-1 where ^ : F ^ F is a multiplicative map and S G Mm(F) is invertible. Moreover, if k < n then ^ is also additive, hence a field embedding. Remark 1.3. Without the assumption n(0) = 0, there are more possibilities. Say, n(A) = 1 © Sym2(/\n-1 A) © f (det A), where Sym2 is the second symmetric power (see [3]) and f : F ^ Mm-1_(n+i)(F) is multiplicative. However, we remark that to classify multiplicative maps n it suffices to assume n(0) = 0. In fact, if n : Mn(F) ^ Mm(F) is multiplicative, then P := n(0) is an idempotent, and from n(X)P = n(X0) = n(0) = P = n(0X) = Pn(X) we deduce that, relative to decomposition Fm = Ker P © Im P we have n(X) = ni(X) © Idr, where r := rk P and n1 : Mn(F) ^ Mm_r (F) is multiplicative with n1(0) = 0. Remark 1.4. If m = (k) there are more possibilities, say n : Mn(F) ^ M(„2)(F), defined by A ^ /\4(A ® A), is multiplicative and maps a rank-2 matrix E11 + E22 into matrix of rank-one but is not of the form in the Theorem. This is because if rk A = r then rk(/\k A) = (r), while n maps a rank-3 matrix E11 + E22 + E33 into a matrix whose rank equals 126. If rank(n(A0)) = 1 there are more possibilities as can be seen by the map n : Mn(F) ^ M(n) (F), defined by A ^ A © 0(n)_n. Proof of Theorem 1.2. If k = n then m = 1, so n : Mn(F) ^ F. Such multiplicative maps were proven to be in accordance with our results by Lemma 1.1. Hence, we may assume in the sequel that k < n. Clearly, n(Id) is an idempotent, and from n(X)n(Id) = n(X • Id) = n(X) = n(Id)n(X) we deduce that, relative to decomposition Fm = Imn(Id) © Ker n(Id) we have n(X) = n1(X) © 0m_r G Mr (F) © 0m_r, where r := rk n(Id) and n1 is multiplicative with n1(0) = 0 and n1(Id) = Idr. As such, if X is invertible, then Idr = n1(Id) = n1(XX_1) = n1(X)n1(X_1), so n1(X) is also invertible and n1(X)_1 = n1(X_1). Let X be any matrix of rank-k. Then, there exist invertible S, T G Mn(F) with SXT = Idk ©0n_k, and in particular, there exist invertible S1, T such that X = S1A0T11. Consequently, 1 = rkn(Ao) = rk(n1(S_1XTf1) © 0m_r) = rk(^1(S1)_1n1(X)n1(T1)_1 © 0m_r; wherefrom rk n(X) = 1 for every X of rank-k. Consequently, n(Idk ©0n_k) is an idem-potent of rank-1, and by appropriate similarity we may assume it equals E11. Given X = X © 0n_k G Mk(F) © 0n_k, we have X = (Idk ©0)X(Idk ©0), wherefrom n(X) = E11n(X)E11 G FE11. Hence, n induces a multiplicative map n : Mk (F) ^ F by n(X)E11 := © 0n_k). It follows by Lemma 1.1 that there exists a nonzero multiplicative map : F ^ F such that n(X) = ^(det xX). In particular, if the rank of X = X © 0n_k is smaller than k, then n(X) = n(xX)E11 = 0. By multiplicativity, n(X) = 0 for every X G Mn(F) with rk X < k — 1. Moreover, given any A G Mn(F), letting A be the compression of A to the upper-left k-by-k block, we have ( ) n((Idk ©0)A(Idk ©0)) = n(A © 0n_k) = «(1) = 1. Consider any A G Mn (F). Then, m m n(A) = Id n(A) Id = ( Y E«) n(A)( Y eJ = Y Eiin(A)Ejj = Y n(P«APj). i=1 3=1 i, j i, j Given indices i = j, there exists Bj« G Mn of rank-k such that Bj« = P3 B3iP« and det(Bji)(ji) = 1; forinstance, if pi = Ete{ti,...,tfc} Ett and P3 = E»e{Sl,...,Sfc} Ess, with t1 < • • • Etv.) Hence, & (det(PiAPj Pj BjiPi )(ii)) = &i( det A(ij))&i(det(Bji)(ji)) = &i( det A(ij)) • &i(1) = &i(det A(ij)). On the other hand, n(PiAPj ) = n(Pi)n(A)n(Pj) = Eiin(A)Ejj = a^ (A)Eij where-from n(PiAPj) • n(Bji) = a 15 except n =16 and possibly n =23 and n = 27, there are geometric trilateral-free (n3) configurations. E-mail address: mwr23@georgetown.edu (Michael W. Raney) In this note we provide new examples of a geometric, triangle-free (233) configuration and a geometric, triangle-free (273) configuration, so that this theorem may now be modifed: Theorem 1.2. For every n > 15 except n = 16, there are geometric trilateral-free (n3) configurations. Additionally, the (273) configuration is also pentalateral-free. It serves as the smallest known example of a geometric (n3) configuration that is both 3-lateral-free and 5-lateral-free; the formerly smallest known example of such a configuration is a (513 ) configuration [2]. 2 The examples Configuration tables and diagrams of both of these new configurations C and C2 are provided below, and rational coordinates for their geometric realizations are given. Verification that the former configuration is trilateral-free, and that the latter configuration is trilateral-free and pentalateral-free, has been conducted using Mathematica. 2.1 C1, a geometric triangle-free (233) configuration /1 1 1 2 2 3 3 4 4 5 5 6 6 7 8 9 9 11 12 12 15 18 21\ 12 8 16 13 17 4 6 7 20 10 14 7 9 10 11 10 19 14 13 15 16 19 221 \3 21 20 19 23 5 22 15 23 18 16 8 12 13 17 11 21 22 14 18 17 20 23/ 21 22 23 Point Coordinates 1 (33/4, 29/4) 4 (4, 6) 7 (3, 5) 10 (3, 4) 13 (3, 3) 16 (445/97, 2) 19 (1,1) 22 (2, 0) Point Coordinates 2 (7, 7) 5 (5, 6) 8 (6, 5) 11 (542/97,4) 14 (455/97, 3) 17 (462/97, 2) 20 (1132/291,1) 23 (1876/485,0) Point Coordinates 3 (2, 6) 6 (2, 5) 9 (1,4) 12 (0, 3) 15 (0, 2) 18 (0,1) 21 (1,0) 2.2 C2, a geometric triangle-free, pentalateral-free (273 ) configuration /1 1 1 2 2 3 3 4 4 4 5 6 7 7 7 10 10 11 12 13 13 15 16 18 19 22 25\ 12 10 20 5 9 6 8 5 14 21 8 9 8 11 23 11 17 14 20 14 16 24 17 21 20 23 261 \3 13 27 19 12 15 25 6 17 24 18 22 9 16 26 12 19 22 23 15 25 27 18 26 21 24 27/ Point Coordinates 1 (0, 8) 4 (2, 7) 7 (1, 6) 10 (0, 5) 13 (0, 4) 16 (1, 3) 19 (3, 2) 22 (5,1) 25 (4, 0) Point Coordinates 2 (3, 8) 5 (3, 7) 8 (4, 6) 11 (1, 5) 14 (2,4) 17 (2, 3) 20 (6, 2) 23 (6,1) 26 (7,0) Point Coordinates 3 (4, 8) 6 (5, 7) 9 (5, 6) 12 (6, 5) 15 (8, 4) 18 (7, 3) 21 (7, 2) 24 (8,1) 27 (8, 0) 3 Motivation for the results Both C1 and C2 have arisen serendipitously in conjunction with the author's study of magic (n3) configurations. An (n3) configuration is said to be magic if it is possible to assign the integers {1,2,... ,n} as labels for its n points, where each integer is used exactly once, in such a manner that the sum of the point labels along each line of the configuration is always the same magic constant, M. Since each point of the configuration is involved in three such sums, we see that nM = 3 ^ i = 3- i(n + 1) M= 2(n +1) Hence n must be odd (and at least 7) for a magic configuration to be possible. The smallest example of a magic configuration turns out to one of the 31 (113) configurations. Its combinatorial table is '1 112 223 3 3 4 4\ 6 785 674 57561 ^11 10 9 11 10 9 11 10 8 9 8/ This configuration is (113)17, according to the (113) configuration labeling scheme initiated in [4] and referenced in [6],[7]. Magic (n3) configurations have not, to the author's knowledge, been previously considered in the literature on configurations, although other magic configurations such as magic stars have been studied [8]. C1 is dual to the magic (233) configuration '1 112223334445556667778 9\ 13 14 15 11 12 15 10 15 16 12 13 14 8 9 14 9 10 13 8 10 11 12 111 22 21 20 23 22 19 23 18 17 20 19 18 23 22 17 21 20 17 21 19 18 16 16 with magic constant § (23 + 1) = 36. Also, C2 is dual to the magic (273) configuration '1 1122233344455566677788899 9N 114 17 18 15 16 18 13 16 17 11 12 17 10 12 18 10 11 16 11 14 15 12 13 15 10 13 14 I \27 24 23 25 24 22 26 23 22 27 26 21 27 25 19 26 25 20 24 21 20 22 21 19 23 20 19/ 2 with magic constant § (27 + 1) = 42. This means that in each case there exists an isomorphism between C and its dual that interchanges the roles of points and lines while preserving incidence structure. We say that the dual configuration of a magic configuration is comagic; hence Ci and C2 are comagic. So for both Ci and C2 it is possible to label its lines in such a manner that the sum of the labels of the three lines incident to any point of the configuration is always the same magic constant, again 3 (n + 1). It turns out that a diagram associated with a comagic configuration may be conveniently constructed. Suppose that (x1 x2 x3)T is a line in the configuration table of the original magic configuration, where x1 < x2 < x3. It follows that the point (x1, x2, x3) G R3 lies in the plane {(x,y,z) G R3 : x + y + z = § (n +1)}. After plotting each corresponding point in this plane, for k = 1,..., n we connect three points with an arc (labeled k) if the three points share k as a coordinate. We thereby produce a diagram within the plane x + y + z = § (n + 1). Next, we project the diagram onto the xz-plane by simply eliminating the y-coordinate. No information about the configuration is lost when doing this, since for any point we may recapture x2 = § (n +1) - x1 - x3. Below is a diagram for C1 achieved in this fashion with each (x1, x3) point indicated. (2,23) (3,23) (5,23) (8,16) (9.16) Observe that this diagram has three nonlinear arcs. After some algebraic manipulation involving shifting seven of the 23 points, we find that it is possible to recast the diagram so that all of the arcs indeed are straight lines. After rescaling the points (via the transformation (x, z) ^ (x - 1, 23 - z)) we arrive at the geometric realization for C1 provided in Section 2.1. We again depict the diagram for C1, this time with its associated magic line labeling. When undergoing this process for the (273) configuration, we discover pleasantly that no shifting of arcs is required. This is a consequence of each line (x1 x2 x3)T satisfying the conditions 1 < xi < 9, 10 < x2 < 18, and 19 < x3 < 27. After lopping off the x2-coordinates and rescaling the resulting points (via the transformation (x, z) ^ (x - 1,z - 19)) we arrive at the geometric realization for C2 provided in Section 2.2. We display the diagram of C2 again with its associated magic line labeling. 27 References [1] A. Al-Azemi and A. Betten, Classification of triangle-free 223 configurations, Inter. Journ. Comb. 2010 (2010) # 767361. [2] M. Boben, B, Grünbaum and T. Pisanski, Multilaterals in configurations, Beitr. Algebra Geom. 52 (2011), 1-13. [3] M. Boben, B. Grünbaum, T. Pisanski and A.Žitnik, Small triangle-free configurations of points and lines, Discrete Comput. Geom. 35 (2006), 405-427. [4] R. Daublebsky von Sterneck, Die configurationen II3, Monatsh. Math. Phys. 5 (1894), 325-330. [5] B. Grunbaum, Configurations of Points and Lines, in Graduate Studies in Mathematics, vol. 103, American Mathematical Society, Providence, 2009. [6] W. Page and H. Dorwart, Numerical patterns and geometric configurations, Math. Mag. 57 (1984), 82-92. [7] B. Sturmfels and N. White, All 113 and 123-configurations are rational, Aequations Math. 39 (1990), 254-260. [8] M. Trenkler, Magic stars, nME Journal 11 (2004), 549-554. d MFA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 261-278 Interlacing-extremal graphs* Irene Sciriha Mark Debono , Martha Borg Department of Mathematics, Faculty of Science, University of Malta, Msida, Malta Patrick W. Fowler, Barry T. Pickup Department of Chemistry, Faculty of Science, University of Sheffield, Sheffield S3 7HF, UK Received 18 November 2011, accepted 13 September 2012, published online 19 November 2012 Abstract A graph G is singular if the zero-one adjacency matrix has the eigenvalue zero. The multiplicity of the eigenvalue zero is called the nullity of G. For two vertices y and z of G, we call (G, y, z) a device with respect to y and z. The nullities of G, G - y, G - z and G - y - z classify devices into different kinds. We identify two particular classes of graphs that correspond to distinct kinds. In the first, the devices have the minimum allowed value for the nullity of G - y - z relative to that of G for all pairs of distinct vertices y and z of G. In the second, the nullity of G - y reaches the maximum possible for all vertices y in a graph G. We focus on the non-singular devices of the second kind. Keywords: Adjacency matrix, singular graphs, nut graphs, uniform-core graphs, nuciferous graphs, interlacing. Math. Subj. Class.: 05C50,05C35, 05C60, 05B20, 92E10, 74E40 1 Introduction Agraph G = G(V, E ) of order n has a labelled vertex set V = {1,2, ...,n}. The set E of m edges consists of unordered pairs of adjacent vertices. We write V(G) for a graph G when the graph G needs to be specified. A subset of V is said to be independent if no two of its vertices are adjacent, i.e., no two are connected by an edge. For a subset Vi of V, G - Vi denotes the subgraph of G induced by V\V1. The subgraph of G obtained by deleting a particular vertex y is denoted by G - y and that obtained by deleting two distinct vertices *This research was supported by the Research Project Funds MATRP01-02 Graph Spectra and Fullerene Molecular Structures of the University of Malta. t Corresponding Author, http://staff.um.edu.mt/isci1/ E-mail addresses: irene.sciriha-aquilina@um.edu.mt (Irene Sciriha), debonomark@gmail.com (Mark Debono), bambinazghira@gmail.com (Martha Borg), P.W.Fowler@sheffield.ac.uk (Patrick W. Fowler), b.t.pickup@sheffield.ac.uk (Barry T. Pickup) y and z is denoted by G - y - z. A graph is said to be bipartite if its vertex set V may be partitioned into two independent subsets Vi and V2. The cycle and the complete graph on n vertices are denoted by Cn and Kn, respectively. The complete bipartite graph Kni,n2 has a vertex partition into two subsets V1 and V2 of independent vertices of sizes n1 and n2, respectively, and has edges between each member of V1 and each member of V2. 1.1 The adjacency matrix The graphs we consider are simple, that is, without loops or multiple edges. We use A(G) (or just A when the context is clear) to denote the 0-1 adjacency matrix of a graph G, where the entry aik of the symmetric matrix A is 1 if {i, k} e E and 0 otherwise. We note that the graph G is determined, up to isomorphism, by A. The adjacency matrix AC of the complement GC of G is J - I - A, where each entry of J is one and I is the identity matrix. The degree of a vertex i is the number of non-zero entries in the ith row of A. If the adjacency matrix A of a n-vertex graph G satisfies Ax = Ax for some non-zero vector x then x is said to be an eigenvector belonging to the eigenvalue A. There are n linearly independent eigenvectors. The eigenvalues of A are said to be the eigenvalues of G and to form the spectrum of G. They are obtained as the roots of the characteristic polynomial ^(G, A) of the adjacency matrix of G, defined as the polynomial det(AI - A) in A. Cauchy's inequalities for a Hermitian matrix M (also collectively known as the Interlacing Theorem) place restrictions on the multiplicity of the eigenvalues of principal submatrices relative to those of M (See [6] for instance). When they are applied to graphs we have: Theorem 1.1. Interlacing Theorem: Let G be an n-vertex graph and w e V. If the eigenvalues of G are A1, A2,..., An and those of G - w are £2,..., £n-1, both in non-increasing order, then A1 > > A2 > £2 > ... > £n-1 > An. 1.2 Cores of singular graphs For the linear transformation A, the kernel, ker( A), of A is defined as the subspace of Rn mapped to zero by A. It is also referred to as the nullspace of A. A graph G is said to be singular of nullity n0 if the dimension of the nullspace ker(A) of A is n0 and no > 0. If there exists a non-zero vector x in the nullspace of the adjacency matrix A, then x is said to be a kernel eigenvector of the singular graph G and satisfies Ax = 0. It is therefore an eigenvector of A for the eigenvalue zero whose multiplicity no is also the number of roots of ^(G, A) equal to zero. A vertex corresponding to a non-zero entry of x is said to be a core vertex CV of G. The core vertices corresponding to x induce a subgraph of G termed the core of G with respect to x. The core structure of a singular graph will be the basis of our classification of all graphs relative to n0. A core graph is a singular graph in which every vertex is a core vertex. The empty graph (K4)c and the four cycle C4 are examples of 4-vertex core graphs of nullity four and two, respectively. A core graph of order at least three and nullity one is known as a nut graph. It is connected and non-bipartite [12]. For singular graphs, the vertices can be partitioned into core and core-forbidden vertices. The set CV of core vertices consists of those vertices lying on some core of G. A core-forbidden vertex (CFV) corresponds to a zero entry in every kernel eigenvector. The set V\CV is the set of CFVs. It follows that, in a core graph, the set of CFVs is empty. Let y and z be two distinct vertices of a graph G. By interlacing, when a vertex y or z is deleted from G, the nullity nG-y or nG-z, that is the multiplicity of the eigenvalue zero of G - y or G - z, respectively, may take one of three values from na - 1 to na + 1. If the two distinct vertices y and z are deleted, then the nullity nG-y-z of G - y - z may take values in the range from nG - 2 to na + 2. Let us call the graph having two particular distinct vertices y and z a device (G, y, z). The set of devices can be partitioned into three main varieties, namely variety 1 when both vertices are CVs, variety 2 when one vertex is a CFV and one a CV and variety 3 when both vertices are CFVs. A device (G, y, z) is said to be of kind (na, nG-y, nG-z, nG-y-z). Since nG-y and nG-z can take three values each and na-y-z can take five values, there are potentially 45 kinds of graphs relative to na. Interlacing further restricts the values of nG-y-z. Moreover, there are kinds of graphs that exclude certain combinatorial properties, such as that of being bipartite, as we shall see in Section 5. In Section 2, we express the characteristic polynomial of ^(G - y, A) as the sum of two terms in AnG and AnG - 1 with coefficients /a(A) and fb(A), respectively, each of which is a polynomial expanded in terms of the entries of the eigenvectors of A forming an orthonormal basis for Rn. By comparing the diagonal entries of the adjugate of (AI - A) and of the spectral decomposition of (AI - A)-1 we obtain, in Section 3, an expression for ^(G - y - z) as the sum of three terms in AVG, AVG - 1 , AVG - 2, respectively, with polynomial coefficients. Moreover, the well known Jacobi's identity (see, for instance, [4]), relating the entries of the adjugate of (AI - A) with the characteristic polynomials of a graph G and those of particular subgraphs of G, is used to determine which kinds are not realized by any graph G. In Section 4, the vertices of a graph are partitioned into three subsets of type lower, middle or upper, respectively, according to the vanishing or otherwise of fa(0) and /5(0). The Interlacing Theorem and Jacobi's identity impose restrictions on the 45 kinds, so that not all are possible. In Sections 5 and 6, we show why there exist exactly twelve kinds of device (G, y, z) and how they are partitioned into the three main varieties. In Section 7, we identify two interesting classes of graphs that in a certain sense have extremal nullities. The first one has the minimum possible nullity nG-y-z, that is nG - 2, for all pairs of distinct vertices y and z in a graph G. A graph G in the second class has the maximum possible nullity nG-y, that is nG + 1, for all vertices y of G. We show that devices within the second class can reach the maximum allowed nG + 2 for the nullity nG-y-z for some but not for all pairs of distinct vertices y and z in a graph G. A characterization is given of the non-singular devices within the second class having the inverse A-1 of the adjacency matrix A with zero entries only on the diagonal. 2 Characteristic polynomials We first need to define some necessary notation. Associated with the n x n adjacency matrix A of a n-vertex graph of nullity nG, there is an ordered orthonormal basis xr, 1 < r < n, for Rn, consisting of eigenvectors of A, with the nG eigenvectors in the nullspace being labelled first. Let the n x 1 column vector be (xy ), where for vertex y, 1 < y < n. If P / V where the ith column of P is the eigenvector x® belonging to the eigenvalue A® in the spectrum of A, diagonalization of A is given by P-1AP = D [A®], where D [A®] is the diagonal matrix having A® as the ith entry on the main diagonal. Expressing A in terms of D and P leads to the spectral decomposition theorem, which can also be applied to (AI - A)-1. This leads to an expression for the characteristic polynomial of the adjacency matrix <(G — y, A) of G — y which is given explicitly in terms of the eigenvector entries [xly}. Together with Jacobi's identity, it will serve as a basis for the characterization of graphs according to those kinds that can exist. Lemma 2.1. (xy )2 <(G — y, A) = £ (A—<(G, A) Proof. The characteristic polynomial of the adjacency matrix <(G — y, A) of G — y is the yth diagonal entry (adj(AI — A))22 of the adjugate of (AI — A). For arbitrary A, the matrix (AI — A) is invertible and <(G — y, A) = ((AI — A)-1)^<(G, A). Since P-1AP = D [A®], adj(AI — A) = (AI — A)-1 = PD[ it follows that < ( G, A) Taking the yth diagonal entry, 1 A - A. ]P 1 <(G — y,A) = (x1 x2 • • • xn )D [-1-] <(G, A) (xy xy xy )D[A — A®] E (xy )2 (A — A®) • /x2\ x2 xy \xn/ (2.1) □ For a graph G with adjacency matrix A of nullity nc, let s(A) denote <(G, A). If the spectrum of A is A1; A2, • • • , An, starting with the zero eigenvalues (if any), we write s(A) = H(A — Ai) = AnG so(A) with so(0)=0. (2.2) Partitioning the range of summation in Equation (2.1), <(G — y, A) = ^ ^ + ^ (^f <(G, A) ^ A ®=1 ®=VG+1 A A® r x 1 2 n x x x 1 1 1 2 x x x 2 2 1 2 n x x x n n n Hence k 2 -, ^ (xk)2s0(A)AnG 4>(G - y, A) = ^(xk)2so(A)AnG-1 + | ^ y\ ^^ (2.3) k=1 k=nG+1 k which we shall express as - y, A) = /6AnG-1 + faAnG. (2.4) 3 Jacobi's Identity Relative to (G, y, z), let us denote by j (A), or j, the entry of the adjugate adj(AI - A) in the yz position, obtained by taking the determinant of the submatrix of (AI - A) after deleting row y and column z and multiplying it by (-1)y+z. We use the convention that Va-y > Vg-z. Throughout the paper, where the context is clear, we may write s0 for so(A), j for j(A), etc. Let s(A), t(A), u(A), v(A), often referred to simply as s, t, u and v respectively, be the characteristic polynomials ^(G, A), ^((G - y), A), ^((G - z), A), ^((G - y - z), A) of the graphs G, G - y, G - z and G - y - z, respectively, that is, the determinants s(A) =det(AI - A(G)) t(A) = det(AI - A(G - y)) u(A) = det(AI - A(G - z)) v(A) = det(AI - A(G - y - z)). (3.1) From Lemma 2.1, t(A) = £(xk)2 n(A - A,) (3.2) k=1 £=k and u(A) = J>k(A - A,). (3.3) k=1 ,=k We shall see that the characteristic polynomial v (A) of G - y - z can also be expressed xy}and {xr} in terms of the eigenvector entries {xy } and {x!J} associated with distinct vertices y and z. Lemma 3.1. For y = z, Jacobi's identity expresses the entry j of the adjugate of AI - A in the yz position, for a symmetric matrix A, in terms of the characteristic polynomials s, u, t and v : j2 = ut - sv Expressing Equations (3.2) and (3.3) as in (2.4), nG n ,h\2 ^ k 2 1 ^ (xk)2s0(A)AnG 1 t(A) = ^>y)2so(A)AnG-1 + £ V y^ _^-= tbAnG-1 + taAnG, (3.4) k=1 k=nG+1 k and t(A) = g (xk)2so(A)AnG-1 + £ (xk (3.5) k=1 k=nG+1 = ub AnG-1 + UaAnG Now we consider pairs of vertices of G. I - A)-1 = A--^-1. n y z k=1 .= k j (A) = E(xkxk) n(A - A.-) (3.6) We can write VG j(A) = J2 XkXk -k= 1 x\: xz so(A)AnG-1 + J2 k=nG+1 xk xkk s o(A)An A Ak (3.7) =jbAnG-1+jaAnG The characteristic polynomial v (A) can be written as v (A) G-1 i „, \no-2 that is v(A) = vaAnG + vbAnG-1 + vcAnG-2, where nG nG ( x® x^ _ x^ x® ) \xzxy xz y J vc SL(Ubtb - j'2) = - S^ E(x® x£ x. x® )2 So i=1 .= 1 u(A)t(A) - j2(A) s(A) , -, nG n (xi x _ xi x. )2 1 / . , . - . . -, V^ V^ (xzxy xyxz) Vb = — (Uatb + Ubta - 2jajb) = So > > ■ ' ' s0 i=1.=nG+1 A A. i n n (xi x. _ x. xi )2 1 / . ,2\ 1 V^ V^ (xy xz xy xz) Va = S0(Uata - ja)=2S0 E E (A - Ai)(A - A.) 0 i=nG+1 .=nG+1 v ' (3.8) 4 Three types of vertex By interlacing, we can identify three types of vertex according to the effect on the nullity on deletion. We call a vertex y lower, middle or upper if the nullity of G - y is na - 1, Va or na + 1, respectively. We shall distinguish among these three types of vertex according to the values of the functions fa and fb in Equation (2.4). In Table 1 we show the entries of the orthonormal eigenvectors {xr } in an ordered basis for Rn as presented in Section 2. We choose a vertex labelling such that the core vertices are labelled first. Note the zero submatrix corresponding to the CFVs. We consider ^(G ^ yA) from Equation 2.3. It has poles at A = u®, 1 < i < h, where, S0AnG for 1 < i < h, the u® are the h distinct non-zero eigenvalues of G. Moreover, the gradient of (xk)2 . A - Ak k=nG+1 is less than 0 for all A = u®. It follows that - y,A) s0AnG has at most (h - 1) roots strictly interlacing the h distinct eigenvalues of A. Note that ^ (xk )2 > 0 k=1 with equality if and only if y is a CFV. Thus at A = 0, fb is non-zero if y is a CV and zero Table 1: Ordered orthonormal basis of eigenvectors of A with * representing a possibly non-zero entry. E(Xy ) -—-— vanishes at A = 0 when y is upper, and does A — Ak k=va + 1 VG not vanish when y is middle. Note that when £ (x k )2 =0, one of the (h - 1) interlacing k= 1 roots may be zero. ( f ) Different cases occur depending on the vanishing or otherwise of the real constant nG n (xk )2 £(xk )2 and £ a y a at A = 0. Equation (2.3) and the analysis in the previous k=i k=nG+i - k paragraph (marked (f)) lead to the result that n0 - 1 < Va-y < Va + 1. This can be generalized for the multiplicity of any eigenvalue of G other than zero by replacing the cores and the nullspace of G by the ^-cores and ^-eigenspace of G (concepts introduced in [10]), thus giving another proof of the Interlacing Theorem. Proposition 4.1. The values of fb and fa of Expression (2.4) for ^(G - y, A) at A = 0 distinguish the three types of vertex as follows: Vertex y Status of y The values of fb and fa Lower CV fb(0) = 0 Middle CF fb(0)=0 and fa (0) = 0 Upper CFV fb(0)=0 and fa (0) = 0 Proof. Let y be a core vertex of a graph of nullity n0 > 0. There exists xk = 0 for some k, 1 < k < n0. Then fb(0) = 0, which is a necessary and sufficient condition for the multiplicity of the eigenvalue zero to be n0 - 1 for G - y. It follows that a vertex is lower if and only if it is a CV. If y is a CFV, then fb(0) = 0. For G - y, the multiplicity of the eigenvalue zero is at n (xk )2 n (xk )2 E(xy ) V—"V (xy ) -—is zero, then A divides > -—, the A - A^ ^ A - Ak k=nG+i k=nG+i multiplicity of the eigenvalue zero is exactly n0 + 1 for G - y and the vertex y is upper. Otherwise the multiplicity of the eigenvalue zero remains n0 for G - y and the vertex y is middle. □ We consider three varieties of devices {(G, y, z)} with pairs (y, z) of vertices, namely variety 1 with both y and z being CVs, variety 2 with z being a CV and y a CFV and variety 3 with both y and z being CFVs. Since a CFV can be upper or middle, varieties 2 and 3 are subdivided further, as seen in Table 3. From Proposition 4.1, for variety 1, ub = 0; tb = 0; for variety 2, ub = 0; jb = tb = vc = 0; for variety 3: ub = jb = tb = vb = vc = 0. Some of these varieties can be further subdivided according to the values at A = 0 of vc, vb and va or ja. From Proposition 4.1, tb(0) = 0 if and only if y is a core vertex. Similarly ub =0 if and only if z is a core vertex. If at least one of z or y is core forbidden, then jb(0) = 0. However, there are 'accidental' cases where jb(0) vanishes when both z and y are CVs, for example in C4 and K2,3 if the vertices y and z are connected by an edge. Indeed this is true for all bipartite core graphs of nullity at least two, since each of u and t has zero as a root. It follows that E2n is a factor of j2 = ut - sv = (jbEn-1 + jaEn)2 and therefore jb(0) = 0. 5 Restrictions on the nullity of G — y — z It is our aim to classify all graphs according to their kind defined by the quadruple (vg , 'G-y , 'G-z , 'G-y-z ). Not all the 45 kinds mentioned in Section 1 exist, as we shall discover. The classification will be given in Table 3 on Page 272. It is best possible since each kind is realized by some graph. 5.1 Restrictions arising from interlacing In a device (G, y, z) of kind ('G, 'G-y, 'G-z, 'G-y-z ), interlacing restricts the values that nG-y-z can take. The following result shows an instance when nG-y-z is determined by interlacing alone. Lemma 5.1. For (vg , 'G-y, 'G-z, 'G-y-z ) — (nG; 'G + 1 'G - 1 'G-y-z ), the nullity VG-y-z of G - y - z is 'G. Hence, ('G, 'G + 1, 'G - 1, nG) is the only kind where the nullities 'G-y and 'G-z differ by two. We say that it belongs to variety 2a. In kinds where the nullities nG-y and nG-z differ by one, interlacing allows nG-y-z to take either the value nG-y or nG-z. All three possible values of nG-y-z are allowed by interlacing when 'G-y = 'G-z. The symmetry about zero of the spectrum of a bipartite graph G (See for instance [8]) requires that the number of zero eigenvalues is 2k, if G has an even number of vertices and 2k +1 if G has an odd number of vertices, for some k > 0. This implies that on deleting a vertex from a bipartite graph, the nullity changes parity. Therefore if the nullity of a graph G and of its vertex-deleted subgraph G - y are the same, then G is not bipartite. Since on deleting a vertex a bipartite graph remains bipartite, it follows that a graph G of a kind where nG = nG-y or nG-y = nG-y-z cannot be bipartite. Lemma 5.2. If a vertex of a graph is middle, then the graph is not bipartite. Figure 1 shows a device (G, y, z) with a middle vertex z which becomes upper in G - y. z Figure 1: A graph with two middle vertices y and z. 5.2 Restrictions arising from Jacobi's Identity Lemma 3.1 requires that ut - sv which is j2 has 2k, k > 0, zero roots. Let gf denote the number of zero roots of the real function f. Therefore, for kinds of graph that imply (i) gut = gsv - 1 and gu = gt or (ii) gut = gsv + 1 and gu = gt, there is a contradiction and those kinds of graphs do not exist. Lemma 5.3. The following kinds of graphs do not exist: (i) (no,no,no - 1,no); (ii) (no, no +1, no +1, no +1); (iii) (no,no,no,no -1). Furthermore, if gut = gsv and gut is odd, then a graph of that kind exists if ut - sv is zero at A = 0, otherwise j2 would have an odd number of zeros. Therefore, if gut = gsv and gut is odd, jb = 0 at A = 0. Lemma 5.4. Graphs with gut = gsv and gut odd exist provided jb = 0 at A = 0. They are non-bipartite and of one of the following kinds: (i) (no,no,no - 1,no -1); (ii) (no, no +1, no, no +1). We shall call kinds (i) and (ii), in Lemma 5.4 above, variety 2b and 3b(i), respectively (See Table 3). Lemma 5.5. If (G, y, z) is a singular graph with gut < gsv and gsv odd, then (G, y, z) is non-bipartite and of kind (no, nc - 1, no - 1, no - 1). Proof. If y and z are CVs, gut < gsv, then (no, no-y, no-z, no-y-z) is (i) (no, no -1, no - 1, no ) or (ii) (no, no - 1, no - 1, no - 1). Now if furthermore, gsv is given to be odd, then no-y-z = nc - 1. It follows that no-y = no-y-z Therefore, G is not bipartite. □ We shall call the graphs in Lemma 5.5 above, variety 1(iii) (See Table 3). 6 Kinds of graphs In this section we determine the properties of a kind (no, no-y, no-z,no-y-z) within each of the three varieties. 6.1 Graphs of variety 1 Graphs of variety 1, are necessarily singular and therefore have at least one core. There are at least two vertices in a core. Lemma 6.1. For a device (G, y, z) of variety 1 and nullity one, j5(0) = 0 for core vertices y and z. Proof. For nG = 1, a non-zero column of the adjugate adj(A) is a kernel eigenvector of G [9]. The non-zero entries occur only at core vertices. Therefore, j5(0) =0. □ There are three types of pairs of vertices (CV,CV) for graphs of variety 1, depending on the nullity of G - y - z. Since nG > 1 and gu = gt = nG - 1, then the nullity gv of G - y - z can take any of the three values nG - 2, nG and nG - 1, corresponding to variety 1(i), 1(ii) and 1(iii), respectively. Theorem 6.2. For a device (G, y, z) of variety 1(iii), j(0) = 0 for core vertices y and z. Proof. For nullity one the result follows from Lemma 6.1. Now consider a graph with na > 1 of variety 1(iii), that is when gv = nG - 1. The number of zeros gut of ut is 2na - 2 and less than that of sv which is odd. If j2, which is ut - sv, is not to have an odd number of zeros, it follows, from j = jbAnG-1 + jaAnG, that j5 = 0 at A = 0. □ For variety 1(i), the vertices y and z are CVs. Moreover, without loss of generality, the vertex z is a CV of the subgraph G - y. Only for variety 1(i) is vc = 0. Definition 6.3. The connected graphs G in the devices {(G, y, z)} with all pairs of vertices (y, z) G V x V being of variety 1(i) are said to form the class of uniform-core graphs. Equivalently, nG-y-z = nG - 2, that is z is a CV of G - y for all vertex pairs (y, z). It is clear that all vertices of a uniform-core graph are CVs, and that they remain so even in a vertex-deleted subgraph G - y for any vertex y of G. Note that this is not the case in general; if y and z are two distinct core vertices of a graph G, then z need not remain a core vertex of G - y. We shall consider uniform-core graphs in more detail in Section 7. 6.2 Graphs of variety 2 In a device (G, y, z) of variety 2, (y, z) is a mixed vertex pair, that is exactly one vertex z of the pair (y, z) is a CV. From Lemmas 5.1 and 5.3, the following result follows immediately. Proposition 6.4. In a device (G, y, z) of variety 2, (i) there is only one kind when y is upper, namely kind (nG, nG + 1, nG - 1, nG) in variety 2a and (ii) only one kind when y is middle, namely kind (na, nG, nG - 1, nG - 1) in variety 2b. From Lemma 5.2, the graphs of variety 2b are non-bipartite. Theorem 6.5. In a device (G, y, z) of variety 2b, the term in A2nG-1 of j2 is identically equal to zero. Proof. In variety 2b, a graph is of kind (no, Va, Va — 1, n0 — 1). The parameter vc vanishes and vb(A) = Ubt- = 0. The number of zeros of ut is the same as that of sv. Therefore, so j2 = ut — sv has at least 2no — 1 zeros. In variety 2b, the term in A2nG-1 in its expansion u6ta is ubta — sovb. Also vc vanishes and vb(A) = —- = 0. Hence, sovb = ubta and the term so in A2nG-1 in the expansion of j2 is identically equal to zero, as expected from the fact that j2 is a perfect square. □ The parameter vb distinguishes between a graph in variety 2a and one in variety 2b. Theorem 6.6. For a graph in variety 2a, vb vanishes at A = 0. For a graph in variety 2b, vb = 0 at A = 0. Proof. For both kinds in variety 2, ub = 0. For an upper vertex, ta =0 at A = 0 and for a middle vertex ta = 0 at A = 0. Since s0 = 0, it follows that for a graph in variety 2a vb = 0 at A = 0 and, for a graph in variety 2b, vb = 0 at A = 0. □ 6.3 Graphs of variety 3 We now consider variety 3 for (CFV,CFV) pairs, when tb, ub, jb, vb and vc all vanish. Interlacing provides three types of vertex pairs depending on whether a CFV in the pair (y, z) is upper or middle. When both vertices are upper (variety 3a), by Lemma 5.3 only variety 3a(i) for gv = n0 and variety 3a(ii), when gv = n0 + 2 are allowed. The values at A = 0 of va or ja suffice to distinguish between graphs of variety 3(i) and 3(ii). Theorem 6.7. For variety 3a(i), both va and ja are non-zero at A = 0. For variety 3a(ii), both va and ja vanish at A = 0. Proof. For variety 3, vb = 0. Variety 3a(i) is (no, n0 + 1, V0 + 1, V0). Since v = vaAnG and n0-y-z = n0, va = 0 at A = 0. Also gj = 2no so that ja = 0 at A = 0. Variety 3a(ii) is (n0, n0+,n0 — 1, n0 — 1). Since gv = n0 + 2, A2 divides va and A divides all of the functions ta, ua and ja. □ For variety 3b, one vertex is upper and one is middle. Interlacing allows only gv = no + 1 and n0, corresponding to variety 3b(i) and variety 3b(ii), respectively. Both vb and jb vanish at A = 0. The value of ja at A = 0 distinguishes between variety 3b(i) and variety 3b(ii). Theorem 6.8. For variety 3b(i), ja vanishes at A = 0. For variety 3a(ii), ja is non-zero at A = 0. Proof. For variety 3b(i), A divides ja, as otherwise ut — sv is not the perfect square j2. variety 3b(ii) gv = n0 requires ja =0 at A = 0. □ For variety 3c, both vertices are middle. The values at A = 0 of ta and ua are nonzero. By Lemma 5.3, gv = n0 + 1 or no, corresponding to variety 3c(i) and variety 3c(ii), respectively. For variety 3c(ii), when gv = n0, va is non-zero at A = 0. Two cases may occur. Either ja =0 at A = 0 or the number of zeros of ja is at least one. The former case is Vertex y Vertex z variety 1 7 variety 1(i) 1 4 variety 1(ii) 1 2 variety 1(iii) 1 15 variety 2a 1 5 variety 2b 17 18 variety 3a(i) 15 17 variety 3a(ii) 5 15 variety 3b(i) 15 16 variety 3b(ii) 11 16 variety 3c(i) 5 6 variety 3c(iiA) 5 17 variety 3c(iiB) Table 2: All varieties and kinds for the same graph G illustrated in Figure 2. denoted by variety 3c(iiA). The latter case is variety 3c(iiB) for which the terms in A2nG-2 and in A2nG -1 of j2 vanish. The remaining case is for variety 3c(i) when gv = nG + 1 and A divides va. Figure 2: A device (G, y, z) of all possible kinds for various (y, z). The graph in Figure 2 exhibits a device (G, y, z) of all varieties and kinds for different choices of (y, z). The classification of devices into kinds and varieties has an application in chemistry in the identification of molecules (with carbon atoms in particular) that conduct or else bar conduction at the Fermi level. In the chemistry paper [3], conductors and insulators are classified into eleven cases that are essentially the twelve kinds of Table 3, with case 7 in [3] corresponding to the kinds (nG, nG ,nG ,nG) in variety 3c(iiA) and (nG, nG, nG ,Vg) in variety 3c(iiB). The latter two varieties are distinguishable by the non-vanishing or otherwise of ja (0). 7 Graphs with analogous vertex pairs In general, vertex pairs in a graph may be of different varieties and kinds. We shall explore two interesting classes of graphs with the same extremal nullity (allowed by interlacing) for all vertex-deleted subgraphs. These emerge in the classification of devices {(G, y, z)} according to their kind (nG, VG-y, nG-z ,VG-y-z). A pair of vertices y and z for which nG-y = nG-z is said to be an analogous vertex pair. Kind Characterization Variety G bipartite Two CVs 1 (9s,9t,9u) = (no gv = no - 2 no — i, no - i) Vc = 0 & tb = 0 & & no > 2 ub = 0 1(i) Allowed gv = no Vc = 0 & & no > tb = 0 & i ub = 0 & Vb(0) = 0 1(ii) Allowed gv = no - 1 Vc = 0 & tb = 0 & & no > i ub = 0 & Vb(0) = 0 1(iii) Forbidden CV and CFV 2 (gs,gt,gu) = (no gv = no no + i,no - i) Vc = 0 & tb = 0 & & no > i ub = 0 & Vb(0) = 0 2a Allowed (gs,gt,gu,gv) = (no,no,no — gv = no - 1 i) Vc = 0 & tb = 0 & & no > i ub = 0 & Vb(0) = 0 2b Forbidden Two CFVs 3 (gs,gt,gu) = (no gv = no no + i, no + i) Vc = 0 & & ta(0 tb = 0 & = 0 & u ub = 0 & a(0) = 0 & Vb(0) = 0 Va(0) = 0 3a 3a(i) Allowed gv = no + 2 Vc = 0 & & ta(0 tb = 0 & = 0 & u ub = 0 & a(0) = 0 & Vb(0) = 0 Va(0) = 0 3a(ii) Allowed (gs,gt,gu) = (no gv = no + 1 no + i, no) Vc = 0 & & ta(0 tb = 0 & = 0 & u ub = 0 & a(0) = 0 & Vb(0) = 0 Va(0) = 0 3b 3b(i) Forbidden gv = no Vc = 0 & & ta(0 tb = 0 & = 0 & u ub = 0 & Vb(0) = 0 a(0) = 0 & Va(0) = 0 3b(ii) Forbidden (gs,gt,gu) = (no gv = no + 1 no , no ) Vc = 0 & & ta(0 tb = 0 & = 0 & u ub = 0 & a(0) = 0 & Vb(0) = 0 Va(0) = 0 3c 3c(i) Forbidden gv = no Vc = 0 & & ta(0 tb = 0 & = 0 & u ub = 0 & a(0) = 0 & Vb(0) = 0 Va(0) = 0 3c(ii) Forbidden gv = no &ja (0) = 0 Vc = 0 & & ta(0 & ja (0 tb = 0 & = 0 & u =0 ub = 0 & a(0) = 0 & Vb(0) = 0 Va(0) = 0 3c(iiA) Forbidden g v = no& ja (0) =0 Vc = 0 & & ta(0 & ja (0 tb = 0 & = 0 & u =0 ub = 0 & a(0) = 0 & Vb(0) = 0 Va(0) = 0 3c(iiB) Forbidden Table 3: A characterization of all devices (G, y, z) according to their variety and kind. The first of these two classes consists of graphs G with the minimum possible nullity Va-y-z for all pairs of distinct vertices y and z, (i.e., no - 2) and therefore also the minimum possible nullities no-y and no-z (i.e., nc - 1). By Definition 6.3, these graphs form precisely the class of uniform-core graphs. On the other hand, the second of the two classes consists of graphs with the maximum possible nullity nc-y-z, that is no + 2, for some pair of distinct vertices y and z, and therefore also the maximum possible nullities no-y and no-z (i.e., no + 1). 7.1 Uniform-core graphs By Definition 6.3, each vertex pair in a uniform-core graph corresponds to a graph of variety (1i). Since the nullity of a graph is non-negative, and nc-y-z = Va - 2 for all vertex pairs y, z of a uniform-core graph G, then the nullity of G is at least two. To understand better the core-structure of uniform-core graphs and be able to characterize them as a subclass of singular graphs, it is necessary to use their core structure with respect to a basis for their nullspace. Let B be a basis for the n-dimensional nullspace of A of a singular graph G (with no isolated vertices) of nullity n > 1. As seen in [11], Hall's Marriage problem for sets, or the Rado-Hall Theorem for matroids, guarantees a vertex-subset S of distinct vertex representatives [1, 11], to represent a system SCores of cores corresponding to the vectors of B. This implies that deleting a vertex v representing a core F eliminates the core F from G - v, which will now have a new system of n - 1 cores. Also any k > 1 cores in a system SCores of no cores cover at least k + 1 vertices. Theorem 7.1. A device (G, y, z) is of variety 1(i) if and only if the two vertices y and z do not lie in one core only, i.e. at least two cores are needed to cover the vertices y and z. Proof. Consider a basis B for the nullspace of A. The vertices y and z lie on at least one core of G. There are two possibilities. Firstly, B has exactly one vector with non-zero entries at positions associated with y and z. In this case no-y-z = no-y = no - 1, which does not correspond to variety 1(i). Secondly, B has at least two vectors with non-zero entries at positions associated with y or z, when no-y-z = no-y - 1 = no - 2, which corresponds to variety 1(i). The two core vertices must represent two distinct cores in a system SCores of no cores corresponding to abasis B for the nullspace [11]. □ A subclass U of uniform-core graphs can be constructed from nut graphs. A graph G e U is obtained from a nut graph H on n vertices and m edges by duplicating each of the n vertices of H. Then G has 2n vertices and 4m edges. Figure 3 shows the uniform- |V (G)| core graph G e U obtained from the smallest nut graph H. The nullity of G is —^--+1. (G) 2 Deletion of any —^---+ 1 vertices reduces the graph to a non-singular graph. Let the vertices of G be labelled 1, 2,..., n, 1', 2', ...n' where {1, 2,...} are the vertices of the nut graph H and {1', 2',...} are the duplicate vertices of {1,2,...} in that order in G. Note that a vertex labelled r for 1 < r < n is adjacent to the original neighbours in H and also to precisely those primed vertices with the same numeric label. For instance, vertex 1 is adjacent to 2 and 7 in H and to 2, 2', 7 and 7' in G. The following result, expressing the adjacency matrix of G e U in terms of the adjacency matrix of H, is immediate. 6 9 Figure 3: The smallest nut graph H and the uniform-core graph G derived from H. Theorem 7.2. If H is the adjacency matrix of the nut graph H, then the adjacency matrix values equal in value to double the eigenvalues of H and an additional n zero eigenvalues corresponding to the n duplicate vertex pairs. If (xi, x2, • • • , xn ) is an eigenvector of H for an eigenvalue /, then (xi, x2, • • • , xn, xi, x2, • • • , xn is an eigenvector of G for an eigenvalue 2/. We shall now characterize uniform-core graphs by requiring that a set of vertex representatives of a system Scores of cores be an arbitrary subset of the vertices for all systems of cores. Theorem 7.3. A graph of nullity is a uniform-core graph if and only if it is a singular graph such that the deletion of any subset of vertices produces a non-singular graph. Proof. Let us relate the nullspace of A to the vertices of a uniform-core graph G of nullity . Let S be any subset of vertices of G labelled {1,2, • • • , } and let B be an ordered basis for the nullspace of A. If all pairs of vertices give a graph of variety 1(i), then no two vertices lie in only one core of SCores. Therefore, it is possible to obtain a new ordered basis B' for the nullspace of A, by linear combination of the vectors in B, such that, for 1 < i < , only the vector i of B' has a non-zero entry at position i [11]. Removal of any vertex in S destroys precisely one eigenvector of B' reducing the nullity by one. Deletion of all the vertices in S destroys all the kernel eigenvectors and leaves a non-singular graph. □ A characterization of the subclass G G U of uniform-core graphs uses the operation NEPS (non-complete extendedp-sum) of a nut graph and K2. The graph product NEPS is described for instance in [2]. Definition 7.4. Given a collection {Gi, G2, • • • , Gk, • • • , Gn } of graphs and a corresponding set B C {0,1}n\{(0, 0,0)}, called the basis, of non-zero binary n-tuples, the NEPS of Gi, G2,Gn is the graph with vertex set V(Gi) x V(G2) x • • • x V(Gn) in which two vertices {wi , w2, • • • , wn} and {yi ,y2, • • • , yn} are adjacent if and only if there exists (ßi, ß2, • • • , ßn) G B such that Wi = yi whenever ß = 0 and Wi is adjacent to y^ whenever ßi = 1. Lemma 7.5. [2] If for 1 < i < n, Ai1, Aì2, • • • , Aini is the spectrum of Gì, of order nì for 1 < i < n, then the spectrum of the NEPS of G1, G2, • • • , Gn with respect to basis B is {E Aßlr ,Aß!2, ••• ^ : ik = 1, 2,...,nfc & k = 1, 2,...,n}. ßeß The following result follows from the construction of a uniform-core graph G G U. Theorem 7.6. A uniform-core graph G G U is the NEPS of a nut graph G1 and K2 with basis {(1,0), (1,1)}. From Lemma 7.5 and Theorem 7.6, the spectrum of the uniform-core graph G G U is Aì + AìAj where {Aì} is the spectrum of the nut graph H and {Aj} = {1, -1} is the spectrum of K2. This agrees with the result in Theorem 7.2. 7.2 Non-singular graphs with a complete weighted inverse We shall now look into the second class of devices. Such a graph G is a device (G, y, z), of variety 3a(ii), for some pair of distinct vertices y and z. Graphs which are devices (G, y, z), of variety 3a(ii), for a particular pair of vertices y and z exist, as shown in the example of Figure 2 for vertex connections 15 and 17. Can a graph G be a device (G, y, z), of variety 3a(ii), for all vertex pairs {y, z}? The question amounts to determining whether it is possible to have n(G - y - z) equal to the maximum allowed nullity relative to n(G), that is n(G) + 2, for all vertex pairs {y, z}. The answer is in the negative. Lemma 7.7. It is impossible that a graph G is a device (G, y, z) of variety 3a(ii)for all pairs of distinct vertices y and z. Proof. Suppose G is a graph which is a device (G, y, z) of variety 3a(ii) for all pairs of distinct vertices y and z. This requires that each of the graphs G - y and G - z is singular and therefore has CVs. Deletion of a CV from G - y, restores the nullity back to n(G). Hence it is impossible to achieve n(G - y - z) = n(G) + 2, for all vertex pairs {y, z}. □ By Lemma 5.3(ii), the kind (no,VG-y ,Vo-z,Vo-y-z)=(nc,nG + 1,nc,nc + 1) is impossible. Hence the only devices (G, y, z) within the second class that have the maximum value of n(G - y) relative to nG, for all vertices y, are of kind (nG, nG + 1, nG + 1, nG). Our focus is on the non-singular graphs of this kind having the inverse A-1 equal to the adjacency matrix of the complete graph with real non-zero weighted edges and no loops. The smallest candidate is K2. Indeed A(K2) = A(K2))-1 = ^ 0 Definition 7.8. Let G be a non-singular graph G with the off-diagonal entries of the inverse A-1 of its adjacency matrix A being non-zero and real, and all the diagonal entries of A-1 being zero. Then G is said to be a nuciferous graph. The motivation for the name nuciferous graph (meaning nut-producing graph) will become clear from Theorem 7.9. To characterize this class of graphs, let us consider the deck {G - v : v G V} of subgraphs, as in the investigation of the polynomial reconstruction problem [10]. Theorem 7.9. Let G be a nuciferous graph. Then G is either K2 or each vertex-deleted subgraph G - v is a nut graph. Proof. Let Q be the n - 1 x n matrix obtained from A 1 by suppressing the diagonal entry from each column. Therefore each entry of Q is non-zero. Let the ith column of Q be q» := (q(i)i, q(2)i,..., q(i-i)i, «(i+i)», q(i+2)i,..., «(n)»)4 for 2 < i < n - 1. The first and last columns are q1 := (q(2)1, q(3)1, ...,q(n)1)t and qn := (q(1)n, q(2)n,..., q(n-1)n)4, respectively. Since AA-1 is the identity matrix I, then A(G - i)q» = 0 for all 1 < i < n. Therefore qi is a kernel eigenvector (with non-zero entries) of G - i for all the vertices i. Hence G - i is a core graph. By interlacing, it has nullity one. It follows that each vertex-deleted subgraph is a nut graph. □ From Lemma 7.7, nuciferous devices (G, y, z) are not of type of variety 3a(ii) for all pairs of distinct vertices y and z. Moreover, from Theorem 7.9, for G = K2, each vertex-deleted subgraph is a nut graph and therefore has nullity one. On deleting a vertex from a nut graph, the nullity becomes zero. Hence a candidate graph G cannot be of variety 3a(ii) for any pair of vertices y and z. Theorem 7.10. Let G be a nuciferous graph G. If G is not K2, then (i) it has order at least eight; (ii) the device (G, y, z) is of variety 3a(i)for all pairs of distinct vertices y and z; (iii) the graph G is not bipartite. Proof. (i) Since nut graphs exist for order at least seven [12], it follows, from Theorem 7.9, that a nuciferous graph G, of order at least three, has at least eight vertices. (ii) From the proof of Lemma 7.7, a nuciferous graph G is of kind (nG, nG + 1, nG + 1, 'G). Thus G is a device (G, y, z) of variety 3a(i) for all pairs of distinct vertices y and z. (iii) From Theorem 7.9, G - y and G - z are nut graphs and therefore cannot be bipartite [12]. Hence G has odd cycles and cannot be bipartite. □ To date, no graph (except K2) has been found to satisfy the condition of Theorem 7.9. An exhaustive search on all graphs on up to 10 vertices and all chemical graphs on up to 16 vertices reveals no counter example. We conjecture the following result. Conjecture 7.11. There are no graphs for which every vertex-deleted subgraph is a nut graph. 8 Chemical implications Graph theory has strong connections with the study of physical and chemical properties of all-carbon frameworks such as those in benzenoids, fullerenes and carbon nanotubes. The eigenvalues and eigenvectors of the adjacency matrix of the molecular graph (the graph of the carbon skeleton) are used in qualitative models of the energies and spatial distributions of the mobile n electrons of such systems. Specifically, graphs and their nullities figure in simple theories of ballistic conduction of electrons by conjugated systems. In the simplest formulation [3] of the SSP (Source and Sink Potential ) [5] approach to molecular conduction, the variation of electron transmission with energy is qualitatively modelled in terms of the characteristic polynomials of G, G - y, G - z, G - y - z, where G is the molecular graph and vertices y and z are in contact with wires. (This is the motivation for the definition of a device in the present paper.) As a consequence, the transmission at the Fermi level (corresponding here to A = 0) obeys selection rules couched in terms of the nullities nG, nG-y, VG-z, and nG-y-z [7], motivating the definition of kinds here. In terms of the varieties defined here, the SSP theory predicts conduction at the Fermi level for connection across the vertex pair (y, z) for 1 (ii), 1 (iii), 3a(i), 3b(ii), 3c(i) and 3c(iiA), and, conversely, insulation at the Fermi level for 1(i), 2a, 2b, 3a(ii), 3b(i) and 3c(iiB). The two classes of graphs with analogous vertex pairs and certain extremal conditions on the nullity of their vertex-deleted subgraphs, explored in Section 7 are envisaged to have interesting developments in spectral graph theory. Moreover, the classification of graphs into varieties and kinds has an application in chemistry in the identification of molecules (with carbon atoms in particular) that conduct or else bar conduction at the Fermi level that has already been investigated in [3]. According to the SSP theory, the first class, the uniform-core graphs, corresponds to insulation at the Fermi-level for all two vertex connections and the second class, the nuciferous graphs, to Fermi-level conducting devices (G, y, z) for all pairs of distinct vertices y and z. The latter class has the additional properties that it consists of devices corresponding to non-singular graphs that are Fermi-level insulators when y = z. Therefore nuciferous graphs have no non-bonding orbital and are conductors for all distinct vertex connection pairs and insulators for all one vertex connections. We conjecture that the only nuciferous graph is K2. References [1] R. A. Brualdi, The mutually beneficial relationship of graphs and matrices, volume 115 of CBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence, RI, 2011. [2] D. M. Cvetkovic, P. Rowlinson and S. Simic, Eigenspaces of Graphs, CUP, Cambridge, 1997. [3] P. W. Fowler, B. T. Pickup, T. Z. Todorova and W. Myrvold, A selection rule for molecular conduction, J. Chem. Phys. 131 (2009), 044104 (7 pages). [4] F. R. Gantmacher, The Theory of Matrices, volume 1, AMS Chelsea, 2010. [5] F. Goyer, M. Ernzerhof and M. Zhuang, Source and sink potentials for the description of open systems with a stationary current passing through. J. Chem. Phys., 126 (2007), 144104. [6] M. Marcus and H. Minc, A Survey of Matrix Theory and Matrix Inequalities, volume 115 of CBMS Regional Conference Series in Mathematics, Allyn and Bacon, Inc., Boston, Mass., 1964. [7] B. T. Pickup and P. W. Fowler, An analytical model for steady-state currents in conjugated systems, Chem. Phys. Lett. 459 (2008), 198-202. [8] A. J. Schwenk and R. J. Wilson, On the eigenvalues of a graph, in: L. W. Beineke, R. J. Wilson (eds), Selected Topics in Graph Theory I, Academic Press, London, 1978. [9] I. Sciriha, On the coefficient of A in the characteristic polynomial of singular graphs, Util. Math., 52 (1997), 97-111. [10] I. Sciriha, Graphs with a common eigenvalue deck, Linear Algebra Appl. 430 (2009), 78-85. [11] I. Sciriha, Maximal core size in singular graphs, Ars Math. Contemp. 2 (2009), 217-229. [12] I. Sciriha and I. Gutman, Nut graphs - maximally extending cores, Util. Math. 54 (1998), 257-272. d MFA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 279-288 Unordered multiplicity lists of wide double paths Aleksandra EriC Faculty of Civil Engineering, University of Belgrade, 11000 Belgrade, Serbia C. M. da Fonseca * Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal Received 20 February 2012, accepted 18 September 2012, published online 19 November 2012 Abstract Recently, Kim and Shader analyzed the multiplicities of the eigenvalues of a ^-binary tree. We carry this discussion forward extending their results to a larger family of trees, namely, the wide double path, a tree consisting of two paths that are joined by another path. Some introductory considerations for dumbbell graphs are mentioned regarding the maximum multiplicity of the eigenvalues. Lastly, three research problems are formulated. Keywords: Graph, tree, matrix, eigenvalues, multiplicities, inverse eigenvalue problem. Math. Subj. Class.: 15A18, 05C50 1 Preliminaries For a given n x n real symmetric matrix A = (aij ), we define the graph of A, and write G(A), as the undirected graph whose vertex set is {1,..., n} and edge set is {ij | i = j and oij =0}. On the other hand, for a given (weighted) graph G, we may define A(G) = (aij ) to be the (real) symmetric matrix whose graph G(A) is G. We focus our attention to the set S (G) = {A g Rnxn | A = At and G (A) = G} , i.e., the set of all symmetric matrices sharing a common graph G on n vertices. Nevertheless, all results can easily be extended to complex Hermitian matrices. If G is a tree, then the matrix A(G) is called acyclic. In particular, if G is a path, we order the vertices of G such that A(G) is a tridiagonal matrix. We will often omit the mention of the graph of the matrix if it is clear from the context. *This research was supported by CMUC - Centro de Matematica da Universidade de Coimbra and FCT -Fundacao para a Ciencia e a Tecnologia, through European program COMPETE/FEDER. E-mail addresses: eric@grf.rs (Aleksandra Eric), cmf@mat.uc.pt (C. M. da Fonseca) Let us denote the (algebraic) multiplicity of the eigenvalue 9 of a symmetric matrix A = A(G) by mA(9). The (n — 1) x (n — 1) principal submatrix, formed by the deletion of row and column indexed by i, which is equivalent to removing the vertex i from G, is designated by A(G\i). Among the linear algebra community, most of the results on multiplicities of eigenvalues are mainly confined to trees motivated by the Parter-Wiener Theorem [16] and to Cauchy's Interlacing Theorem. For a more detailed account on the subject the reader is referred to [16]. We remark that the Parter-Wiener Theorem was reformulated in the survey work [7], by the second author, motivated by the earlier seminal work of C. Godsil on matchings polynomials [9, 10, 11]. The same approach produced a result for the multiplicities of an eigenvalue of a matrix involving certain paths of the underlying graph, with many interesting applications to general graphs [4]. Theorem 1.1. [6, 8] Let P be a path that does not contain any edge of any cycle in G. Then mA(G\P )(9) > mA(G) (9) — 1 . (1.1) Since a tree has no cycles, the inequality (1.1) is true for any path in a tree, which generalizes a result for the standard adjacency acyclic matrices [9]. The inequality (1.1) can provide us an upper bound for the multiplicity of an eigenvalue of a graph. The next result was established by R.A. Beezer in [3, Lemma 2.1] and it gives a lower bound. It was originally stated for standard adjacency matrices, but it can be proved for weighted adjacency matrices. Lemma 1.2. Let us suppose that H1,...,Hk be graphs, and let v1,...,vt be additional vertices. Construct a graph H by adding new edges that have one endpoint in the set {v1,..., vt} and the other endpoint in a vertex of some Hi. If A1 A 0 Ak \ C \ C T e S (H ) : D / where Ai e S (Hi), for i = 1... k, D is a real diagonal block, and C has t rows, then 0 mA(A) m Ai (A) — t. (1.2) i=1 We remark that (1.2) is a special case of Cauchy-type interlacing theorems for block Hermitian matrices. In fact, if A is an eigenvalue of the upper block decomposition A1 © • • • © Ak, a block vector calculation shows that the dimension the eigenspace of A of A is at least as great as the dimension of the intersection of the eigenspace of A of B and the null space of C (for more details the reader is referred to [15]). Interestingly, Lemma 1.2 provides us an algorithm construct matrices of certain graphs where the maximum multiplicity is attained. For example, let us consider the (4,3)-tadpole graph T Considering the path joining vertices 1 and 4 in (1.1) we see that, for any eigenvalue A of A G S (T ), mA( A) < 2 (see also [2, 8]). On the other hand, from (1.2), setting A i for the Jacobi matrix whose graph is the path joining vertices 1 and 2 (an edge) and A2 for the matrix whose graph is the cycle containing vertices 4, 5, 6, and 7, we have m a (A) > mAl (A) + mÄ2 (A) - 1 > 1 + 2 - 1=2 . Therefore, if we want to construct a matrix in S (T) with an eigenvalue, say %/2 of maximal multiplicity, we may set Ai 11 11 and / 0 1 0 - 1 A2 = 1 0 0 1 1 0 0 1 , - 1 0 1 0 Then any matrix of the form / -1 1 0 0 0 0 0 \ 1 1 * 0 0 0 0 0 * * * 0 0 0 A= 0 0 * 0 1 0 -1 0 0 0 1 0 1 0 0 0 0 0 1 0 1 \ 0 0 0 1 0 1 0 / has a/2 (and, in this case, - %/2 too) as an eigenvalue of maximal multiplicity 2. In general, for a given (m, n)-tadpole Gm,n, if A is an eigenvalue of A G S(Gm,n) but not eigenvalue of A(Cm), then A is simple, from (1.1). In this note, we show how to use inequality (1.1) to generalize a result on the maximum multiplicity of an eigenvalue of a ^-binary tree due to Kim and Shader [18] to a more general family of trees: a wide double path. A wide double path is consists of two paths that are joined by another path. Unordered multiplicity lists for two classes of wide double paths are established generalizing previous results. At the end three new research problems are pointed out. First, we show how combining both bounds (1.1) and (1.2) in order to give necessary and sufficient conditions for an eigenvalue has maximal multiplicity in the case of a dumbbell graph. 2 Dumbbell graphs A dumbbell graph is obtained by joining two cycles by a path. We will assume that the length of this path is greater than 2. Nevertheless, all results are true for lower lengths, with slight modifications. Dumbbells graphs are a special class of bicyclic graphs, i.e., connected graphs in which the number of edges equals the number of vertices plus one. These graphs are well considered in graph theory, combinatorics, and optimization literature [12, 13, 14, 19, 20, 21]. Much attention has recently been paid to the spectral properties of these non-acyclic graphs [22, 23]. We start this section establishing an upper bound for the multiplicity of an eigenvalue of a dumbbell graph. Proposition 2.1. The maximum multiplicity of an eigenvalue of a dumbbell graph is 3. Proof. Let P be the path intersecting both pendant cycles of the dumbbell graph D. Observe that the (disconnect) subgraph D\P is the union of two paths. Then, from (1.1), mA(D) (A) < mA(D\P) (A) + 1 < 2 + 1 = 3 , for any eigenvalue A of D. □ Next we characterize the matrices where an eigenvalue attains the maximum multiplicity. Proposition 2.2. Let D be a dumbbell graph and let P be the path intersecting both pendant cycles Ci and C2 at the vertices vi and vi, respectively. If A is an eigenvalue of A G S (D) of multiplicity 3, then A is an eigenvalue of A(Ci \ vi) and of A(C2 \ vi). Proof. Again, by Theorem 1.1, if mA(D)(A) = 3, then mA(D\P)(A) > 3 - 1 = 2 . Since D\P = (Ci \ vi) U (C2 \ vi) and both Ci \ vi and C2 \ vi are paths, A must be an eigenvalue of each A(Ci \ vi) and of A(C2 \ vi). □ Corollary 2.3. Let D bea dumbbell graph and let P = vi v2 • • • vi be the path intersecting both pendant cycles Ci and C2 at the vertices vi and vi, respectively. If A is an eigenvalue of A G S (D) of multiplicity 3, then A is an eigenvalue of both A(Ci) and A(C2). Proof. Considering the path P' = v2 • • • vi and (1.1), we have mA(D\P')(A) > 2 . Now we only have to observe that D \ P' = Ci U (C2 \ vi) and mA(C<2\Ve)(A) = 1. □ Note that we can conclude a result more general than Corollary 2.3. In fact, A should be an eigenvalue of any tadpole graph obtaining by joining, for example, the path vž • • • vi, for any i = '2,...,l — 1, to the cycle C2 at vi. The next result is a straightforward consequence of (1.1). Proposition 2.4. Let D be a dumbbell graph and let P be the path intersecting both pendant cycles Ci and C2 at the vertices vi and vi, respectively. If A is an eigenvalue of A G S (D) but neither an eigenvalue of A(Ci \ vi) nor of A(C2 \ vi) or neither an eigenvalue of A(Ci) nor of A(C2), then mA( A) = 1, i.e., A is a simple eigenvalue of A. As we mentioned before, Lemma 1.2 provides an interesting algorithm producing matrices of certain graphs with eigenvalues of maximum multiplicity. For, let us consider the real number A. For a given path of order ki, let Ai be a tridiagonal matrix of order ki, with eigenvalue A, using (according Jean Favard) the simple et élégant Wendroff's algorithm [24], appeared a long time ago but somehow has not received so far the appropriate consideration by the linear algebra community. Now, using the general algorithm established in [5], it is possible to construct periodic Jacobi matrices, say A2 and A3, whose cycles are of orders k2 and k3, respectively, with A being an eigenvalue of both matrices. Let us set Ai A2 T T yT \ A = A3 G S (H ), x 0 0 \ y 0 0 where x is the 0,1 vector with 1's in the position 1 and ki + 1 and 0 elsewhere, and, analogously, y is the 0,1 vector with 1's in the position ki and ki + k2 + 1 and 0 elsewhere. Then A is an eigenvalue of A of multiplicity 3. In fact, from (1.2), mA( A) > mAl (A) + mÄ2 (A) + m a 3 (A) - 2=1 + 2 + 2 - 2 = 3 . 3 Maximum multiplicities We now turn back our attention to a family of binary trees. Recall that a binary tree is a tree such that the degree of each vertex is no more than three. In this section we will consider the family constituted by the trees of the following form: take five paths Pi,... ,P5 and two vertices u and v ; join any terminal vertex of Pi, P2, and P5 to u; the other terminal vertex of P5 and any terminal vertex of P3 and of P4 are added to v. These trees can also be seen as consisting of two paths that are joined by another path. Therefore, we will call them wide double paths. The paths Pi,..., P4 are the legs (or branches) of such tree. In [18] Kim and Shader studied several spectral properties of the so-called ^-binary trees. It is a particular case of the trees under discussion now: P5 has size 1 (i.e., a single vertex) and the longest legs among the four legs are connected to different terminal vertices. Theorem 3.1. Let T be a wide double path and A G S (T ). Then the maximum multiplicity of an eigenvalue of A is 3. Proof. We only have to apply (1.1), for example, to the path Pi - u - P5 - v - P3. □ Theorem 3.2. For a given wide double path T, A is an eigenvalue of A G S (T) of maximum multiplicity if and only if A is an eigenvalue of each path Pi,...,P5. Proof. Set Aj = A(Pj), for i = 1,..., 5. Let us assume first that mA(A) = 3. Considering the path Pi - u - P5 - v - P3 in T, from (1.1), it follows mA2 (A) + m A4 (A) > 3 - 1 = 2 . Thus, mA2 (A) = mAi (A) = 1. Analogously, we prove mAl (A) = mA3 (A) = 1. It remains to prove that mA5 (A) = 1. In fact, since P5 can be obtained from T deleting the paths P1 - u - P2 and P3 - u - P4, we have again, from (1.1), 1 > mA5 (A) > mA(A) - 1 > mA(A) - 2=1, where A = A(H), with H being the generalized star with center u and legs P1, P2, and P5. Conversely, if mAi (A) = 1, for i = 1,..., 5, then 5 3 > mA(A) > ^ mAi (A) - 2 = 3 . i=1 from Theorem 3.1and(1.2). □ Let us set li for the order of the path Pi, with i = 1,..., 5, in a wide double path W. Corollary 3.3. The number n3 of eigenvalues with multiplicity 3 of a wide double path is at most min{^1,.. .,t5}. Corollary 3.4. [18, Theorem 2(a)] Let T be a ^-binary tree and A e S (T). Then there are no eigenvalues of multiplicity 4 or more, and the number n3 of eigenvalues with multiplicity 3 is at most one. Furthermore, if A e ct(A) with mA( A) = 3, then the diagonal entry of A corresponding to the axis vertex of T is A. We now investigate the eigenvalues of multiplicity 2. The first result is a consequence of Lemma 1.2 and Theorem 3.2. Lemma 3.5. Let T be a wide double path and let A e S (T ). If A e ct(A) is an eigenvalue of exactly four of the paths P1,...,P5, then mA( A) = 2. As before, n2 denotes the number of eigenvalues of multiplicity 2 of a given matrix. Theorem 3.6. Let W be a wide double path and r = min{£j + £j | i = 1, 2, j = 3,4}. Then n2 ^ r - 2n3 . (3.1) Proof. By Theorem 3.2, if A1,..., A„3 are the distinct eigenvalues of A of multiplicity 3, then they must be (simple) eigenvalues of both A(P2) and A(P4). Taking into account Lemma 3.5, the inequality (3.1) follows. □ We remark that Theorem 3.2 is in fact much more general. An analogous result can be proved for any generalized caterpillar, i.e., a tree for which removing the legs produces a path, and the maximum multiplicity of any eigenvalue is equal to the number of legs minus one. In particular we have the following result. Lemma 3.7. Let S be a generalized star and A e S (S ). Then mA( A) = 2 if and only if A is an eigenvalue of each leg. Theorem 3.8. Let T be a wide double path and A e S (T ). Then mA( A) = 2 if and only if A is a simple eigenvalue of the paths P1, P2, and of the generalized star with center v and legs P3, P4, P5, or is a simple eigenvalue of the paths P3, P4, and of the generalized star with center u and legs P1,P2,P5. Proof. Let us assume that mA(A) = 2. From (1.1), for any path P in T, mA(T\P)(A) ^ 1. Therefore, if A is not an eigenvalue of Pi (P2), then it must be an eigenvalue of both P3 and P4. Moreover, if S denotes the generalized star with center u and legs P1 , P2, P5, then mA(s) (A) > 1. The other assertion is set in a similar fashion. The converse follows from (1.1) and (1.2). □ We now address the question on the number n1 of simple eigenvalues of A e S(W). Since n = ni + 2n2 + 3n3 = £i + • • • +£5 +2, on the one hand, we have ni ^ n . (3.2) In fact, the equality is attained when we construct A(L1),... ,A(L5), with distinct eigenvalues. On the other hand, ni = £1 + I2 + £3 + £4 + £5 +2 - 2n2 - 3n3 > £1 + £2 + £3 + £4 + £5 +2 - 2r + n3 > |£i - £2| + |£3 - £4| + £5 + 2 . (3.3) Interestingly, we observe that, from the lower bound (3.3), any matrix in S( W) must have at least £5 + 2 simple eigenvalues. This generalizes [18, Corollary 3]. 4 Unordered multiplicity lists Recall that if m1 > • • • > mk, with m1 + • • • + mk = n, are the multiplicities of the distinct eigenvalues of an n-by-n symmetric matrix A, then (m1,..., mk ) is the unordered multiplicity list (or list) of the eigenvalues of A. By unordered multiplicity list of a graph G we mean the set of unordered multiplicity lists for all matrices in S (G). Without loss of generality, we will assume that £1 ^ £2 and £3 ^ £4. Moreover, we convention that for a finite sequence of real numbers a1,..., aj, with i < 0, is empty. We are able now to find the unordered multiplicity lists of the wide double path under discussion. Theorem 4.1. Let W be a wide double path of order n, with £1 > £2 and £3 > £4. Then the set of unordered multiplicity lists of W consists of the positive integer lists of the form (3,..., 3, 2,..., 2,1,..., 1), (4.1) n- n2 ni with 0 ^ n3 ^ min{£2, £4, £5}, 0 ^ n2 ^ £2 + £4 - 2n3, and n1 = n - 2n2 - 3n3. Proof. From our discussion in the previous section, it is clear that any unordered multiplicity list of W is of the form (4.1). Conversely, let S1 (resp., S2) be the generalized star with center vertex u (resp., v) and legs L1, L2, L5 (resp., L3, L4, L5). Now, for k = 0,..., min{£2, £4, £5},p = 0,..., £2 -k, and q = 0,..., £4 - k, let us consider the £1 + £5 - k + p +1 distinct real numbers ßi,...,ߣ 4 — k-q, 0b ... , £4+^5 +p+q+1 strictly interlacing with the 4 + 4 - k + p (distinct) real numbers Ai,.. ., Xi5, ai,. .., a^-fc, &£2-k-p+i,.. ., <äi2-k , and the 4 +4 — k + q + 1 distinct real numbers . .., a^2-k-p,Ml, ... ,M^3-^2+^5+q+p+i strictly interlacing with the 4 + 4 — k + q (distinct) real numbers . . . , A^5 ,ßb .. . , ß^3 - k, ߣi-k-q+1, . . . , ߣi-k . Now we consider A e S(W) such that a(A(L5)) = {Ai,..., A k, Ak+i,..., A^5 } a(A(Li)) = {Ai,...,Ak ,ai,...,a£2-k-p,af2-k-p+i,...,a£i-k} a(A(L2)) = {Ai,..., A k ,ai,... ,a£2-k-p,ä£2-k-p+i,... ,äi2-k } a(A(Si)) = {Ai, Ai,..., Ak, Ak, ai,..., a^-k-p, ßi,..., ße^-k-q, di,..., Oil-ii +e5+p+q+i } a(A(L3)) = {A 1, . . . , Ak, ß1, . . . , ße4-k-q, ße 4-k-q+1, . . . , ße3-k } a(A(L4)) = {A 1, . . . , Ak, . . . , ße4-k-q, ße 4-k-q+1, . . . , Ä4-k } a(A(S2)) = {Al, Al,..., Ak, Ak, ..., ße4-k-q, al,..., ae2-k-p, Mi,..., Me3-e2+e5+q+p+iL The existence of the Jacobi matrices is granted by [24] and the two generalized stars by [17, Theorem 11]. It is clear that the unordered multiplicity list of A is (3,..., 3, 2,..., 2,1,...,1), 13 12 ti (4.2) with t3 = k, t2 = 4 + 4 — 2k — p — q, and ti = (4 — 4) + (4 — 4) + 4 + 2 + k + 2(p + q). Note that 0 < k + 2(p + q) < 2(4 + 4). □ Observe that with 1 = 4 < 4, 4, < 4,4, we are able to recover the results in [18] for ^-binary trees. Moreover, Theorem 4.1 can also be applied for 4 =0 [1, 17]. Finally, a tree is minimal provided there is a matrix such that number of distinct eigenvalues is equal to the diameter (counting edges) plus one. From Theorem 4.1, we conclude that the wide double paths are minimal, for 4, 4 < 4,4 < 4. In fact, with t3 = 0, t2 = 4 + 4, and tl = (4 — 4) + (4 — 4 ) + 4 +2, since the number of distinct eigenvalues is 4 + 4 + 4 + 2 and the diameter is 4 + 4 + 4 + 1. 5 Open problems In this paper, we provided the solution for the inverse eigenvalue problem of a wide double path, generalizing the results for the very particular case of the family of the so-called binary trees, recently established. The approach adopted here is different, offering a general result with a more concise proof. A natural generalization a wide double path is when we have more than 2 legs adjacent to the "central" vertices u and v. Let us suitably call such trees as wide double generalized stars. Problem 1. Characterize the unordered multiplicity lists of a wide double generalized star. It seems this is not a difficult problem to handle and an elegant characterization similar to Theorem 4.1 may be achieved. A more hard problem is related an analogous characterization for binary trees. Problem 2. Characterize the unordered multiplicity lists of a binary tree. Our results may also be seen as the starting point for a more meaningful study: Problem 3. What are the unordered multiplicity lists of trees having maximum multiplicity 3? Of course, from our approach, the previous question can be extended to general graphs. Probably new techniques will need to be developed for this attractive and vast area of research. Some computational experiments allow us to assert that there will be some surprising multiplicity lists. References [1] F. Barioli and S.M. Fallat, On the eigenvalues of generalized and double generalized stars, Linear Multilinear Algebra 53 (2005), 269-291. [2] F. Barioli, S.M. Fallat and L. Hogben, Computation of minimal rank and path cover number for certain graphs, Linear Algebra Appl. 392 (2004), 289-303. [3] R.A. Beezer, Trees with very few eigenvalues, J. Graph Theory 14 (1990), 509-517. [4] A. Eric and C.M. da Fonseca, Some consequences of an inequality on the spectral multiplicity of graphs, submitted. [5] R. Fernandes and C.M. da Fonseca, The inverse eigenvalue problem for Hermitian matrices whose graphs are cycles, Linear Multilinear Algebra 57 (2009), 673-682. [6] C.M. da Fonseca, A note on the multiplicities of the eigenvalues of a graph, Linear Multilinear Algebra 53 (2005), 303-307. [7] C.M. da Fonseca, On the multiplicities of eigenvalues of a Hermitian matrix whose graph is a tree, Ann. Mat. Pura Appl. 187 (2008), 251-261. [8] C.M. da Fonseca, A lower bound for the number of distinct eigenvalues of some symmetric matrices, Electron. J. Linear Algebra 21 (2010), 3-11. [9] C.D. Godsil, Spectra of trees, Ann. Discrete Math. 20 (1984), 151-159. [10] C.D. Godsil, Algebraic Combinatorics, Chapman and Hall, New York and London, 1993. [11] C.D. Godsil, Algebraic matching theory, Electron. J. Combin. 2 (1995), #R8. [12] J.L. Gross and J. Chen, Algebraic specification of interconnection network relationships by permutation voltage graph mappings, Math. Systems Theory 29 (1996), 451-470. [13] P. Hansen, A. Hertz, R. Kilani, O. Marcotte and D. Schindl, Average distance and maximum induced forest, J. Graph Theory 60 (2009), 31-54. [14] P. Hell and X. Zhu, Adaptable chromatic number of graphs, European J. Combin. 29 (2008), 912-921. [15] R.A. Horn, N.H. Rhee and W. So, Eigenvalue inequalities and equalities, Linear Algebra Appl. 270 (1998), 29-44. [16] C.R. Johnson, A. Leal Duarte and C.M. Saiago, The Parter-Wiener theorem: refinement and generalization, SIAM J. Matrix Anal. Appl. 25 (2003), 352-361. [17] C.R. Johnson, A. Leal Duarte and C.M. Saiago, Inverse eigenvalue problems and lists of multiplicities of eigenvalues for matrices whose graph is a tree: The case of generalized stars and double generalized stars, Linear Algebra Appl. 373 (2003), 311-330. [18] I.-J. KimandB.L. Shader, Unordered multiplicity lists of a class of binary trees, Linear Algebra Appl. (2011), doi:10.1016/j.laa.2011.07.006 [19] S. Kratsch and P. Schweitzer, Isomorphism for graphs of bounded feedback vertex set number, in: H. Kaplan (ed.), Algorithm Theory - SWAT 2010, Lecture Notes in Computer Science 6139, Springer, 2010, 81-92. [20] J.H. Kwak and J. Lee, Isomorphism classes of cycle permutation graphs, Discrete Math. 105 (1992), 131-142. [21] S.-M. Lee, K.-J. Chen and Y.-C. Wang, On the edge-graceful spectra of cycles with one chord and dumbbell graphs, Congr. Numer. 170 (2004), 171-183. [22] J. Wang, F. Belardo, Q. Huang and E.M. Li Marzi, Spectral characterizations of dumbbell graphs, Electron. J. Combin. 17 (2010), #R42. [23] J. Wang, Q. Huang, F. Belardo and E.M. Li Marzi, A note on the spectral characterization of dumbbell graphs, Linear Algebra Appl. 431 (2009), 1707-1714. [24] B. Wendroff, On orthogonal polynomials, Proc. Amer. Math. Soc. 12 (1961), 554-555. d MFA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 289-299 GCD-Graphs and NEPS of Complete Graphs Walter Klotz Institut fur Mathematik, Technische Universität Clausthal, Germany Torsten Sander Fakultät für Informatik, Ostfalia Hochschule für angewandte Wissenschaften, Germany Received 21 February 2012, accepted 20 August 2012, published online 19 November 2012 Abstract A gcd-graph is a Cayley graph over a finite abelian group defined by greatest common divisors. Such graphs are known to have integral spectrum. A non-complete extended p-sum, or NEPS in short, is well-known general graph product. We show that the class of gcd-graphs and the class of NEPS of complete graphs coincide. Thus, a relation between the algebraically defined Cayley graphs and the combinatorially defined NEPS of complete graphs is established. We use this link to show that gcd-graphs have a particularly simple eigenspace structure, to be precise, that every eigenspace of the adjacency matrix of a gcd-graph has a basis with entries -1,0,1 only. Keywords: Integral graphs, Cayley graphs, graph products. Math. Subj. Class.: 05C25, 05C50 1 Introduction Given a set B C {0,1}n and graphs G1,..., Gn, the NEPS (non-complete extended p-sum) of these graphs with respect to basis B, G = NEPS(Gi,..., Gn; B), has as its vertex set the Cartesian product of the vertex sets of the individual graphs, V (G) = V (G1) x ■ ■ ■ x V (Gn). Distinct vertices x = (x1,..., xn), y = (y1,..., yn) G V (G) are adjacent in G, if and only if there exists some n-tuple (ß1,..., ßn) G B such that xi = yi, whenever ßi = 0, and xj, yi are distinct and adjacent in Gi, whenever ßi = 1. In particular, NEPS(G1; {(1)}) = G1 and NEPS(G1; 0) = NEPS(G1; {(0)}) is the graph without edges on the vertices of G1 . The NEPS operation generalizes a number of known graph products, all of which have in common that the vertex set of the resulting graph is the Cartesian product of the input vertex sets. For example, NEPS(G1, ...,Gn; {(1,1,..., 1)}) = G1 ... G„ is the E-mail addresses: klotz@math.tu-clausthal.de (Walter Klotz), t.sander@ostfalia.de (Torsten Sander) product of Gì,..., Gn (cf. [10], "direct product" in [15]). As can be seen, unfortunately, the naming of graph products is not standardized at all. The "Cartesian product" of graphs in [15] is even known as the "sum" of graphs in [10]. With respect to this seemingly arbitrary mixing of sum and product terminology, let us point out that here the term "sum" (and also the "p-sum" contained in the NEPS acronym) indicates that the adjacency matrix of the constructed product graph arises from a certain sum of matrices (involving the adjacency matrices of the input graphs). Refer to [10] or [11] for the history of the notion of NEPS. We remark that the NEPS operation can be generalized even further, see e.g. [12] and [21]. Next, we consider the important class of Cayley graphs [13]. These graphs have been and still are studied intensively because of their symmetry properties and their connections to communication networks, quantum physics and other areas [8], [13]. Let r be a finite, additive group. A subset S C r is called a symbol (also: connection set, shift set) of r if —S = {—s : s G S} = S, 0 G S. The undirected Cayley graph over r with symbol S, denoted by Cay(T, S), has vertex set r; two vertices a, b G r are adjacent if and only if a — b G S. Let us now construct the class of gcd-graphs. The greatest common divisor of nonnegative integers a and b is denoted by gcd(a, b), gcd(0, b) = gcd(b, 0) = b. If x = (x1,... ,xr ) and m = (m1,..., mr ) are tuples of nonnegative integers, then we set gcd(x, m) = (dì,. .. ,dr ) = d, di = gcd(x,, m,) for i = 1,... ,r. For an integer n > 1 we denote by Zn the additive group of integers modulo n, the ring of integers modulo n, or simply the set {0,1,..., n — 1}. The particular choice will be clear from the context. Let r be an (additive) finite abelian group represented as a direct sum of cyclic groups, r = Zmi e ... e Zmr , mi > 1 for i = 1,..., r. Suppose that d, is a divisor of m,, 1 < d, < m,, for i = 1,..., r. For the divisor tuple d = (dì,..., dr ) of m = (m1,..., mr) we define Sr(d) = {x =(xi,...,xr ) G r : gcd(x, m) = d}. Let D = {d(1),..., d(k)} be a set of distinct divisor tuples of m and define k Sr(D) = (J Sr(dj)). 3 = 1 Observe that the union is actually disjoint. The sets Sr (D) shall be called gcd-sets of r. We define the class of gcd-graphs as the Cayley graphs Cay(r, S) over a finite abelian group r with symbol S a gcd-set of r. The most prominent members of this class are perhaps the unitary Cayley graphs Xn = Cay(Zn, Un), where Un = SZn(1) is the multiplicative group of units of Z„ (cf. [16], [17], [22]). The main goal of this paper is to show in Section 2 that every gcd-graph is isomorphic to a NEPS of complete graphs. Conversely, every NEPS of complete graphs is isomorphic to a gcd-graph over some abelian group. This relation is remarkable since it allows us to define gcd-graphs either algebraically (via Cayley graphs) or purely combinatorially (via NEPS). The characterization of gcd-graphs as NEPS of complete graphs reveals some new access to structural properties of gcd-graphs. As a first application, we show in Section 3 that every gcd-graph has simply structured eigenspace bases for all of its eigenvalues. This means that for every eigenspace a basis can be found whose vectors only have entries from the set {0,1, -1}. It is known that other graph classes exhibit a similar eigenspace structure, although not necessarily for all of their eigenspaces [9], [20], [25]. Finally, we present some open problems in Section 4. 2 Isomorphisms between NEPS of complete graphs and gcd-graphs We are going to show in several steps that gcd-graphs and NEPS of complete graphs are the same. Lemma 2.1. Let V = Zmi © • • • © Zmr and d = (d1,... ,dr ) a tuple of positive divisors of m = (m1,..., mr ). Define b = (bi) G {0,1}r by b 11 if di < mi, I 0 if di = mi. Then we have Cay(r, Sr(d)) = NEPS(Cay(Zmi, Szmi (di)),..., Cay(Zmr, K)); {b}). Proof. Both Cay(r, Sr(d)) and the above NEPS have the same vertex set r. It remains to show that they have the same edge set. Let x, y G r with x = (x1,..., xr), y = (y1,... ,yr ) and suppose that x = y. Now x and y are adjacent in Cay(r, Sr(d)) if and only if gcd(xi — yi, mi) = di for i = 1,..., r. The latter condition means that in case di < mi the vertices xi and yi are adjacent in Gi = Cay(Zmi, SZm. (di)), and in case di = mi we have xi = yi. But this is exactly the condition for adjacency of x and y in NEPS(G1,..., Gr ; {b}). □ The following lemma allows us to break down the Cayley graphs that form the factors of the NEPS mentioned in Lemma 2.1. Each factor can be transformed into a gcd-graph over a product of cyclic groups of prime power order. Using Lemma 2.1 once again, we obtain a representation of the original graph as a NEPS of NEPS of gcd-graphs over cyclic groups of prime power order. Lemma 2.2. Let the integer m > 2 and a proper divisor d > 1 of m be given as products of powers of distinct primes, r m = ^ mi, mi = pa, ai > 0 for i = 1,. .., r, i=1 r d = L! di, d- = pßi , 0 < ßi < a- for i = 1, ..., r. i=1 If we set r = Zmi © • • • © Zmr and d = (d1,..., dr ), then there exists an isomorphism Cay(Zm, Szm(d)) ~ Cay(r,Sr(d)). Proof. By the Chinese remainder theorem [23] we know that every z G Zm is uniquely determined by the congruences z = zi mod mi, zi G Zmi for i = 1,..., r. This gives rise to a bijection Zm ^ r by virtue of z ^ (zi,..., zr) =: z. We show that this bijection induces an isomorphism between Cay(Zm, SZm (d)) and Cay(r, Sr(d)). Let x, y G Zm, x = y. Note that X and y are vertices of Cay(r, Sr (d)). The vertices x and y are adjacent in Cay (Zm, SZm (d)) if and only if gcd(x—y, m) = d. This is equivalent to gcd(xi — yi, mi) = di for every i = 1,..., r. Now this means gcd(x — y, m) = d, with m = (m1,..., mr ), which is the condition for adjacency of x and y in Cay(r, Sr(d)). □ Next we shall prove a lemma that helps us consolidate the nesting of NEPS operations into a single NEPS operation. As a result, we then know that every single-divisor tuple gcd-graph is isomorphic to a NEPS of gcd-graphs over cyclic groups of prime power order. Lemma 2.3. Let H = NEPS(H (1),...,H(t); B ) (2.1) be a NEPS of graphs H(j) with respect to basis B such that each graph H(j) is itself a NEPS of graphs G(j) with respect to basis B(j), H(j) = NEPS(Gij),..., Gj); B(j)) for j = 1,...,t. (2.2) Then there exists a set B' C {0,1}r, r = r1 + ... + rt, such that H ~ NEPS(GÌ1),..., Gri),..., Git),..., Gr); B'). (2.3) Proof. We show that in (2.1) the graph H(1) can be replaced by G^,..., G^). More precisely, we construct a set B such that H ~ NEPS(G(11),...,Gr1),H(2) ,...,H(t); B), B C {0,1}ri +t-1 . (2.4) An analogous procedure can be repeated for H(2),..., H(t) until we end up with the representation (2.3) of H. In the original representation (2.1) every vertex x of the vertex set V(H) has the form x = (x(1),... ,x(t)), x(j) G V (H(j)) for j = 1,... ,t. (2.5) By (2.2) each coordinate x(j) is itself an rj-tuple, in particular x(1) = (x11),...,xr1)), x(1) G V (g(1)) for i = 1,...,r1 . Expansion of x(1) in (2.5) yields x = (x(1), . . . , x(1), x(2), . . . , x(t)), x((1) G V (g(1)) for i = 1, ...,r1, x(j) G V (H(j)) for j = 2,.. .,t. This is the representation of vertices for (2.4). (2.6) Now we adapt the basis set B to the new representation of vertices of H such that adjacencies remain unchanged. Let the distinct vertices x and y of H be given in their original representation according to (2.5) and in their new representation X, y according to (2.6). X = (x(1),...,x(t)), y = (y(1),...,y(t)), X = (x11), .. ., x^, x(2),.. ., x(t)), y = (y(1), ..., y(1), y(2),. .., y(t)). For each b = (b1,..., bt) G B we define a set B(b) C {0, 1}ri+t-1 such that x, y adjacent with respect to b ^ x, y adjacent with respect to B(b). (2.7) Case 1: b1 = 0. For x and y to be adjacent with respect to b we must have x(1) = y(1). If this is satisfied, then x and y are adjacent, if and only if (x(2),..., x(t)) and (y(2),..., y(t)) are adjacent with respect to (b2,..., bt). We achieve (2.7) by setting b = (0,..., 0, b2,..., bt) (first r1 entries equal to zero) and B(b) = {b}. Case 2: b1 = 1. Now x and y are adjacent with respect to b, if and only if x(1) and y(1) are adjacent in H(1) and x(2),..., x(t) and y(2),..., y(t) are equal or adjacent with respect to b2,..., bt, respectively. By (2.2) vertices x(1) and y(1) of H(1) are adjacent, if and only if they are adjacent with respect to some b(1) = (b^,..., b^)) G B(1). In this case we satisfy (2.7) by setting B(b) = {(b(11),..., b« b(2),..., b(t)) : b(1) G B(1)}. Finally, we collect the new basis tuples in B = U{B(b): b G B} and thus achieve (2.4). □ The next step towards our goal is to show that a single-divisor gcd-graph over a cyclic group of prime power order is actually isomorphic to a NEPS of complete graphs. We denote the complete graph on n vertices by Kn. For our purposes, we assume that the vertex set of Kn is Zn = {0,1,..., n - 1}. Lemma 2.4. Let m = pa be a prime power, d = pß a divisor of m, 0 < ß < a. Then the gcd-graph over Zm with respect to d is isomorphic to a NEPS of a copies of the complete graph Kp, i.e. Cay(Zm, Szm (d)) ~ NEPS(Kp,..., Kp; B) for some B C {0,1}a. Proof. In case ß = a we have Cay(Zm, SZm (m)) : So we may now assume ß < a. Let us denote G = Cay(Zm, SZm(d)) and H = basis B is not yet fixed). For every x G Zm let (x0 representation of x, a-1 x = ^^ xjp®, 0 < xj < p for i = 0,..., a — 1. i=0 - NEPS(Kp,...,Kp; {(0,..., 0)}). = NEPS(Kp,..., Kp; B) (where the ,..., xa-1) be defined by the p-adic We shall assume that the vertex set of Kp is Zp. Define the bijection ( : Zm ^ Zp © • • • © Zp = Zpa by f(x) = (x0,..., xa_i). We now construct a basis set B C {0,1}a such that (f induces an isomorphism between G and H. Observe that for every z G Zm, gcd(z, m) = d ■ Zi = 0 for every i < ß and zß = 0. This leads to the definition of B as follows: B = {(bo,...,ba_i) G {0,1}a : bi =0 for every i <ß,bß = 1}. Let x, y G Zm, x = y, f(x) = (xo,.. .,xa_i ), f(y) = (y o,... ,ya_i ). Now x and y are adjacent in G if and only if gcd(x — y,m) = d, which means xi — yi =0 for every i < ß and xß — yß = 0. Thanks to our choice of B, this is exactly the condition for f(x) and f(y) being adjacent in H. □ Theorem 2.5. Let G be an arbitrary gcd-graph, G = Cay(r, Sr(D)), r = Zmi © • • • © Zmr, D = {d(1),..., d(k)} a set of divisor tuples of m = (m1,..., mr ). If n = p1 • • • pt is the prime factorization of n = m1 • • • mr, then G ~ NEPS (Kp,,..., Kpt ; B ) = H for some B C {0,1}1. Proof. Each divisor tuple in D gives rise to a graph G(j) = Cay(r, Sr(d(j))), j = 1, . . . , k. By application of the preceding lemmas of this section we know that G(j) ~ NEPS(Kpi,..., Kpt ; B(j)) = H(j) for some B(j) C {0,1}4. The graphs G(j) constitute an edge disjoint decomposition of G. Now, for every divisor tuple d(1),..., d(k) G D, we perform the decomposition process outlined by the lemmas in exactly the same way, in the sense that the vertex numberings of the resulting graphs H are coherent. Then the graphs Halso constitute an edge disjoint decomposition of G: kk E(G) = y E(G(j)), E(H) = y E(H(j)) j=1 j=1 The binary sets B(j), 1 < j < k, are also pairwise disjoint. The disjoint union of the edge sets E (H(j)), 1 < j < k, is generated in the NEPS of Kp,,..., KPt by k B = y Bj). j=1 With this choice of B the isomorphisms between the subgraphs G(j) and H(j), 1 < j < k, extend to an isomorphism between G and H. □ Theorem 2.6. Let G be a NEPS of complete graphs, G = NEPS(Kmi,..., Kmr ; B). Then G is isomorphic to a gcd-graph over r — Zmi © • • • © Zmr. Proof. The vertex set of G can be represented by r = Zmi © • • • © Zmr. Edges of G are generated by the binary r-tuples b = (bi) of the basis set B. Vertices x = (x1,..., xr) = y = (y1,... ,yr) are adjacent in G with respect to b, if xi = yi, whenever bi = 0, and xi = Vi, whenever bi = 1. Let the set D(b) consist of all positive divisor tuples d = (di,..., dr) of m = (mi,..., mr) such that di = mi, whenever bi = 0, and di a proper divisor of mi, whenever bi = 1. Then x and y are adjacent with respect to b, if and only if gcd(x — v, m) G D(b). If we define D = U{D(b) : b G B}, then the gcd-graph Cay(r, Sr (D)) is isomorphic to G. □ Theorems 2.5 and 2.6 imply the following corollary. Corollary 2.7. Let n = pi ■ ■ ■ pt be the prime factorization of the integer n > 2. Every gcd-graph with n vertices is isomorphic to a gcd-graph over r — 0 ■ ■ ■ 0 Zpt. We conclude this section with some examples. Example 2.8. We generalize the definition of a Hamming graph given in [15]. The Hamming graph G = Ham(mi,..., mr; D) has vertex set V(G) = Zmi 0 ... 0 Zmr. Distinct vertices are adjacent in G, if their Hamming distance is in D. It can be easily shown that G is a NEPS of the complete graphs Kmi,..., Kmr. Example 2.9. Sudoku graphs arise from the popular game of Sudoku. The Sudoku graph Sud(n) models the number restrictions imposed when filling out an n2 ® n2 Sudoku puzzle. Each vertex represents a cell of the Sudoku puzzle. Two vertices are adjacent if the two corresponding cells are required to contain different numbers (which is the case when they lie in the same row, column or block of the puzzle). It has been shown that Sudoku graphs are NEPS of complete graphs [25]. Example 2.10. This is an example that demonstrates the application of Theorem 2.5. Let r = Z4 0 Zi8 and D = {(1, 6), (4, 2), (2,9)}. We want to represent the graph Cay(r, Sr (D)) as a NEPS of complete graphs. To start with, let us consider the graph Cay(r, Sr ((1,6))). Application of Lemma 2.1, Lemma 2.2, once again Lemma 2.1, then Lemma 2.3, Lemma 2.4, and finally once again Lemma 2.3 gives us: Cay(Z4 0 Zi8,S((1, 6))) ~ NEPS(Cay(Z4, S(1)), Cay(Zi8, S(6)); {(1,1)}) ~ NEPS(Cay(Z4, S(1)), Cay(Z2 0 Zg, S((2, 3))); {(1,1)}) ~ NEPS(Cay(Z4, S(1)), NEPS(Cay(Z2, S(2)), Cay(Zg, S(3)); {(0,1)}); {(1,1)}) ~ NEPS(Cay(Z4, S(1)), Cay(Z2, S(2)), Cay(Zg, S(3)); {(1, 0,1)}) ~ NEPS(NEPS(K2, K2; {(1,0), (1,1)}), NEPS(K2; {(0)}), NEPS(K3,K3; {(0,1)}); {(1,0,1)}) ~ NEPS(K2,K2,K2,K3,K3; {(1,0,0,0,1), (1,1,0,0,1)}). Note that for the sake of simplicity we have dropped the subscripts of the symbol sets since the respective groups are clear from the context. Regarding the application of Lemma 2.3 note that, trivially, G ~ NEPS(G; {(1)}). Cay(Z4 0 Zi8,S((4, 2))) ~ NEPS(K2, K2, K2, K3, K3; {(0, 0, 0,1,0), (0, 0,0,1,1)}), Cay(Z4 0 Zi8,S((2, 9))) ~ NEPS(K2, K2, K2, K3, K3; {(0,1,1, 0,0)}). The graph Cay(r, Sr(D)) is the disjoint union of the graphs Cay(r, Sr(d)) with d G D which we have considered above, so we arrive at: Cay(r,Sr(D)) ~ NEPS(K2, K2, K2, K3, K3; {(1,0,0,0,1), (1,1,0, 0,1), (0, 0, 0,1,0), (0,0,0,1,1), (0,1,1, 0, 0)}). 3 Eigenspace bases of gcd-graphs The eigenvalues and eigenspaces of an undirected graph G are the eigenvalues and eigenspaces, respectively, of any adjacency matrix of G. The multiset of all eigenvalues of a graph is called its spectrum. According to Harary and Schwenk [14], a graph G is defined to be integral if all of its eigenvalues are integers. Integral graphs have been a focus of research for some time; see [4] for a survey. In particular, many notable results on integrality of Cayley graphs have been obtained. Integral cubic and quartic Cayley graphs on abelian groups have been characterized in [1] and [2], respectively. Circulant graphs are the Cayley graphs over Zn, n > 1. So [26] showed that the integral circulant graphs with n vertices are exactly the gcd-graphs over Zn. This result was extended in [18] to groups of the form Z2 0 ... 0 Z2 0 Zn, n > 2. A complete characterization of integral Cayley graphs over abelian groups has recently been achieved by Alperin and Peterson [3]. The eigenvalues of G = NEPS(G1,..., Gn; B) are certain sums of products of the eigenvalues of the Gi, cf. [10]: Theorem 3.1. Let G1,..., Gn be graphs with n1,..., nr vertices, respectively. Further, for i = 1,..., r let Ai1,..., Ain. be the eigenvalues of Gi. Then, the spectrum of the graph G = NEPS(G1,..., Gn ; B ) with respect to basis B consist of all possible values n = Y^ Aßl Aßn Mil,...,i„ = A 1i i ' ... ' Anin (ßl,...,ß„)£B with 1 < ik < nk for 1 < k < n. A first consequence is that every NEPS of integral graphs is integral. It is easily checked that the complete graph Kn on n > 2 vertices has the simple eigenvalue n - 1 and the eigenvalue -1 with multiplicity n - 1. Hence NEPS of complete graphs are integral. Using Theorem 2.5, we now readily confirm the following result of [18]: Proposition 3.2. Every gcd-graph is integral. An interesting property of a graph is the ability to choose an eigenspace basis such that its vectors have entries from a very small set only. This may be possible only for certain or for all of its eigenvalues. For example, in [9] a construction is given for a basis of the eigenspace of eigenvalue -2 of a generalized line graph whose vectors contain only entries from {0, ±1, ±2}. Imposing an even greater restriction on the admissible entries, we call an eigenspace basis simply structured if it consists of vectors containing only entries from {0, 1, -1}. Accordingly, an eigenspace is considered as simply structured if it has a simply structured basis. Observe that the eigenvalue belonging to a simply structured eigenspace is necessarily integral. For a trivial example of a simply structured eigenspace basis, consider a connected r-regular graph. Here the all ones vector constitutes a basis of the eigenspace corresponding to the eigenvalue r. Moreover, for several graph classes, the eigenspaces corresponding to the eigenvalues 0 or -1 are simply structured, cf. [5],[20],[24]. It is somewhat remarkable if all of the eigenspaces of a graph are simply structured. In [25] is has been shown that Sudoku graphs are NEPS of complete graphs (recall Example 2.9) and admit simply structured eigenspace bases for all eigenvalues. As we shall see, this is true for any NEPS of complete graphs. For this we require the following theorem [11]: Theorem 3.3. If X and Y are graphs of orders n and m with linearly independent eigenvectors x(1),..., x(n) and y(1),..., y(m), respectively, then the nm tensor products x(i) ( y(j) (i = 1, .. ., n; j = 1,.. ., m) form a set of linearly independent eigenvectors of any NEPS of X and Y. This fact readily extends to NEPS with more factors. Corollary 3.4. Any NEPS of graphs for which all eigenspaces are simply structured inherits that very property. Proof. Using the notation of the previous theorem, it is obvious that x(i) ( y(j) has only entries from {0,1, -1} if the same holds for x(i) and y(j). This remains true for an arbitrary number of factors. □ We can now prove the following result: Proposition 3.5. All eigenspaces of a gcd-graph are simply structured. Proof. Consider the complete graph Kn, n > 2. The all-ones vector (1,1,..., 1) forms a basis of the eigenspace of eigenvalue n - 1. A basis of the eigenspace of eigenvalue -1 is formed by the vectors x(1) = (-1,1,0, 0,..., 0, 0), = (-1,0,1, 0,..., 0, 0), x (2) x(n-1) = (-1,0,0, 0,..., 0,1). Thus the result follows from Corollary 3.4 and Theorem 2.5. □ 4 Open problems Let us conclude with a number of open problems we think are worth investigating in the future: 1. Does every integral Cayley graph over a finite abelian group have a simply structured eigenspace basis for every eigenvalue? 2. Find a small class of integral graphs such that every integral Cayley graph over an abelian group is a NEPS of some graphs of this class. 3. It has been shown by So [26] that integral Cayley graphs over Z pa, p prime, are uniquely determined by their spectrum. Find more groups r such that cospectral integral Cayley graphs Cay(r, S1), Cay(r, S2) are necessarily isomorphic. 4. Try to determine or estimate the number g(n) of nonisomorphic gcd-graphs on n vertices. In [18] we showed that for a prime p > 5 we have g(p2 ) = 6. Observe that g(2a) is the number of nonisomorphic cubelike graphs on 2a vertices, cf. [19]. 5. Determine graph invariants for gcd-graphs such as connectivity, clique number, and chromatic number, cf. [6], [7]. References [1] A. Abdollahi and E. Vatandoost, Which Cayley graphs are integral?, Electron. J. Comb. 16 (2009), R122, 1-17. [2] A. Abdollahi and E. Vatandoost, Integral quartic Cayley graphs on Abelian groups, Electron. J. Comb. 18 (2011), P89, 1-14. [3] R. C. Alperin and B. L. Peterson, Integral Sets and Cayley Graphs of Finite Groups, Electron. J. Comb. 19 (2012), P44, 1-12. [4] K. Balinska, D. Cvetkovic, Z. Rodosavljevic, S. Simic and D. Stevanovic, A survey on integral graphs, Univ. Beograd, Publ. Elektrotehn. Fak. Ser. Mat. 13 (2003), 42-65. [5] S. Barik, S. Fallat and S. Kirkland, On Hadamard diagonalizable graphs, Linear Algebra Appl. 435 (2011), 1885-1902. [6] M. Basic and A. Ilic, On the clique number of integral circulant graphs, Appl. Math. Letters 22 (2009), 1406-1411. [7] M. Basic and A. Ilic, On the chromatic number of integral circulant graphs, Comp. Math. Appl. 60 (2010), 144-150. [8] M. Basic, M. D. Petkovic and D. Stevanovic, Perfect state transfer in integral circulant graphs, Appl. Math. Lett. 22 (2009), 1117-1121. [9] L. Brankovic and D. Cvetkovic, The eigenspace of the eigenvalue —2 in generalized line graphs and a problem in security of statistical databases, Publ. Elektroteh. Fak., Univ. Beogr., Ser. Mat. 14 (2003), 37-48. [10] D. Cvetkovic , M. Doob and H. Sachs, Spectra of graphs. Pure and Applied Mathematics, Academic Press, 1980. [11] D. Cvetkovic , P. Rowlinson and S. Simic, Eigenspaces of graphs. Encyclopedia of Mathematics and its Applications, Cambridge University Press, 1997, new edition 2008. [12] D. Cvetkovic and M. Petric, Connectedness of the non-complete extended p-sum of graphs, Zb. Rad., Prir.-Mat. Fak., Univ. Novom Sadu, Ser. Mat. 13 (1983), 345-352. [13] G. Hahn and G. Sabidussi (Eds.), Graph Symmetry: Algebraic Methods and Applications, NATO Science Series C, Springer Netherlands, 1997. [14] F. Harary and A. J. Schwenk, Which graphs have integral spectra?, Lect. Notes Math. 406, Springer Verlag (1974), 45-50. [15] W. Imrich and S. Klavzar, Product graphs. Structure and recognition., Wiley-Interscience Series in Discrete Mathematics and Optimization, 2000. [16] D. Kiani, M. M. H. Aghaei, Y. Meemark and B. Suntornpoch, Energy of unitary Cayley graphs and gcd-graphs, Linear Algebra Appl. 435 (2011), 1336-1343. [17] W. Klotz and T. Sander, Some properties of unitary Cayley graphs, Electron. J. Comb. 14 (2007), R45, 1-12. [18] W. Klotz and T. Sander, Integral Cayley graphs defined by greatest common divisors, Electron. J. Comb. 18 (2011), P94, 1-15. [19] L. Lovasz, Spectra of graphs with transitive groups, Period. Math. Hung. 6 (1975), 191-195. [20] M. Nath and B. K. Sarma, On the null-spaces of acyclic and unicyclic singular graphs, Linear Algebra Appl. 427 (2007), 42-54. [21] M. Petric, Connectedness of the generalized direct product of digraphs, Univ. Beograd. Publ. Elektrotehn. Fak. (Ser. Mat.) 6 (1995), 30-38. [22] H. N. Ramaswamy, C. R. Veena, On the Energy of Unitary Cayley Graphs, Electron. J. Comb. 16 (2009), N24, 1-8. [23] H. E. Rose, A course in number theory, Oxford Science Publications, Oxford University Press, 1994. [24] J. W. Sander and T. Sander, On the Kernel of the Coprime Graph of Integers, Integers 9 (2009), 569-579. [25] T. Sander, Sudoku graphs are integral, Electron. J. Comb. 16 (2009), N25, 1-7. [26] W. So, Integral circulant graphs, Discrete Math. 306 (2005), 153-158. d MFA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 301-304 The bipartite graphs of abelian dessins d'enfants Ruben A. Hidalgo * Departamento de Matemàtica, Universidad Tècnica Federico Santa Maria Valparaiso, Chile Received 25 April 2012, accepted 31 July 2012, published online 21 November 2012 Abstract Let S be a closed Riemann surface and let ß : S ^ C be a regular branched holomor-phic covering, with an abelian group as deck group, whose branch values are contained in the set {to, 0,1}. Three dessins d'enfants are provided by jß-1([0,1]), jß-1([1, to]) and jß-1([0, to]). In this paper we provide a description of the bipartite graphs associated to these dessins d'enfants using simple arguments. Keywords: Dessins d'enfants, Belyi curves, Algebraic curves, Riemann Surfaces. Math. Subj. Class.: 11G32, 14H37, 30F10 1 Introduction As a consequence of the Riemann-Roch theorem, there is a bijective correspondence between isomorphism classes of closed Riemann surfaces and isomorphism classes of complex algebraic curves. A closed Riemann surface S is called a Belyi curve if there is a non-constant meromorphic function ß : S ^ C whose branch values are contained in {to, 0,1}. The functionß is called a Belyi function for S and (S,ß) is called a Belyi pair. If the branch orders of ß at 0, 1 and to are p, q and r, respectively, then we say that the Belyi pair (S,ß) is of type (p, q, r). If (S,ß) is a Bely pair, then the pre-image D1 = ß-1([0,1]) (D2 = ß-1([1, to]) and D3 = jß-1([0, to]), respectively) defines a dessin d'enfant on S (see [7]), that is, a bipartite map on S (the pre-image of 0 are the white vertices and the pre-image of 1 are the black vertices of D1). Conversely, by the Uniformization Theorem, each dessin d'enfant D on a closed orientable surface induces a unique (up to isomorphisms) Riemann surface structure S on it and a Belyi mapß : S ^ C so that D andjß-1([0,1]) are equivalent bipartite maps on S .A famous result due to Belyi [1, 2] states that a closed Riemann surface S is a Belyi curve if and only if S can be defined by an algebraic curve over Q. This relationship was •Partially supported by Projects Fondecyt 1110001 and UTFSM 12.11.01. E-mail address: ruben.hidalgo@usm.cl (Rubén A. Hidalgo) observed by Grothendieck in his famous Esquisse d'un programme [3] to propose a study of the structure of the absolute Galois group Gal(Q/Q) by its action on the dessins d'enfants. As a consequence, a natural link between Galois theory, Belyi pairs and dessins d'enfants appears and, moreover, Galois invariants should be expressed in a purely combinatorial form. Unfortunately, the action of Gal(Q/Q) on dessins d'enfants is not well understood. A particular class of dessins d'enfants are those produced by Belyi pairs (S,ß) for which ß is a regular branched holomorphic cover; in which case we say that S is a quasiplatonic curve. Wolfart [8] noticed that quasiplatonic curves (and also the corresponding regular dessins d'enfants) are definable over their field of moduli. In the particular case when the deck group of ß is an abelian group, we say that S is an abelian quasiplatonic curve, that (S,ß) is an abelian Belyi pair (the corresponding dessins d'enfants are called abelian dessins d'enfants). In this case, if (p, q, r) is the type of (S,ß), then we say that signature (0; p, q, r) is an abelian triangular signature. In [4] it was noticed that every abelian Belyi pair (and the corresponding abelian dessins d'enfants) can be defined over Q, that is, they are fixed points for the action of Gal(Q/Q). In this paper we describe, using simple arguments, the underlying bipartite graphs of the abelian dessins d'enfants (see Theorem 1). Next, we proceed to describe a couple of classical examples. If n, m, d are positive integers, then we denote by the bipartite graph obtained from the complete bipartite graph Knm by replacing each edge by d edges. In this way, Kn,m = Kn,m. (1) If (S,ß) is an abelian Belyi pair of type (k, k, k), where k > 2 is an integer, and whose deck group of ß is Z2, then (S ,ß) is isomorphic to (Fk,ßk), where Fk is the classical Fermat curve {xk + yk + zk = 0} c P2, ßk([x : y : z]) = -(y/x)k and the deck group of ßk is the abelian group generated by a([x : y : z]) = [wkx : y : z] and b([x : y : z]) = [x : Mky : z], where Mk = e2m/k. The fixed points of a (respectively, b and ab) are given by the k points in(respectively,jß-1(0) andjß-1(1)). The abelian dessins d'enfants Di = ß-1([0,1]), D2 = ß-1([1, ro]) and D3 = ß-1([0, ro]) have as bipartite graph the complete bipartite graph Klk [5,6]^ (2) If (C,ß) is an abelian Belyi pair, with deck group H, then either H s Zn or H s Z2. The associated bipartite graphs of the corresponding abelian dessins d'enfants are in the first case equal to K n and Kn 1 and in the second case equal to k2 2. Theorem 1 below generalizes the above to the case of abelian dessins d'enfants of any type. Theorem 1. Let (S ,ß) be an abelian Belyi pair of type (p, q, r) and let d be the degree of ß. Then the bipartite graphs associated to the three abelian dessins d'enfants are given by r> = K'pq/d C = Kqr/d C = Kpr/d G1 = Kd/p,d/q, G2 = Kd/q,d/r, G3 = Kd/p,d/r ' Remark 2. Particular classes of abelian dessins d'enfants are those provided by the maximal ones with respect to its type (as the case provided by classical Fermat curves). An abelian dessin d'enfant associated to an abelian Belyi pair (S,ß), with abelian group H as deck group of ß, is called an homology dessin d'enfant (and (S,ß) is called an homology Belyi pair) if there is no an abelian Belyi pair (R, n), with abelian group G as deck group of n, so that S = R/L for some non-trivial subgroup L < G acting freely on R with H = G/L. If (S ,ß) is an homology Belyi pair, then equations over Q for S and ß were found in [4]. Clearly every abelian dessin d'enfant is covered by an homology dessin d'enfant of the same type. If the genus of S is at least two, then a homology Belyi pair (S ,ß) of type (p, q, r) can be uniformized as follows. Let r be a Fuchsian group of signature (0; p, q, r) and let F be its derivative subgroup. Then (S,ß) is equivalent to (H2/F,jßr), where ßr is the natural quotient map H2/F ^ H2/r. In this way, not only the bipartite graphs may be described, but also the corresponding dessins d'enfants. 2 Proof of Theorem 1 Let (S,ß) be an abelian Belyi pair of type (p, q, r) and let H be the abelian group being the deck group ofß (so d = |H|, the order of H). Let D1 be the dessin d'enfant whose edges are the pre-images underß of the arc [0,1], the black vertices are the pre-images of 0 and the white vertices are the pre-images of 1. The number of black vertices is equal to |H| /p, the number of white vertices is equal to |H|/q and the number of faces is |H|/r. The degree of a black vertex is p, the degree of a white vertex is q and the degree of a face is r. Let x1 € S (respectively, y1 € S ) be such thatß(x1) = 0 (respectively, ß(y1) = 1). Let Zp s (a) < H (respectively, Zq s (b) < H) be the H-stabilizer of x1 (respectively, the H-stabilizer of y1). As H acts transitively onjß-1(0) (respectively, ß-1(1)) and H is abelian, we may see that: 1. the H-stabilizer of every point in jß-1(0) (respectively, ß-1(1)) is (a) (respectively, (b)); 2. H = (a, b); 3. (b) (respectively, (a)) acts transitively onjß-1(0) (respectively, ß-1(1)). As there is a black vertex and a white vertex connected with an edge, condition (3) above ensures that every black vertex and every white vertex is connected by an edge. Again from (3), the (b)-stabilizer of x1 (respectively, the (a)-stabilizer of y1) is its cyclic subgroup of (b) (respectively, (a)) of order pq/|H|. It follows that every pair of black and white vertices are connected with pq/|H| edges. All the above information permits to obtain that the graph associated to D1 is the bipartite graph n = K'pq/!H\ G1 = K|H|/p, | H|/q. Similarly, let D2 (respectively, D3) be the dessin d'enfant obtained as the pre-image of the arc [1, (respectively, 0]) and the corresponding graph Q2 (respectively, Q3). Then, working in the same way as for D1 one obtains that = Kqr/ | H| , = K pr / |H| . G2 = K|H| /q, |H|/r, G3 = K |H| / p,|H|/r ' Remark 3. It is well know that a signature (0; p, q, r) is an abelian triangular signature if and only if lcm(p, q, r) = lcm(p, q) = lcm(p, r) = lcm(q, r), where lcm stands for the least common multiple. In that case we may write p = AA12A13, q = AA12A23, r = AA13A23, where gcd(A12, A13) = gcd(A12, A23) = gcd(A13, A23) = 1 and A = gcd(p, q, r), where gcd stands for the greatest common divisor. In [4] we proved that if (S,ß) is an homology Belyi pair, then H s ZA x Zu, where U = lcm(p, q, r) = AA12A13A23. In particular, the bipartite graphs in Theorem 1 are given by In the general situation, that is, for any abelian Bely pair of type (p, q, r) with ß of degree d, the bipartite graphs of the three dessins d'enfants are given by where l = A2A12A13A23/d. Acknowledgments The author would like to thank the referee for his/her careful reading and the suggested corrections to the first version. References [1] G. V. Belyi, Galois extensions of a maximal cyclotomic field, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 267-276. [2] G. V. Belyi, A new proof of the three-point theorem, Mat. Sb. 193 (2002), 21-24. [3] A. Grothendieck, Esquisse d'un programme, in: L. Schneps and P. Lochak (eds.), Geometric Galois actions 1. The Inverse Galois Problem, Moduli Spaces and Mapping Class Groups, London Math. Soc. Lecture Note Ser. 242, pages 5-48, with an English translation on pp. 243-283, Cambridge Univ. Press, Cambridge, 1997. [4] R.A. Hidalgo, Homology closed Riemann surfaces, Quarterly Journal of Math. (2011), doi: 10.1093/qmath/har026 [5] G. A. Jones and M. Streit, Galois groups, monodromy groups and cartographic groups, in: L. Schneps and P. Lochak (eds.), Geometric Galois Actions 2. The Inverse Galois Problem, Moduli Spaces and Mapping Class Groups. London Math. Soc. Lecture Note Ser. 243, pages 25-65. Cambridge Univ. Press, Cambridge, 1997. [6] G. Ringel and J. W. T. Youngs, Das Geschlecht des vollständigen dreifarbbaren Graphen, Comm. Math. Helv. 45 (1970), 152-158. [7] D. Singerman and J. Wolfart, Cayley Graphs, Cori Hypermaps, and Dessins d'Enfants, Ars Math. Contemp. 1 (2008), 144-153. [8] J. Wolfart, ABC for polynomials, dessins d'enfants and uniformization—a survey, in: Elementare und analytische Zahlentheorie, Schr. Wiss. Ges. Johann Wolfgang Goethe Univ. Frankfurt am Main, 20, Franz Steiner Verlag Stuttgart, Stuttgart, 2006, pp. 313-345. p = KA12 G1 = KAA23/l,AAa/l, p = KIA 13 G3 = KAA23/l,AA12/l, d MFA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 305-322 On stratifications for planar tensegrities with a small number of vertices Oleg Karpenkov TU Graz, Kopernikusgasse 24, A8010, Graz, Austria Jan Schepers Departement Wiskunde, Katholieke Universiteit Leuven Celestijnenlaan 200B, 3001, Leuven, Belgium Brigitte Servatius Mathematics Department, Worcester Polytechnic Institute 100 Institute Road, Worcester, MA 01609-2280, USA Received 16 January 2012, accepted 30 November 2012, published online 8 December 2012 Abstract In this paper we discuss several results about the structure of the configuration space of two-dimensional tensegrities with a small number of points. We briefly describe the technique of surgeries that is used to find geometric conditions for tensegrities. Further we introduce a new surgery for three-dimensional tensegrities. Within this paper we formulate additional open problems related to the stratified space of tensegrities. Keywords: Tensegrities, equilibrium, surgeries. Math. Subj. Class.: 52C30, 05C10 1 Introduction In this paper we study the stratified spaces of tensegrities with a small number of points. We work mostly with planar tensegrities. In the case of 4 and 5 point configurations we give an explicit description of all the strata and present a visualization of the entire stratified space. Further we give a geometric description of the strata for 6 and 7 points and use the technique of surgeries to find new geometric conditions adding to the list of already known ones. In particular, we introduce a new surgery for tensegrities in R3. E-mail addresses: karpenkov@tugraz.at (Oleg Karpenkov), janschepers1@gmail.com (Jan Schepers), bservat@math.wpi.edu (Brigitte Servatius) 1.1 Configuration space of tensegrities The first steps in the study of rigidity and flexibility of tensegrities were made by B. Roth and W. Whiteley in [9] and further developed by R. Connelly and W. Whiteley in [3], see also the survey about rigidity in [13]. N. L. White and W. Whiteley in [12] started the investigation of geometric conditions for a tensegrity with prescribed bars and cables. In the preprint [7] M. de Guzman describes several other examples of geometric conditions for tensegrities. Let us recall standard definitions of tensegrities (as in [2], [4], etc.). See also [10] for a collection of open problems and a good bibliography. Definition 1.1. Fix a positive integer d. Let G = (V, E) be an arbitrary graph without loops and multiple edges. Let it have n vertices vi,... ,vn. • A configuration is a finite collection P of n labeled points (pi,p2,... ,pn), where each point p i (also called a vertex) is in a fixed Euclidean space Rd. • The embedding of G with straight edges, induced by mapping v j to p j is called a tensegrity framework and it is denoted as G(P ). • We say that a load or force F acting on a framework G(P ) in Rd is an assignment of a vector fi in Rd to each vertex i of G. • We say that a stress w for a framework G(P) in Rd is an assignment of a real number wi,j = wj,i (we call it an edge-stress) to each edge pipj- of G. An edge-stress is regarded as a tension or a compression in the edge pipj. For simplicity reasons we put wi,j = 0 if there is no edge between the corresponding vertices. We say that w resolves a load F if the following vector equation holds for each vertex i of G: fi wi,j (pj - pi) = {j|j=i} By p j -pi we denote the vector from the point pi to the point p j. • A stress w is called a self-stress if, the following equilibrium condition is fulfilled at every vertex pi: E wi,j (pj - pi)=°. {j|j=i} • A couple (G(P), w) is called a tensegrity if w is a self-stress for the framework G(P). • If wi,j < 0 then we call the edge pipj- a cable, if wi,j > 0 we call it a strut. Let W (n) denote the linear space of dimension n2 of all edge-stresses wi,j. Consider a framework G(P) and denote by W(G, P) the subset of W(n) of all possible self-stresses for G(P). By definition the set W(G, P) is a linear subspace of W(n). Definition 1.2. The configuration space of tensegrities corresponding to the graph G is the set Qd(G) := {(G(P),w) | P G (Rd)n, w G W(G,P)}. The set {G(P ) | P g (Rd)n} is said to be the base of the configuration space, we denote it by Bd(G). 1.2 Stratification of the base of a configuration space of tensegrities Suppose we have some framework G(P) and we want to find a cable-strut construction on it. Then which edges can be replaced by cables, and which by struts? What is the geometric position of points for which given edges may be replaced by cables and the others by struts? These questions lead to the following definition. Definition 1.3. A set W(G, Pi) is said to be equivalent to a set W(G, P2) if there exists a homeomorphism £ between W(G, P1) and W(G, P2), such that for any self-stress w in W(G, P1) the self-stress £(w) satisfies sgn (£(w)) = sgn (w). Henceforth we call a set W (G, P) a linear fiber. The described equivalence relation on linear fibers gives us a stratification of the base Bd(G) = (Rd)n. A stratum is by definition a maximal connected set of points with equivalent linear fibers. In the paper [4] we prove that all strata are semialgebraic sets (which implies for instance that they are path connected). The idea of this paper is to make the first steps in the study of particular configuration spaces of tensegrities. We present the techniques to find geometric conditions and open problems for further study that already arise in very simple situations of 9 point configurations. Let us, first, make the following three general remarks. GR1. The majority of the strata of codimension k can be defined by algebraic equations and inequalities that define the strata of codimension 1. The exceptions here are mostly in high codimension (the simplest one is as follows: for two points connected by an edge there is no codimension 1 stratum, but there is one codimension 2 stratum corresponding to coinciding points; actually it is interesting to find the complete list of such exceptions). So the most important case to study is the codimension 1 case. GR2. A stratification of a subgraph is a substratification of the original graph (i.e., each stratum for a subgraph is the union of certain strata for the original graph), hence below we skip the description of B2(G) for graphs with 5 vertices other than K5. GR3. For any stratum there exists a certain subgraph that locally identifies the stratum (i.e., for any point x of the stratum there exists a neighborhood B(x) such that any configuration in B(x) has a nonzero self-stress for the subgraph if and only if this point is on the stratum). According to general remarks GR1 and GR2 the most interesting case is to study the strata of codimension 1 for the complete graph on n vertices (denoted further by Kn). It is possible to find some of the strata of Kn directly. For the other strata one, first, should find an appropriate subgraph that locally identifies the stratum, and then find appropriate surgeries (explained in Section 3) to reduce the complexity of the subgraph to find geometric conditions. This paper is organized as follows. In Section 2 we study the stratification of configuration spaces of tensegrities in the plane with a small number of vertices. In Subsections 2.1 and 2.2 we briefly describe the trivial cases of two and three point configurations. Further in Subsections 2.3 and 2.4 we study the four and the five point cases. In each of the cases we describe the geometry and the number of strata. In addition we introduce the adjacency diagram of full dimension and codimension 1 strata. In Subsections 2.5 and 2.6 we describe geometric conditions for the codimension 1 strata of 6, 7, and 8 point tensegrities. In Section 3 we present the technique of surgeries to find geometric descriptions for the strata. In Subsection 3.1 we describe surgeries that do not change graphs, and in Subsection 3.2 we show a couple of surgeries in the two-dimensional case. We introduce a new three-dimensional surgery in Subsection 3.3. In conclusion, we formulate several open questions in Subsection 3.4. 2 Stratification of the space B2 (Kn) for small n In this section we study the geometry of tensegrity stratifications for graphs with a small number of vertices. The cases of n = 2, 3,4,5 are studied in full detail. Starting from n = 6 there are some gaps in the understanding of tensegrities. Still for n = 6,7,8 the complete description of the geometric conditions for the strata is known, we briefly describe several results on them here (see [4] for more information). 2.1 Case of two points Consider, first, the case of two points (n = 2). There are only two graphs on two points: a complete one K2 and a graph without edges (denote it by G0,2). All the fibers of the base B2(G0j2) = R4 are of dimension 0, and, therefore, they are equivalent. Hence the stratification is trivial. The complete graph K2 here has only one edge. If two points of the graph do not coincide then the stress at this edge should be zero. When two points coincide then the stress at the edge can be arbitrary, and we have a one-dimensional set of solutions (i.e., a fiber). So the base B2(K2) = R4 has a codimension 2 stratum (a 2-dimensional plane). The complement to this stratum is a stratum of codimension 0. 2.2 Three point configurations There are four different types of graphs here: let 3 be the graph with i edges for i = 0,1, 2, 3. In cases G0 3 and G13 the base stratifications are the following direct products: B2(Go,3) = B2(Go,2) X R2 and B2(Gi,3) = B2^) x R2. So B2(G0 3) is trivial and B2(G13) has a 4-dimensional subspace and its complement as strata. The base B2(G23) contains five strata. One of them corresponds to the configuration where three points coincide: the fiber here is 2-dimensional, this stratum is isometric to R2. There are three strata where one of the edges of the graph vanishes: they are isometric to R4 \ R2. Finally, the complement to the union of these strata is the only stratum of maximal dimension. There are no nonzero tensegrities for a configuration in this stratum. For the complete graph on three vertices we have, for the first time, codimension 1 strata. There are three codimension 1 strata, all of them correspond to the following configuration: three points are in one line. Different strata correspond to having a different point between the two others. Let us briefly describe one of such strata. Let Pi = (xi; yi) be the points of the graph (i = 1, 2, 3). Then the condition that the three points are in a line is defined by a quadratic equation: (x2 - xi)(y3 - yi) - (x3 - xi)(y2 - yi) = 0 This quadric divides the space into two connected components: corresponding to positively and negatively oriented triangles. To sum up we present for B2 (K3) the following table. Dimension of a stratum 0 1 2 3 4 5 6 Number of such strata 0 0 1 0 3 3 2 2.3 Stratification of B2 ( K4 ) In this subsection we restrict ourselves to the complete graph K4 (for its subgraphs we apply the reasoning of GR2 above). A plane configuration of four points in general position admits a unique tensegrity (up to a multiplicative constant), which is called an atom. In [8] it was proved that any self-stress for Kn is a sum of self-stressed atoms in Kn (i.e., a sum of certain K4 c Kn with scalars). For K4 there are exactly 14 strata of general position. The strata of codimension 1 correspond to three of four points of the graph lying in a line. Actually in this case there is no jump of dimension of the fiber: there is also a unique (up to scalar) solution corresponding to the three points in a line. But the stresses on the edges from the fourth point are all zero, and hence a fiber of this stratum is not equivalent to general fibers. The number of such strata is 24. In codimension 2 we have two different types of strata corresponding to • four points in a line: the dimension of a fiber is 2 (twelve strata); • two points coincide: the dimension of a fiber is 1 (twelve strata). In codimension 3 there is one type of strata with configurations of four points in a line, two of which coincide. Six of them with the double point in the middle and twelve of them with the double point not in the middle. In codimension 4, there are two types of strata: • three points coincide (4 strata); • two pairs of points coincide (3 strata). And, finally, there is a codimension 6 stratum when all four points coincide. We remark that for none of the strata the fiber is 3-dimensional. The cardinalities of strata are shown in the following table. Dimension of a stratum 0 1 2 3 4 5 6 7 8 Number of strata 0 0 1 0 7 18 24 24 14 2.3.1 The space of formal configurations Let us draw schematically the adjacency of the strata of maximal dimension via strata of codimension 1. The dimension of the stratified space is 8, let us reduce it to two via factoring by proper affine transformations. We will use the following simple proposition. Proposition 2.1. Invertible affine transformations of the plane do not change the equivalence class of a fiber W (G, P ). In other words if P is a configuration and T an invertible affine transformation of the plane then W (G, P ) ~ W (G, T (P )). □ So instead of studying the stratification itself we restrict to the set of formal configurations with respect to proper affine transformations of the plane. Definition 2.2. We say that a four point configuration v1,v2,v3,v4 is formal in one of the following cases: i) nondegenerate case: a configuration Px,y,+ with vertices vi = (0,0), v2 = (1,0), v3 = (x,y), v4 = (x,y+1) for arbitrary (x,y). ii) nondegenerate case: a configuration Px,y,- with v1 = (0,0), v2 = (1,0), v3 = (x,y), v4 = (x, y—1) for arbitrary (x,y). iii) degenerate case: a configuration Pa,+ with v1 = (0,0), v2 = (1,0), v3 = (0,1), v4 = (A, 1) for an arbitrary A. iv) degenerate case: a configuration Pa,- with v1 = (0,0), v2 = (1,0), v3 = (0, —1), v4 = (A, — 1) for an arbitrary A. v) closure: we add two formal configurations with vertices v1 = (0,0), v2 = (1,0), v3 = (1,0), v4 = (1, ±to). We denote the set of all formal configurations by A4. In some sense the space A4 is the space of all codimension 0 and codimension 1 configurations factored by the group of proper affine transformations. Proposition 2.3. For any codimension 0 and codimension 1 configuration there exists a unique formal configuration to which the first configuration can be affinely deformed. □ The space A4 is endowed with a natural topology of a quotient space. Proposition 2.4. There is a natural topology of a sphere S2 for the set A4. Proof. Let us introduce a topology of the unit sphere S2 for A4. Consider the configurations of case i) on the plane z = 1: we identify the point Px,y,+ with the point (x, y, 1). Consider the projection of this plane to the upper unit hemisphere S2 from the origin. So we have a one to one correspondence between the configurations of case i) and the upper hemisphere. Similarly we take the plane z = —1 for the case ii) identifying the point (—x, —y, —1) with the configuration Px,y- and projecting it to the lower hemisphere. For the equator of the unit sphere we use all the other cases as asymptotic directions. First, we associate the configuration Pa,+ with the point (cos(n — arccotanA), sin(n — arccotanA), 0). Let us explain the topology at one of such points of the equator. Suppose we start with Px,y,+ . The transformation sending the first three points to (0,0), (1,0), and (0,1) is linear with matrix ( 1 —x/y A V 0 1/y ) • Then the image of the fourth point of Px,y,+ is (— x/y, 1+1/y). While x tends to infinity and x/y tends to A the last point tends to (—A, 1), and hence the configuration Px,y,+ tends to P-a,+, as in the above formula. Secondly, we associate PAj- with the point (cos(- arccotan A), sin(- arccotan A), 0) in a similar way. Finally, we glue P+TO and P-TO to the points (1,0,0) and (-1,0,0) respectively. □ So, the codimension 0 and 1 stratification of B2(K4) can be derived from the stratification of the sphere. We show the stereographic projection of A4 from the point (0,0, -1) to the plane z =1 on Figure 1. There are four types of strata of codimension 1, they correspond to the fact that certain three points are in a line. They separate the plane into 14 connected components. In each of the connected components we draw a typical type of configuration: (v1, v2, v3, v4). Here v1 is blue, v2 is purple, v3 is red and v4 is green. Remark 2.5. Different geometric conditions are represented by different colors in the picture, the correspondence is as follows. • Light blue strata (6 strata forming a circle) correspond to configurations with vi, v2, and v3 in a line. • Dark blue strata (6 strata) contain configurations with v1, v2, and v4 in a line. • Light green strata (6 strata) contain configurations with v1, v3, and v4 in a line. • Dark green strata (6 strata) correspond to configurations with v2, v3, and v4 in a line. We have 24 strata of codimension 1 in total. • The dashed black line is the projection of the equator. It corresponds to the degenerate case of parallel segments. The dashed line is not a stratum, it has the same fiber as all the points in its neighborhood. While one passes the dashed line the red-green segment "rotates" around the blue-purple segment. Remark 2.6. The 14 connected components of the plane are in one-to-one correspondence with the 14 faces of a cuboctahedron (accordingly, the 12 points on these circles correspond to its vertices). Thus, the four circles are those circumscribed around the equatorial regular hexagons of the cuboctahedron. The vertices of this polytope lie on a sphere, hence, through stereographic projection the four circumcircles in question project in fact to circles in the image plane. 2.4 Stratification of B2(K5) 2.4.1 General description of the strata We have 264 strata of general position. As in the two previous cases the strata of codimension 1 correspond to three points of the graph lying in a line. The number of such strata is 600. The following strata are of codimension 2: • twice three points in a line: 270 strata; • four points in a line: 120 strata; • two points coincide: 420 strata. In codimension 3 we have the following cases: • three points in a line and one double point: 60 strata; • four points in a line two of which coincide: 180 strata; • five points in a line: 60 strata. For codimension 4 we have the following list: • one triple point: 20 strata; • five points in a line two of which coincide: 120 strata; • two double points: 30 strata. In codimension 5 we get: • five points in a line three of which coincide: 30 strata; • five points in a line with two pairs of points coinciding: 45 strata. In codimension 6 there are the following strata: • a triple point and a double point: 10 strata; • one point and one point of multiplicity four: 5 strata. And, finally, there is a codimension 8 stratum when all five points coincide. The cardinalities of the strata are shown in the following table. Dimension of a stratum 0 1 2 3 4 5 6 7 8 9 10 Number of strata 0 0 1 0 15 75 170 300 810 600 264 2.4.2 Visualization of #2(^5) Let us now describe the structure of the stratification B2(K5). Like in case of B2(K4) we introduce a set A5 which represents the adjacency of strata of full dimension and of codimension 1. By definition we put A5 = A4 X R2, i.e., we consider all the four point configurations of A4, and to each configuration we add the fifth point. We take the product topology for A5. So at each point of A4 we attach an R2-fiber. It will soon become clear that for any full dimension stratum of A4 the corresponding fibration is trivial, but the adjacency is not. On Figures 2 and 3 we show A5 in the following way. We draw the stratification of A4 and inside each connected component we show the typical fiber of the component. The first four points are represented by purple, blue, green, and red points. The lines passing through any pair of them divide the fiber into 18 connected components, that correspond to strata of full dimension. At each such component we write a letter of the Latin alphabet (we consider capital and small letters as distinct). • Two regions denoted by the same letter and lying in neighboring connected components of A4 separated by light red, dark red, and black strata are in the same stratum. • Two regions denoted by the same letter and lying in neighboring connected components of A4 separated by light blue, dark blue, light green, and dark green strata are in distinct strata which are adjacent to the same codimension 1 stratum. • Two regions denoted by a distinct letter and lying in neighboring connected components of A4 are not in one stratum and are not adjacent to the same codimension 1 stratum. The light blue, dark blue, light green, and dark green strata represent the same geometric conditions as in Remark 2.5 above. For the remaining strata we have: • The dark red stratum symbolizes that the line through the red and blue points is parallel to the line through the green and purple points. • The light red stratum symbolizes that the line through the red and purple points is parallel to the line through the green and blue points. • The black stratum symbolizes that the line through the red and green points is parallel to the line through the purple and blue points. Remark2.7. The configuration space B2(K5) has several obvious symmetries. First, there is the group of permutations S5 that acts on the points of B2(K5); these symmetries are hardly seen from Figures 2 and 3 since the representation is not S5-symmetric. Secondly, there is a symmetry about the origin that sends configurations from B2(K5) to themselves, on Figures 2 and 3 we used capital and small letters to indicate this symmetry (for instance, the strata of "a" contain centrally symmetric configurations to the configurations of the strata "A"). As in the case of 4 point configurations we skip the subgraphs of K5, see the second general remark above (GR2). Figure 2: Stratification of B2(K5) (Left part). Figure 3: Stratification of B2(K5) (Right part). 2.5 Essentially new strata in B2 ( K6 ) The stratification of B2(K6) is much more complicated, at this moment we do not even know how many strata of distinct dimension are present in the stratification. According to GR1 the first step in studying the stratification of B2(K6) is to study all possible distinct types of strata of codimension 1. In the examples of Kn for n < 6 we only have strata corresponding to the following geometric condition: three points are in a line. For the case of 6 points we get two additional types of strata: six points on a conic, and three lines passing through three pairs of points have a unique point of intersection. So the following are three codimension 1 strata (appeared in [12] by N. L. White and W. Whiteley): • three points in a line; • the lines v1v2, v3v4, and v5v6 meet in one point (or all parallel); • all the six points are on a conic. We conclude this subsection with the following problems. Problem 2.8. Find a description of B2(K6), B3 (K4 ) and B3(K5) similar to the ones for B2(K4) and B2(K5) shown in the previous subsections. 2.6 A few words about the case n > 6 In [4] we have studied strata of the 7 and 8 point configurations. There are 4 distinct types of codimension 1 strata for 7 points and 17 types for 8 points. The 4 types of codimension 1 strata for 7 points are defined by the following geometric conditions: • three points in a line; • the lines v1v2, v3v4, and v5v6 meet in one point (or all parallel); • the lines v1v2, v3v4, and v5p (where p is the intersection of the lines v2v6 and v3v7) have a common nonempty intersection; • the six points v1, v2, v3, v4, v5, and p (where p is the intersection of the lines v1v6 and v3v7) are on a conic. For the list of strata of 8 point configurations we refer to [4]. It turns out that the geometric conditions of any codimension 1 stratum can be obtained by the following procedure. Consider the points of configuration P ; for each two pairs of points (vi;vj) and (vk,v;) of this configuration consider the point of intersection of the lines v^ and vfcv;. This leads to a bigger configuration of points including P and the above intersections, we denote it by U (P). This operation can be iteratively applied infinitely many times, which results in a universal set Uw(P) = U Um(P). m=0 Any condition for a codimension 1 stratum is always as follows: three certain points of Uw(P) are in a line (for the details, see for instance [9] and [4]). Example 2.9. The condition the lines v1v2, v3v4, and v5v6 meet in one point in terms of points of U 1(P ) = U (P ) is as follows. The points v1, v2, and p = v3v4 n v5v6 are in a line. Remark 2.10. For simplicity reasons we omit discussions of cases where certain lines vivj and vk v; are parallel, due to the fact that this situation is never generic for codimension 1 strata. In general one may think that if the lines vivj and vkv; are parallel, then their intersection point is in the line at infinity in the projectivization of R2. Remark 2.11. At first glance, the condition six points are on a conic is of different nature. Nevertheless, it is a relation on the points of the configuration in UX(P) described by Pascal's theorem: The intersections of the extended opposite sides of a hexagon inscribed in a conic lie on the Pascal line. See also Example 2.15 below. Problem 2.12. Describe all the possible different types of strata for 9 points. Problem 2.13. How to calculate the number of different types of strata for n points with arbitrary n? It is also interesting to have an answer for the following question: how many iterations does one need to perform (i.e., find the minimal m for Um(P)) to describe all conditions for the codimension 1 strata of n-point configurations P ? Problem 2.14. Which configurations of Um(P) define the same geometric condition? This problem is a kind of question of finding generators and relations for the set of all conditions. Let us show one type of such "relations" in the following example. Example 2.15. Consider the condition: six points vi, v2,..., v6 are on a conic. This condition is described by configurations contained in U 1(P) via Pascal's theorem: {p = vCT(i)vCT(2) H vCT(4)vCT(5) q = vCT(2)vCT(3) n vCT(5)vCT(6) , r = vCT(3)vCT(4) n vCT(6)vCT(1) where a is an arbitrary permutation of the set of six elements. So, there are 60 different configurations of U 1(P) defining the same geometric condition. 3 Further study of strata: surgeries We now look into subgraphs contained in a particular stratum and ask the basic question on the dimension of the fiber. Even graphs of very low connectivity admit non-zero tensegrities, for disconnected or one-connected graphs we may simply examine the connected or 2-connected components. Also 2-connected graphs may be decomposed via the 2-sum, see [11]: Consider graphs G1 and G2, their configurations P1 and P2 admitting tensegrities with p1q1 a cable in G1(P1) and p2q2 a strut in G2 (P2 ). We form the 2-sum G1 0 G2 by identifying p1 with p2 and q1 with q2 and removing the identified edge. We can inherit a configuration P from P1 and P2 by fixing P1 and properly dilating, rotating and translating P2. It is clear that dimW(G1 0G2,P) = dimW(G1,P1) +dimW(G2,P2) - 1. Since 2-sum decomposition is canonical, we can describe geometric conditions for 2-connected graphs by geometric conditions on their 3-blocks. For example the geometric condition for G in Figure 4 is that the lines v1v2, v3v4, and v5v6 meet in one point. 2 u7 V8 Vi V2 Figure 4: The 2-sum of a triangular prism with K4 V4 Vi V3 V4 V3 V4 V5 V6 V5 V6 V2 Vi V2 Vi V3 V2 Figure 5: Examples of subgraphs of K6 admitting tensegrities at codimension 1 strata of B2(Ke). 3.1 Subgraphs related to codimension 1 strata As we have already mentioned in GR3, for any codimension 1 stratum there exists at least one subgraph of Kn that generically does not admit tensegrities but at this stratum admits a one-dimensional family of tensegrities. Let us show such subgraphs for the codimension one strata of B2(K6) and B2 (K7). Example 3.1. In the case of K6 we have three strata of different geometrical nature. The first triangular subgraph (Figure 5, left) is related to the strata with three points in a line. The second (Figure 5, middle) corresponds to the strata whose three pairs of points generate lines passing through one point. The last one (Figure 5, right) corresponds to the configurations of six points on a conic. Example 3.2. In the case of K7 there are the following new examples of subgraphs, corresponding to the main 4 different types of strata. From the left to the right we have the following geometric conditions • v1, v2, and v3 are in a line; • the lines v1v2, v3v4, and v5v6 meet in one point; • the lines v1v2, v3v4, and v5p (where p = v2v6 n v3v7) have a common point; • the six points v1, v2, v3, v4, v5, and p (where p = v1v6 n v3v7) are on a conic. Note that the example for three points in a line is actually the 2-sum of a triangle with two atoms, so the only way for a non-zero self-stress on the edges is to have v1, v2, and v3, V5 V3 V6 V7 V1 v4 v4 V7I 'v2 V1 V3 V4 V5 V6 V2 V1 V3 V4 V6 V3 V7 V2 V1 V2 v V V V 4 3 4 3 V i Figure 6: Examples of subgraphs of K7 admitting tensegrities at codimension 1 strata of B2(Kr). the vertices of the triangle, in a line. Remark 3.3. Geometric conditions for the graphs with 8 and fewer vertices are given in [4]. Several of those geometric conditions were described before in terms of bracket polynomials in [12] by N. L. White and W. Whiteley. We also refer to the paper [1] by E. D. Bolker and H. Crapo for the relation of bipartite graphs with rectangular bar constructions. 3.2 Surgeries on subgraphs that change geometric conditions in a predictable way In this subsection we present several surgeries that allow to guess the geometric conditions for new strata (characterized by certain subgraphs) via other strata (characterized by these graphs modified in a certain way). We call such modifications of graphs surgeries. 3.2.1 Surgeries that do not change geometric conditions Let G be a graph, denote by Ge the graph with an edge e removed. Proposition 3.4. (Edge exchange) Consider a graph G and a subgraph H, and let ei and e2 be two edges of H. Let P be a configuration for which dim W (H, P) = 1. Suppose also that the self-stresses of H do not vanish at the edges e1 and e2. Then we have dim W (Gei ,P ) = dim W (Ge2 ,P ). □ In the situation of Proposition 3.4 the strata of Gei (P) and Ge2 (P) are defined by the same geometrical conditions. 3.2.2 Two two-dimensional surgeries that change geometric conditions The first surgery is described in the following proposition. Proposition 3.5. Consider the frameworks G(P ), G[(P[ ), and G2(P/ ) as on the figure: If none of the triples of points (p, v2, v3), (q, v2,v3), (p, v2, v4), (q, v3, v4) and (v2, v3, v4) are on a line then we have dim W (G{,P( ) = dim W (G2 ,P2j ). Example 3.6. Let us consider a simple example of how to get a geometric condition for the graph to admit a tensegrity knowing all geometric conditions for 6-point graphs. Let us apply Surgery I to the points v5, v6, v7. We have: V4 vi v 4. Vi V5 P V3 V2 . The geometric condition to admit a tensegrity for the graph on the right is: the lines v1v2, v3v4 and v5p intersect in a point. Hence the geometric condition for the original graph is: the lines v1v2, v3v4 and v5p intersect in a point, where p = v2v6 n v3v7. Now let us show the second surgery. Proposition 3.7. Consider the frameworks G(P), Gl1 (P11 ), and G21 (Pi,1 ) as on the following figure: If none of the triples of points (p,q,v i), (p, v1;v4), (r, v1;v4), (q, v i,v4), (s,vl7v4), or (r, s,v4) lie on a line then we have dim W(Gl1 ,P(I) = dim W(G!,1, P,). □ Remark 3.8. Both surgeries were shown in [4]. There is a certain analogy of the first surgery to AY exchange in matroid theory (see for instance [13] and [5] for the connections between matroids and rigidity theory), but it is not exactly the same. Remark 3.9. Actually these surgeries are valid in the multidimensional case as well under the condition that certain points are in one plane. 3.3 A new tensegrity surgery in R3 We conclude this paper with a single surgery for tensegrities in R3. Proposition 3.10. Consider a graph G and frameworks G(P ), G1(P1), and G2(P2) as follows: 61 / \ 63 62 64 Gl (Pi ) 61 / \ 63 62 64 G2 (P2 ) Denote the plane v2v3v4 by n1. Suppose that the couples of edges e1 and e2, e3 and e4, e5 and e6 define planes n2, n3, and n4, different from n1. Assume that n2 n n3 n n4 is a one point intersection. If G1(P1) and G2(P2) have nonzero stress on the edges connecting v1, v2, v3, and v4 then n1 n n n n3 n n4 = v1. In this case we additionally have dim W (G1 ,P1) = dim W (G2,P2). Proof. The first statement follows since v1 only has valency 3 in G2(P2), so v1, v2, v3, and v4 need to be coplanar to have a nonzero edge-stress. Now we explain how to map W(G1,P1) to W(G2, P2). The inverse map is simply given by the reverse construction. By the conditions v1 is the intersection point of the planes n1, n2, and n3. We add the uniquely defined plane atom on v1;v2,v3, v4 to G1(P1) that cancels the edge-stress on v2v3. Since the plane n1 does not coincide with the plane n2 spanned by the forces on e1 and e2, the edge-stress on v2v4 is also canceled. By the same reasons the edge-stress on v3v4 is canceled as well. This uniquely defines a self-stress on G2(P2). □ 3.4 Some related open problems The next goal in this approach is to continue to study the geometry of the strata. Ideally one would like to find techniques that will give geometric conditions for a graph via its combinatorics. This question seems to be a very hard open problem. The study of surgeries is the first step to solve it at least in codimension 1. For a start we propose the following open question. Problem 3.11. Find all geometric conditions for the strata of 9 point tensegrities. The surgeries introduced in this section were extremely useful for the study of 8 point configurations (see in [4]). We think that it is not enough to know only these surgeries to find all the geometric conditions. This gives rise to another question. Problem 3.12. Find other surgeries on graphs that predictably change the geometric conditions. As far as we know there is no systematic study of strata for tensegrities in R3 or higher dimensions: these cases are much more complicated than the planar case. At least the stratification of B3 (K5 ) should have a description similar to that of B2 (K4), since 5 points in general position in R3 admit a unique non-zero self-stress. Additionally one should examine the rigidity properties of subgraphs in a stratum. For K4 we have 14 strata of full dimension. For 8 of them the convex hull is a triangle, in 5 of the strata the points are in convex position. A tensegrity for the convex position has 4 struts (cables) and two cables (struts), while in the non-convex case there are three cables and three struts. All of these tensegrities are (infinitesimally) rigid and struts and cables may be exchanged without destroying rigidity. However, when viewed as graphs embedded in R3 only half of them are rigid. For the convex case, there must be cables on the convex hull and two struts. In the non-convex case there must be a triangle of struts on the convex hull and three cables in the interior, termed a spider web by R. Connelly. None of these are proper forms in the sense of B. Grunbaum. They are minimally rigid, but in the convex case they have members intersecting in a vertex other than a vertex of the graph, in the non-convex case there is a vertex without a strut. B. Grunbaum in his lectures on lost mathematics [6] asks about the number of proper forms given n struts. On 3 struts, there is only one tensegrity which is minimally rigid with edges only intersecting at vertices and such that every vertex is endpoint of at least one strut. For 4 struts there are at least 20 forms, but it is not known how many there are. The number of forms on n struts is bounded by the number of strata on B3(Kn). For the hierarchies of the various kinds of rigidity see [3]. Acknowledgments. We are grateful to the unknown reviewer for Remark 2.6 and other useful suggestions. Oleg Karpenkov is supported by the Austrian Science Fund (FWF), grant M 1273-N18. Jan Schepers is a Postdoctoral Fellow of the Research Foundation -Flanders (FWO). References [1] E. D. Bolker and H. Crapo, Bracing rectangular frameworks I, SIAM J. Appl. Math. 36 (1979), 473-490. [2] R. Connelly, Rigidity, in: P. M. Gruber and J. M. Wills (eds.), Chapter 1.7 of Handbook of convex geometry, vol. a, North-Holland Publishing Co., Amsterdam, 1993, pp. 223-271. [3] R. Connelly and W. Whiteley, Second-order rigidity and prestress stability for tensegrity frameworks, SIAM J. Discrete Math. 9 (1996), 453-491. [4] F. Doray, O. Karpenkov and J. Schepers, Geometry of configuration spaces of tensegrities, Discrete Comput. Geom. 43 (2010), 436-466. [5] B. Jackson and T. Jordan, Connected rigidity matroids and unique realizations of graphs, J. Combin. Theory Ser. B 94 (2005), 1-29. [6] B. Grunbaum, Lectures on Lost Mathematics, lectures were given in 1975; the notes were digitized and reissued at the Structural Topology Revisited conference in 2006, http://hdl. handle.net/177 3/157 00,2010. [7] M. de Guzman, Finding Tensegrity Forms, preprint, 2004. [8] M. de Guzman and D. Orden, From graphs to tensegrity structures: Geometric and symbolic approaches, Publ. Mat., Barc. 50 (2006), 279-299. [9] B. Roth and W. Whiteley, Tensegrity frameworks, Trans. Amer. Math. Soc. 265 (1981), 419446. [10] B. Servatius, Tensegrities, PAMM 7 (2007), 1070101-1070102. [11] B. Servatius and H. Servatius, On the 2-sum in rigidity matroids, European J. Combin 32 (2011), 931-936. [12] N. L. White and W. Whiteley, The algebraic geometry of stresses in frameworks, SIAM J. Alg. Disc. Math. 4 (1983), 481-511. [13] W. Whiteley, Rigidity and scene analysis, in: J. E. Goodman and J. O'Rourke (eds.), Handbook of Discrete and Computational Geometry, chapter. 49, CRC Press, New York, 1997, 893-916. d MFA Also available at http://amc.imfm.si ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 323-350 Relations between graphs Jan HubiCka Computer Science Institute of Charles University, Univerzita Karlova v Praze Malostranské nam. 25, 118 00 Praha 1, Czech Republic Jürgen Jost Max Planck Institute for Mathematics in the Sciences Inselstrasse 22, D-04103 Leipzig, Germany Department of Mathematics, University of Leipzig, D-04081 Leipzig, Germany Santa Fe Institute, 1399 Hyde ParkRd., Santa Fe, NM 87501, USA Yangjing Long Max Planck Institute for Mathematics in the Sciences Inselstrasse 22, D-04103 Leipzig, Germany Peter F. Stadler Bioinformatics Group, Department of Computer Science, and Interdisciplinary Center for Bioinformatics, University of Leipzig, Härtelstraße 16-18, D-04107 Leipzig, Germany Max Planck Institute for Mathematics in the Sciences Inselstrasse 22, D-04103 Leipzig, Germany Santa Fe Institute, 1399 Hyde ParkRd., Santa Fe, NM 87501, USA Fraunhofer Institut für Zelltherapie und Immunologie - IZI Perlickstraße 1, D-04103 Leipzig, Germany Department of Theoretical Chemistry, University of Vienna Wahringerstraße 17, A-1090 Wien, Austria Center for non-coding RNA in Technology and Health University of Copenhagen, Gr0nnegardsvej 3, DK-1870 Frederiksberg C, Denmark Ling Yang School of Mathematical Sciences, Fudan University No. 220 Handan Rd., 200433, Shanghai, China Received 18 May 2012, accepted 24 October 2012, published online 14 January 2013 E-mail addresses: Jan.Hubicka@mff.cuni.cz (Jan Hubicka), jost@mis.mpg.de (Jürgen Jost), ylong@mis.mpg.de (Yangjing Long), studla@bioinf.uni-leipzig.de (Peter F. Stadler), yanglingfd@fudan.edu.cn (Ling Yang) Abstract Given two graphs G = (VG, EG) and H = (VH, EH), we ask under which conditions there is a relation R C VG x VH that generates the edges of H given the structure of the graph G. This construction can be seen as a form of multihomomorphism. It generalizes surjective homomorphisms of graphs and naturally leads to notions of R-retractions, R-cores, and R-cocores of graphs. Both R-cores and R-cocores of graphs are unique up to isomorphism and can be computed in polynomial time. Keywords: Generalized surjective graph homomorphism, R-reduced graph, R-retraction, binary relation, multihomomorphism, R-core, cocore. Math. Subj. Class.: 05C60, 05C76 1 Introduction 1.1 Motivation Graphs are frequently employed to model natural or artificial systems [3, 11]. In many applications separate graph models have been constructed for distinct, but at least conceptually related systems. One might think, e.g., of traffic networks for different means of transportation (air, ship, road, railroad, bus). In the life sciences, elaborate network models are considered for gene expression and the metabolic pathways regulated by these genes, or for the co-occurrence of protein domains within proteins and the physical interactions of proteins among each other. Let us consider an example. Most proteins contain several functional domains, that is, parts with well-characterized sequence and structure features that can be understood as functional units. Protein domains for instance mediate the catalytic activity of an enzyme and they are responsible for specific binding to small molecules, nucleic acids, or other proteins. Databases such as SuperFamily compile the domain composition of a large number of proteins. We can think of these data as a relation R c D x P between the set D of domains and the set of P proteins which contain them. Protein-protein interaction networks (PPIs) have been empirically determined for several species and are among the best-studied biological networks [16]. From this graph, which has P as its vertex set, and the relation R we can obtain a new graph whose vertex set are the protein domains D, with edges connecting domains that are found in physically interacting proteins. This "domain interaction graph" conveys information e.g. on the functional versatility of protein complexes. On the other hand, we can use R to construct the domain-cooccurrence networks (DCNs) [14] as simple relational composition R o R+. In examples like these, the detailed connections between the various graphs have remained unexplored. In fact, there may not be a meaningful connection between some of them, e.g. between PPIs and DCNs, while in other cases there is a close connection: the domain interaction graph, for example, is determined by the PPI and R. A second setting in which graph structures are clearly related to each other is coarse-graining. Here, sets of vertices are connected to a single coarse-grained vertex, with coarsegrained edges inherited from the original graph. In the simplest case, we deal with quotient graphs [15], although other, less stringent constructions are conceivable. Similarly, we would expect that networks that are related by some evolutionary process retain some sort di • • Pi di d2 d3 d4 d2 •■ d3 d4 • P4 Graph G Relation R Graph G * R Figure 1: The graph G * R is determined by the graph G and the relation R. of structural relationship. 1.2 Main definitions A well-defined mathematical problem is hidden in this setting: Given two networks, can we identify whether they are related in meaningful ways? The usual mathematical approach to this question, namely to ask for the existence of structure-preserving maps, appears to be much too restrictive. Instead, we set out here to ask if there is a relation between the two networks that preserves structures in a less restrained sense. The idea is to transfer edges from a graph G to a graph H with the help of a relation R between the vertex set V of G and the vertex set B of H. In this context, R is simply a set of pairs (v, b), with v G V, b G B. Since graphs can be regarded as representations of binary relations, we can also consider G as a relation on its vertex set, with (x, y) G G if and only if x and y are connected by an edge of G. We then have the composition G o R given by all pairs (x, b) for which there exists a vertex y G V connected by an edge of G to x and (y, b) G R. This, however, like R is a relation between elements of different sets. In order to equip the target set B with a graph structure, we simply connect elements u and v in B if they stand in relation to connected elements of G. In the following, we give a formal definition, and we shall then relate it to the composition of relations just described. A directed graph G is a pair G = (VG, EG) such that EG is a subset of VG x VG. We denote by VG the set of vertices of G and by EG the set of edges of G. We consider only finite graphs and allow loops on vertices. An undirected graph (or simply a graph) G is any directed graph such that (u, v) G EG if and only if (v,u) G EG. We thus consider undirected graphs to be special case of directed graphs and we still allow loops on vertices. A simple graph is an undirected graph without loops. Definition 1.1. Let G = (VG, EG) be a graph, B a finite set, and R C V x B a binary relation, where for every element b G B, we can find an element v G VG such that (v, b) G R. Then the graph G * R has vertex set B and edge set Eg-,.r = {(u, v) G B x B| there is (x, y) G EG and (x, u), (y, v) G R} . (1.1) An example of the * operation is depicted in Fig. 1. Graphs with loops are not always a natural model, however, so that it may appear more appealing to consider the slightly modified definition. Definition 1.2. Let G = (Va, Ea) be a simple graph, B a finite set, R a binary relation, where for every element b e B, we can find an element v e Va such that (v, b) e R. Then the (simple) graph G * R has vertex set B and edge set Ea*R = {(u, v) e B x B|u = v and there is (x, y) e Ea and (x, u), (y, v) e R} . (1.2) We shall remark that these definitions remain meaningful for directed graphs, weighted graphs (where the weight of edge is a sum of weights of its pre-images) as well as relational structures. For simplicity, we restrict ourselves to undirected graphs (with loops). Most of the results can be directly generalized. Graphs can be regarded as representations of symmetric binary relations. Using the same symbol for the graph and the relation it represents, we may re-interpret definition 1.1 as a conjugation of relations. R+ is the transpose of R, i.e., (u, x) e R+ if and only if (x, u) e R. The double composition R+ o G o R contains the pair (u, v) in B x B if and only if there are x and y such that (u, x) e R+, (y, v) e R, and (x, y) e Ea. Thus G * R = R+ o G o R. (1.3) Simple graphs, analogously, correspond to the irreflexive symmetric relations. For any relation R, let R1 denote its irreflexive part, also known as the reflexive reduction of R. Since definition 1.2 explicitly excludes the diagonals, it can be written in the form G*R = (R+ o G o R)1. (1.4) We have G * R = (G * R)1, and hence Ea*R C Ea*R. The composition G * R is of particular interest when G is also a simple graph, i.e., G = G1. The main part of this contribution will be concerned with the solutions of the equation G * R = H. The weak version, G * R = H, will turn out to have much less convenient properties, and will be discussed only briefly in section 7. Throughout this paper we use the following standard notations and terms. For relation R C X x Y we define by R(x) = {p e Y|(x,p) e R} the image of x under R and R-1(p) = {x e X |(x,p) e R} the pre-image of p under R. The domain of R is defined by dom R = {x e X|3p e Y s.t. (x,p) e R}, and the image of R is defined by imgR = {p e Y|3x e X s.t. (x,p) e R}. We say that the domain of R is full if for any x e X we have R(x) = 0. Analogously, the image is full if for any p e Y we have R-1(p) = 0. Let R C X X Y is a binary relation, then R is injective, if for all x and z in X and y in Y it holds that if (x, y) e R and (z, y) e R then x = z. R is functional, if for all x in X, and y and z in Y it holds that if (x, y) e R and (x, z) e R then y = z. We denote by Ia the identity map on G, i.e., {(x, x)|x e Va}. Let G = (Va, Ea) be a graph and let W C Va. The induced subgraph G[W] has vertex set W and (x, y) is an edge of G[W] if x, y e W and (x, y) e Ea. A graph Pk is a path of length k. Similarly, is an (elementary) cycle of length k with vertex set {0,1,..., k - 1}. Finally, Kk is the complete (loopless) graph with k vertices. An isolated vertex is a vertex with degree 0. Note that the vertex with a loop is not isolated in this sense. 1.3 Matrix multiplication The operation * can also be formulated in terms of matrix multiplication. To see this, consider the following variant of the operation on weighted graphs. Definition 1.3. If G is a weighted graph, we use w(x, y) to denote the weight between x and y. Given a finite set B and a binary relation R C VG x B, G®R is defined as a weighted graph H with vertex set B, for any u, v e B, w(u, v) = J2(x,u)eR,(y,v)eR w(x, y). Ignoring the weights, operations * and © are equivalent. Using the language of matrices, G© R = H can be interpreted as matrix multiplication: Wg®r = R+WgR (1.5) where R is the matrix representation of the relation R, i.e., Rxu = 1 if and only if (x, u) e R, otherwise Rxu = 0, R+ denotes the transpose of R, and WG is the matrix of edge weights of G. 1.4 Graph homomorphisms and multihomomorphisms The notion of relations between graphs is in many ways similar (but not equivalent) to the well studied notion of graph homomorphisms. The majority of our results focus on similarities and differences between those two concepts. We give here only the basic definitions of graph homomorphisms. For more details see [7]. A homomorphism from a graph G to a graph H is a mapping f : VG ^ VH such that for every edge (x, y) of G, (f (x), f (y)) is an edge of H. Note that homomorphisms require loops in H whenever (x, y) e EG and f (x) = f (y). In contrast, f is a weak homomorphism if (x, y) e Eg implies that either f (x) = f (y) or (f (x), f (y)) e EH. Every homomorphism from G to H induces also a weak homomorphism, but not conversely [9]. Since every homomorphism preserves adjacency, it naturally defines a mapping f1 : Eg ^ Eh by setting f 1((x, y)) = (f (x), f (y)) for all (x, y) e Eg. If f is surjective, we call f a vertex surjective homomorphism, and if f1 is surjective, we call f an edge surjective homomorphism. f is surjective homomorphism if it is both vertex- and edge-surjective [7]. A map f : VG ^ VH is, of course, a special case of a relation. This is seen by setting F = {(x, f (x))|x e VG}. Hence, there is a surjective homomorphism from G to H if and only if there is a functional relation F such that G * F = H. Another important connection to the graph homomorphisms is the following simple lemma. Lemma 1.4. If G * R = H, and the domain of R is full, then there is a homomorphism f from G to H contained in R. Proof. If G * R = H, then take any functional relation f C R, we have G * f C H, where f is a homomorphism from G to H. □ Analogously, there is a weak surjective homomorphism from G to H if and only if there is a functional relation F such that G * F = H, and there is a weak homomorphism from G to H if there is a functional relation F C VG x VH such that G * F is a subgraph of H. The existence of relations between graphs thus can be seen as a proper generalization of graph homomorphisms or weak graph homomorphisms, respectively. Finally, a full homomorphism from a graph G to a graph H is a vertex mapping f such that for distinct vertices u and v of G, we have (u, v) an edge of G if and only if (f (u), f (v)) is an edge of H, see [4]. Relation between graphs can be regarded also as a variant of multihomomorphisms. Multihomomorphisms are building blocks of Hom-complexes, introduced by Lovasz, and are related to recent exciting developments in topological combinatorics [10], in particular to deep results involved in proof of the Lovasz hypothesis [1]. A multihomomorphism G ^ H is a mapping f : VG ^ 2Vh \ {0} (i.e., associating a nonempty subset of vertices of H with every vertex of G) such that whenever {ui, u2} is an edge of G, we have (v1; v2) G EH for every v1 G f (u1) and every v2 G f (u2). The functions from vertices to sets can be seen as representation of relations. A relation with full domain thus can be regarded as surjective multihomomorphism, a multihomomor-phism such that pre-image of every vertex in H is non-empty and for every edge (u, v) in H we can find an edge (x, y) in G satisfying u G f (x), v G f (y). 1.5 Examples Similarly to graph homomorphisms, the equation G * R = H (or G * R = H respectively) may have multiple solutions for some pairs of graphs (G, H), while there may be no solution at all for other pairs. As an example, consider K2 (two vertices x, y connected by an edge) and C3 (a cycle of three vertices u, v, w). Denote R1 = {(u, x), (v, y)}, R2 = {(v, x), (w, y)}, R3 = {(w,x), (u, y)}, then it is easily seen that C3 * Ri = K2 for each 1 < i < 3, i.e. the equation C3 * R = K2 has more than one solution. On the other hand, there is no relation R such that K2 * R = C3. Otherwise, each vertex of C3 is related to at most one vertex of K2, since C3 is loop free; hence there exists a vertex in K2 which has no relation to at least two vertices in C3, w.l.o.g., one can assume (x, u), (x, v) G R; then the definition of * implies that there is no edge between u and v, which causes a contradiction. Because relations do not need to have full domain (unlike graph homomorphisms), there is always an relation from a graph G to its induced subgraph G[W]. Relations with full domain are not restricted to surjective homomorphisms. As a simple example, consider paths P1 with vertex set {x, y} and P2 with vertex set {u, v, w}, respectively, and set R = {(x, u), (x, w), (y, v)}. One can easily verify P1 * R = P2 by direct computation. Here, R is not functional since x has two images. 1.6 Outline and main results This paper is organized as follows. In section 2 the basic properties of the strong relations between graphs are compiled. It is shown that relations compose and every relation can be decomposed in a standard way into a surjective and an injective relation (Corollary 2.3). We discuss some structural properties of graph preserved by the relations. Equivalence on the class of graphs induced by the existence of relations between graphs is the topic of section 3. We consider two forms: the strong relational equivalence, where relations are required to be reversible, and weak relational equivalence. Equivalence classes of strong relational equivalence are characterized in Theorem 3.8. To describe equivalence classes of the weak relational equivalence we introduce the notion of an R-core of a graph and show that it is in many ways similar to the more familiar construction of the graph core (Theorem 3.17). We explore in particular the differences between core and R-core and an effective algorithm to compute the R-core of given graph is provided. Section 4 is concerned with the partial order induced on relations between two fixed graphs G and H. Focusing on the special case G = H the minimal elements of this partial order are described. In Theorem 4.7 we give a, perhaps surprisingly simple, characterization of those graphs G for which all relations of G to itself are automorphisms. R-retraction is defined in section 5 in analogy to retractions. It naturally gives rise to a notion of R-reduced graphs that we show to coincide with the concept of graph cores. By reversing the direction of relations, however, we obtain the concept of a cocore of a graph, which does not have a non-trivial counterpart in the world of ordinary graph homomor-phisms, and explore its properties. The computational complexity of testing for the existence of a relation between two graphs is briefly discussed in section 6. In Theorem 6.1 we describe the reduction of this problem to the surjective homomorphism problem. Finally, in section 7 we briefly summarize the most important similarities and differences between weak and strong relational composition. 2 Basic properties 2.1 Composition Recall that the composition of binary relations is associative, i.e., suppose R C W x X, S C X x Y, and T C Y x Z. Then R o (S o T) = (R o S) o T. Furthermore, the transposition of relations satisfies (R o S)+ = S + o R+. Interpreting the graph G as a relation on its vertex set, we easily derive the following identities: Lemma 2.1 (Composition law). (G * R) * S = G * (R o S). Proof. (G * R) * S = S + o (R+ o G o R) o S = (S + o R+) o G o (R o S) = (R o S)+ o G o (R o S) = G * (R o S). □ Now we show that every relation R can be decomposed, in a standard way, to a relation RD duplicating vertices and a relation RC contracting vertices. Lemma 2.2. Let R C X x Y be a relation. Then there exists a subset A of X, a set B, an injective relation with full domain RD C A x B and a functional relation RC C B x Y, such that R = o RD o RC, where is the identity on X restricted to A. Proof. Put A = dom R. Then the relation removes vertices in X \ dom R. It remains to show, therefore, that any relation R C X x Y with full domain can be decomposed into an injective relation RD C X x B and a functional relation RC C B x Y .To see this, set B = R and declare (x, a) e RD if and only if a = (x,p) e R for some p e Y, and (ß, q) e RC if and only if ß = (y, q) e R for some y e X .By construction RD is injective and RC is functional. Furthermore, (x0,po) e RD o RC if and only if there is a e R that is simultaneously of the form (x0,p) and (x,p0), i.e., x = x0 and p = p0. Hence (x0,p0) e R. □ Note that this decomposition is not unique multiple copies of R. More precisely, let B (x, (a, i)) e Rd (1 < i < k) if and only if a = . For instance, we could construct B from = R x {1, 2, • • • , k}, then we would set = (x,p) e R for some p e Y, etc. The set B as constructed in the proof of Lemma 2.2 has minimal size. To see this, it suffices to show that, given B there is a mapping from B onto R. Since RD is injective and RC is functional we may set a G B ^ (R-1(a),RC(a)). Since R = o RD o RC we conclude that the mapping is surjective, and hence |B | > |R|. According to Lemma 2.1, the decomposition of R in Lemma 2.2 can be restated as follows: Corollary 2.3. Suppose G * R = H. Then there is a set B, an injective relation RD C dom R X B with full domain, and a surjective relation RC C B x img R such that G [dom R] * Rd * RC = H. In diagram form, this is expressed as We shall remark that from the fact the relations compose it follows that the existence of a relation implies a quasi-order on graphs that is related to the homomorphism order. This order is studied more deeply in [8]. 2.2 Structural properties preserved by relations In this subsection we investigate structural properties of H that can be derived from knowledge about certain properties of G and the fact that there is some relation R such that G * R = H. 2.2.1 Connected components Proposition 2.4. Let G * R = H and denote by H1, • • • , Hk the connected components of H. Then there are relations Ri C VG x VHi for each 1 < i < k such that G * Ri = Hi and R = U k = 1 Rj. Furthermore, set Gj = G[R 1(Vhì)]. Then there are no edges between Gj and Gj for arbitrary i = j. Proof. Define the restriction of R to the connected components of H as Ri = {(x, y) G R|y G VHi}. Clearly, R is the disjoint union of the Ri and G * Ri C Hi. The definition of * implies H = G * R = (Ui Ri)+ ◦ G o R^ = Ui Uj R+ ◦ G o Rj. Since Ri and Rj relate vertices of G to different connected components of HU we have R+ o G o Rj = 0. It follows that H = (J R+ o G o Rj = |J i R+ o G o Ri = |J i G * Ri. Hence G * Ri = Hi. Any edge between Gi and Gj would generate edges between Hi and Hj, thus causing a contradiction to our assumptions. □ Denote by b0 (G) the number of connected components of G, then from Proposition 2.4 we arrive at: Corollary 2.5. Suppose both G and H do not have isolated vertices. If G * R = H and R has full domain, then b0(G) > b0(H ). Proof. Our notations is the same as in Proposition 2.4. We claim for arbitrary connected component C of graph G, there exists a unique i, such that C is a connected component of Gi. Otherwise one can find two vertices x, y G C, x and y adjacent, such that x G VGi and y G VGj, since G has no isolated vertices, which contradicts E(Gi, Gj) = 0. Thus bo(G) > b0(H) is easily followed. □ From corollary 2.5, we know that H is connected whenever G is connected. The connectedness of G, however, cannot be deduced from the connectedness of H. For example, consider G = P1 U P1 with vertex set jxi, x2, x3, x4} and edges jxi, x2} and {x3, x4}, and H = P2 with vertex set {v1,v2, v3}. Set R = {(x1, v1), (x2,v2), (x3, v2), (x4, v3)}. One can easily verify that G * R = H. On the other hand, H is connected but G has 2 connected components. The point here is, of course, that R is not injective. 2.2.2 Colorings Graph homomorphisms of simple graphs can be seen as generalizations of colorings: A (vertex) k-coloring of G is a mapping c : VG ^ {1, 2,..., k} such that adjacent vertices have distinct colors, i.e., c(u) = c(v) whenever (u,v) G EG. Every k-coloring c can be also seen as a homomorphism c : G ^ Kk. The chromatic number x is defined as the minimal of colors needed for a coloring, see e.g. [7]. Thus, if R is a functional relation describing a vertex coloring, then G * R C Kk. Conversely, G * R C Kk, where R has full domain, then from Lemma 1.4, there exists a homomorphism from G to Kk, which is a coloring of G. Lemma 2.6. If G is a simple graph and R has full domain, then x(G) < x(G * R). Proof. Suppose G*R = H and the domain of R is full, from Lemma 1.4 we know G ^ H, so x(G) < x(G * R). □ 2.2.3 Distances Observation 2.7. If Pk * R = G, G is a simple graph and the domain of R is full, Pk with the vertex set 0,1, • • • , k, then there is a walk [v0, v1;..., vk] in G, where (i, vi) G R for 0 < i < k. Observation 2.8. If Ck * R = G, G is a simple graph and the domain of R is full, then there is a closed walk [v0, v1;..., vk-1] in G, where (i, vi) G R for 0 < i < k - 1. Let dG(x, y) denote the canonical distance on graph G, i.e., dG(x, y) is the minimal length of a path in graph G that connects vertices x and y; if there is no path connects vertices x and y, then the distance is infinite. Lemma 2.9. Suppose there exists a relation R with full domain s.t. G * R = H, x, y G VG, u, v G VH and (x, u) G R, (y, v) G R. If x = y, then dH(u, v) < dG(x, y); If x = y and x is not an isolated vertex, then dH (u, v) < 2. Proof. If x = y and x is not isolated, pick a vertex z of graph G which is adjacent to vertex x, and find a vertex w e H satisfying (z, w) e R. Then (w, u) e EH and similarly (w, v) e Eh. So dH(u, v) < 2. If x = y, choose the shortest path P = x,xi,x2, • • • , xk, y between x and y, and find corresponding vertices ui, u2, • • • , uk e H such that(xi? ui) e R for any 1 < i < k - 1 it is easily seen that (u, ui) e EH, (ui,ui+i) e EH and (uk,v) e EH, then d(u, v) < d(x,y). □ The eccentricity e of a vertex v is the greatest distance between v and any other vertex. The radius of a graph G, denoted by rad(G), is the minimum eccentricity of any vertex. The diameter of a graph G, denoted by diam(G), is the maximum eccentricity of any vertex in the graph, i.e., the largest distance between any pair of vertices. Corollary 2.10. Suppose G * R = H, G and H are connected graphs, and R has full domain, then rad(H) < max{rad(G), 2}. An analogous result holds for the diameters. In particular, if G is not a complete graph, then diam(G) > diam(G * R). Corollary 2.11. There is a relation from the path of length k, Pk, to the path of length l, Pi, if and only if either k > l or k = 1, l = 2. Proof. For k > l there is a surjective homomorphism f from Pk to Pl and hence by Lemma 1.4 there is also a relation from Pk to Pl. In Section 1.5 we already showed a relation from Pi to P2. To show that Pi * R = P2 is the only case with k < l we first observe that Lemma 2.9 excludes the existence of relation from Pk to Pl for 1 < k < l. Now suppose R satisfies Pi * R = Pk for k > 2. Since Pk has at least 4 vertices, either one of the vertices of Pi has at least 3 images so that Pi * R has a vertex with degree at least 3, or both of the vertices in Pi have at least 2 images, in which case all vertices of Pi * R have degree at least 2. In both cases Pi * R cannot be a path. □ In particular, {Pi,P2} is the only pair of paths such that there is a relation between them in both directions. 2.2.4 Complete graphs The complement graph H of a simple graph H has the same vertex set as H, and two vertices are connected in H if and only if they are not connected in H. Note that in this subsection we do not require that the domain of R is full. Proposition 2.12. Let H be a simple graph. Then there exists a relation R such that Kk * R = H if and only if H is the disjoint union of at most k complete graphs. Proof. Denote the connected components of H by Hi,..., Hm. If m < k and every connected component of H is a complete graph, let R = {(i,u)|i =1, ••• ,m,u e VHi} and by the definition of complement graph, for any i = 1, • • • , m, all the vertices in Hi are independent in H, and u is adjacent to v whenever u e VHi and v e VHj for distinct i, j. Hence it is easily seen that Kk * R = H. Conversely, if Kk * R = H, denote the vertices in Kk by 1, • • • , k, s.t. dom R = {1, • • • , m}. We claim that R is injective, otherwise H would have loops. Thus VH is the disjoint union of R(1), • • • , R(m). For any two distinct vertices u, v in R(i), u and v are independent in H and for distinct i and j every vertex in R(i) are adjacent with every vertex in R(j) whenever R(i) = 0. Therefore for any i, R(i) is the vertex set of a connect 2.2.5 Subgraphs Relations between graphs intuitively imply relations between local subgraphs. In this section we make this concept more precise. Denote by the closed neighborhood of x in G. Furthermore, we let NG[x] := VG \ NG [x] be the set of vertices that are not adjacent (or identical) to x in G and denote by Gx := G[Ng [x]] the induced subgraph of G that is obtained by removing the closed neighborhood of a vertex x. Analogously, for a subset S C VG we define as the induced subgraph obtained by removing all vertices in S and their neighbors. Then we have the following result about relations between local subgraphs. Proposition 2.13. Suppose G*R = H and S and D are subsets of VG and VH, respectively, such that G[S] * R|(SxD) = H[D], R|(SxD) has full domain on S, and there is no isolated vertex in D. Then S * R = D, where R = R|(sxd) is the corresponding restriction of R. Proof. Obviously, S * R is an induced subgraph of D. We have to show the reverse inclusion. Given u g Vp and x g R-1 (u), we first show that there are two possibilities: 1. x is a vertex of S. 2. x is an isolated vertex of S. Assume that is not the case, i.e., that x G % and that x is either a non-isolated vertex of S or x is in the neighborhood of some vertex of S. In either case there is y G S connected by an edge to x. Consequently there is also v g D, such that v g R(y), connected by an edge to u. It follows u G Vp, a contradiction. Now consider an arbitrary edge (u, v) G Ep. We have (x, y) G EG such that u G R(x) and v G R(y). It follows that x and y are not isolated and thus x, y are vertices of S. Consequently S * R has precisely the same edges as D. Because D has no isolated vertices and thus every vertex is an endpoint of some edge, we know that the vertex set of S * R is same as the vertex set of D. □ This result is of particular practical use in the special case where S and D consist of a single vertex. When looking for a relation R such that G * R = H one can remove a vertex including its neighborhood from G as well as the prospective image including the neighborhood from H and solve the problem on the subgraphs. component of H, which is a complete graph. □ Ng[x] := {z G Vg|z = x V (x,z) G Eg} (2.2) (2.3) 3 Relational equivalence Graphs G and H are homomorphism equivalent (or hom-equivalent) if there exists homomorphisms G ^ H and H ^ G. It is well known that every equivalence class of the homomorphism order contains a minimal representative that is unique up to isomorphism: the graph core [7]. We define similar equivalences implied by the existence of (special) relations between graphs. In this section, we require all relations to have full domain unless explicitly stated otherwise. With this condition we will show that these equivalences produce a rich structure closely related to but distinct from the structure of homomorphism equivalences. This may come as a surprise: the equivalence implied by the existence of surjective homomorphisms is not interesting. Consider two graphs G and H and suppose there are surjective homomorphisms f : G ^ H and g : H ^ G. Since every vertex in VG has at most one image under f, we have |VG| > |VH |. Analogously |VH | > |VG|, and hence | VG | = | VH |. Thus f and g are both bijective, and G is isomorphic to H. 3.1 Reversible relations Definition 3.1. A relation R is reversible with respect to graph G if (G * R) * R+ = G. We write NG(x) := {z e VG|(x, z) e EG} for the open neighborhood of vertex x in graph G. Proposition 3.2. Suppose R = RD o RC, where RD and RC are constructed as in the proof of Proposition 2.2. Then R is reversible with respect G if and only if for every a and ß satisfying RG(a) = RG(ß) we have Ng*rd (a) = Ng*rd (ß). Proof. We set G1 = G * RD, then from Lemma 2.1 we have G1 * RC = H. If RC (a) = Rc(ß) implies Ng, (a) = Ng, (ß), then H * R+ = G1. Since G1 * R+D = H, we have H * R+ * R+ = H * R+ = G, i.e., R is reversible. Conversely, since R is reversible, i.e., H * R+ = G, setting G2 = H * R+ gives G2 * RD = G. Hence G1 * RC * R+ = G2 and G2 * RD * RD = G1. From 1Gl C RC * R+ we conclude G1 C G2, and similarly 1G2 C RD * RD yields G1 D G2. Hence G1 = G2. R+ is injective, hence a, ß e VG2 = VGl has the same open neighborhood whenever the pre-image of a and ß under R+ coincide, i.e. RC(a) = RC(ß). □ Rd is an injective relation, hence one can easily get Ng*rd (a) = RD(NG(x)) provided that (x, a) e Rd. On the other hand, if we define R to be the image of RD as in the proof of Proposition 2.2, then RC (a) = RC(ß) implies there are two distinct vertices x, y e VG, s.t. (x, u), (y, u) e R, where u = RC(a) = RC(ß), and verse visa. Using Proposition 3.2 we thus obtain Proposition 3.3. A relation R is reversible with respect to G if and only if for every two vertices x and y such that R(x) n R(y) = 0 we have NG(x) = NG(y). 3.2 Strong relational equivalence Definition 3.4. Two graphs G and H are (strongly) relationally equivalent, G ^ H, if there is a relation R such that G * R = H and H * R+ = G. Lemma 3.5. Relational equivalence is an equivalence relation on graphs. G H Gthin = Hthin Figure 2: Non-isomorphic graphs G and H with isomorphic thin graphs. Proof. The relation — is reflexive since G * IG = G. Symmetry also follows directly from the definition. Suppose G * R = H and H * R+ = G and H * Q = K and K * Q+ = H, i.e., (G * R) * Q = K and (K * Q+) * R+ = G, i.e., G * (R o Q) = K and K * (Q+ o R+) = K * (R o Q)+ = G, i.e., - is also transitive. □ Definition 3.6. The thinness relation S of G is the equivalence relation on VG defined by (x, y) G S if and only if NG(x) = NG(y). A graph G is called thin if every vertex forms its own class in S. Thin graphs are also known as "point determining graphs" [13]. We denote by S the corresponding partition of VG, and write RS C VG x S for the relation that associates each vertex with its S-equivalence class, i.e., (x, ß) G RS if and only if x G ß. Definition 3.7. The thin graph of G, denoted by Gthin, is the quotient graph G/S, i.e., Gthin has vertex set S and two equivalence classes a and t of S are adjacent in Gthin if and only if (x, y) is an edge of G with x G a and y G t. As noted e.g. in [6, p.81], Gthin is itself a thin graph. Furthermore, RS is a full homo-morphism of G to Gthin, see [4]. Thinness and the quotients w.r.t. the thinness relation play an important role in particular in the context of product graphs, see [9]. In this context it is well known that G can be reconstructed from Gthin and the knowledge of the S-equivalence classes. In fact, we have Gthin * Rs + = G. (3.1) Theorem 3.8. G and H are in the same equivalence class w.r.t. — if and only if their thin graphs are isomorphic. Proof. Assume G — H. From Equation(3.1) we know that G — Gthin, H — Hthin, so Gthin — Hthin. Now we claim that Gthin and Hthin are isomorphic. Suppose Gthin*R = Hthin, then the pre-image of R is unique. Otherwise, there exist distinct vertices x, y G VGthin such that R(x) = R(y), then NGthin(x) = NGthin(y), contradicting thinness. Likewise, the pre-image of R-1 is unique, i.e., the image of R is unique. Hence R is one-to-one. □ The example in Fig. 2 shows that thin graphs can be isomorphic while G and H themselves are not isomorphic. Relational equivalence thus is coarser than graph isomorphism (surjective homomorphic equivalence) but stronger than homomorphic equivalence. Figure 3: G and H are weakly relationally equivalent but have non-isomorphic thin graphs. 3.3 Weak relational equivalence Definition 3.9. Two graphs G and H are weak relationally equivalent, G ^w H, if there are relations R and S such that G * R = H and H * S = G. Lemma 3.10. Weak relational equivalence is an equivalence relation on graphs. Proof. By definition is symmetric. Because G * IG = G, relation is reflexive. Suppose G G' and G' G". Thus there are relations R, S, R', and S', such that G' = G * R, G'' = G' * R', G = G' * S, and G' = G'' * S'. By the composition law (Lemma 2.1) it follows that G'' = G * (R o R') and G = G'' * (S' o S), i.e, G -w G''. Hence is transitive. □ Strong relational equivalence implies weak relational equivalence. To see this, simply observe that the definition of the weak form is obtained from the strong one by setting S = R+. The converse is not true, as shown by the graphs G and H in Fig. 3: It is easy to see that their thin graphs are different and thus G and H are not strongly relationally equivalent. However, are relationally equivalent. To get relation from G to H contract vertices 2 and 3 and keep other vertices on place, i.e., R = {(1,1), (2, 2), (3, 2), (4,4), (5, 5), (6, 6), (7, 7)}. To get relation from H to G, duplicate 5 and 7 and contract them together to 3, S = {(1,1), (2, 2), (4,4), (5, 5), (6, 6), (7, 7), (5, 3), (7, 3)}. Consequently, weak relational equivalence is coarser than strong relational equivalence. 3.4 R-cores A graph is an R-core, if it is the smallest graph (in the number of vertices) in its equivalence of ^w. This notion is analogous to the definition of graph cores. In this section we show properties of R-cores that are similar to the properties of graph cores. To this end we first need to develop a simple characterization of R-cores. Again we start from a decomposition of relations. Consider a relation R such that G * R = H. We seek for pair of relations R1 and R2 such that R = R1 o R2. In contrast to Lemma 2.2, however, we now look for a decomposition so that the graph G' = G * R1 is smaller (in the number of vertices) than G. G' The existence of such a decomposition follows from a translation of the well-known Hall Marriage Theorem [12] to the language of relations. We say that the relation R C A x B satisfies the Hall condition, if for every S C A we have |S| < |R(S) |. Theorem 3.11 (Hall's theorem). If G * R = H and R satisfies the Hall condition, then R contains a monomorphism f : G ^ H. Proof. The Hall Marriage Theorem is usually described on set systems. For set systems satisfying the Hall condition, the theorem guarantees the existence of a system of distinct representatives, see i.e. [12]. Relations can be seen as set systems (defined by the images of individual vertices). Furthermore, in our setting the system of distinct representatives directly corresponds to a monomorphism contained in the relation R. □ Lemma 3.12. If G * R = H and relation R does not satisfy the Hall condition, then there are relations R1 and R2 such that R = R1 o R2, and the number of vertices of graph G' = G * Ri is strictly smaller than the number of vertices of G. Proof. Without loss of generality assume that VG n VH = 0. If R does not satisfy the Hall condition, then there exists a vertex set S c VG such that |S| > |R(S)|. Now we define relations R1 and R2 as follows: JR(x) for x € ^ fx for x G rR^ R1(x)=S tU . R2(x) = ip/ N tu • (33) x otherwise, R(x) otherwise. Obviously Ri o R2 = R and |VG,| = |VG| - (|S| - |R(S)|) < |VG|. □ This immediately gives a necessary, but in general not sufficient, condition for a graph to be an R-core. Corollary 3.13. If G is an R-core, then every relation R such that G * R = G satisfies the Hall condition and thus contains a monomorphism. Proof. Assume that there is a relation R that does not satisfy the Hall condition. Then there is a graph G', |VG/1 < |VG|, and relations R1 and R2 such that G * R1 = G' and G' * R2 = G. Consequently G' is a smaller representative of the equivalence class of a contradiction with G being R-core. □ To see that the condition of Corollary 3.13 is not sufficient consider a graph consisting of two independent vertices. Next we show that R-cores are, up to isomorphism, unique representatives of the equivalence classes of . Figure 4: Construction of an embedding from GR-core to G. Proposition 3.14. If both G and H are R-cores in the same equivalence class of then G and H are isomorphic. Proof. Because both G and H are R-cores, we know that | VG | = | VH |. Consider relations R1 and R2 such that G*R1 = H and H*R2 = G. Applying Lemma 3.12 we know that R1 satisfies the Hall condition. Otherwise there would be a graph G' with |VG' | < |VG| so that G' is relationally equivalent to both G and H contradicting the fact that G and H are R-cores. Similarly, we can show that R2 also satisfies the Hall condition. From Theorem 3.11 we know that there is a monomorphism f from G to H, and monomorphism g from H to G. It follows that number of edges of G is not larger than the number of edges of H and vice versa. Because G and H have the same number of edges and same number of vertices, G and H must be isomorphisms. □ It thus makes sense to define a construction analogous to the core of a graph. Definition 3.15. H is an R-core of graph G if H is an R-core and H ^w G. All R-cores of graph G are isomorphic as an immediate consequence of Prop. 3.14. We denote the (up to isomorphism) unique R-core of graph G by GR-core. Lemma 3.16. GR-core is isomorphic to a (not necessarily induced) subgraph of G. Proof. Take any relation R such that GR-core *R = G. By the same argument as in Corollary 3.13, there is a monomorphism f : GR-core — G contained in R. Consider the image of f on G. □ Theorem 3.17. GR-core is isomorphic to an induced subgraph of G. Proof. Fix R1 and R2 such that GR-core * R1 = G and G * R2 = GR-core. R = R1 oR2 is a relation such that GR-core * R = GR-core. By Corollary 3.13, R contains a monomorphism f : GR-core —y GR-core. Because such a monomorphism is a permutation, there exists n such that fn, the n-fold composition of f with itself, is the identity. Put R1 = Rn-1 o R1. Because Rn contains the identity and Rn = R[ o R2, it follows that for every x G VGR-core, there is a vertex I(x) G VG such that I(x) G Rl(x) and x G R2(1 (x)). We show that for two vertices x = y, we have I(x) = I(y) and thus both I and I-1 are monomorphisms. Assume, that is not the case, i.e., that there are two vertices x = y such that I(x) = I(y). Consider an arbitrary vertex z in the neighborhood of x. It follows that I(z) must be in the neighborhood of I(x) and consequently z is in the neighborhood of y. Thus the neighborhoods of x and y are the same. By Theorem 3.8, however, we know that the R-core is a thin graph (because weak relational equivalence is coarser than strong relational equivalence), a contradiction. Finally observe that I is an embedding from GR-core to G. For every edge (x, y) G EGR-core we also have edge (I(x), I(y)) G EG because I is contained in relation R^. Similarly because I-1 is contained in relation R2, every edge (I(x), I(y)) G EG corresponds toanedge (x,y) G egr.OT6. □ We close the section with an algorithm computing the R-core of a graph. In contrast to graph cores, where the computation is known to be NP-complete, there is a simple polynomial algorithm for R-cores. Observe that the R-core of a graph containing isolated vertices is isomorphic to the disjoint union of the R-core of the same graph with the isolated vertices removed and a single isolated vertex. The R-core of a graph without isolated vertices can be computed by Algorithm 1. Algorithm 1 The R-core of a graph Graph G with loops allowed and without isolated vertices, vertex set denoted by V, neighborhoods NG(i), i G V. 1: for i G V do Input: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 W(i) = 0 found = FALSE for j G V \ {i} do if N (j ) C N (i) then W (i) := W (i) U N (j ) end if if N (i) C N (j) then found = TRUE end if end for if W (i) = N (i) A found then delete i from V N(i) = 0 end if end for return The R-core G[V] of G. The algorithm removes all vertices v G G such that (1) the neighborhood of v is union of neighborhood of some other vertices v1, v2,..., vn and (2) there is vertex u such that Ng(V) C Ng(U). It is easy to see that the resulting graph H is relationally equivalent to G. Condition (1) ensures the existence of a relation R1 such that H * R1 = G, while the condition (2) ensures the existence of a relation R2 such that G * R2 = H. We need to show that H is isomorphic to GR-core. By Theorem 3.17 we can assume that GR-core is an induced subgraph of H that is constructed as an induced subgraph of G. We also know that there are relations R1 and R2 such that GR-core * R1 = H and G * R2 = GR-core. By the same argument as in the proof of Theorem 3.17 we can assume both R1 and R2 to contain an (restriction of) identity. Now assume that there is a vertex v g Vh \ VGR-core. We can put u = R2 (v) and because R2 contains an identity we have NG(v) C NG(u). We can also put {v1; v2... vn} to be set of all vertices such that v g R1(vi ). It follows that the neighborhood of v is the union of neighborhoods of v1; v2,..., vn and consequently we have v G VH, a contradiction. 4 The partial order Rel(G, H) 4.1 Basic properties For fixed graphs G and H we consider partial order Rel(G, H). The vertices of this partial order are all relations R such that G * R = H. We put R1 < R2 if and only if R1 C R2. This definition is motivated by Hom-complexes, see [10]. In this section we show the basic properties of this partial order and concentrate on minimal elements in the special case of Rel(G, G). Proposition 4.1. Suppose G * R' = H, G * R" = H and R' C R", then any relation R with R' C R C R'' also satisfies G * R = H. Proof. From R' C R C R'' we conclude G*R' C G*R C G*R''. Hence G*R' = G*R'' implies G * R = H. □ Hence it is possible to describe the partial order Rel(G, H) by listing minimal and maximal solutions R of G * R = H w.r.t. set inclusion. For example, if G is P3 with vertices v0, v1; v2, v3 and H is P1 with vertices x0, x1, it is easily seen that R'' = {(v0, x0), (v2, x0), (v1; x1), (v3, x1)} is a maximal solution of G * R = H and R' = {(v0,x0), (v1,x1)} is a minimal solution, because R' C R'', then all the relations R with R' C R C R'' satisfy G * R = H. We note that minimal and maximal solutions need not be unique. 4.2 Solutions of G * R = G For simplicity, we say that a relation R is an automorphism of G if it is of the form R = {(x, f (x))|x G VG} and f : VG ^ VG is an automorphism of G. We shall see that conditions related to thinness again play a major role in this context. Recall that G is thin if no two vertices have the same neighborhood, i.e., NG(x) = NG(y) implies x = y. Here we need an even stronger condition: Definition 4.2. A graph G satisfies condition N if NG (x) C NG(y) implies x = y. In particular, graph satisfying condition N is thin. Proposition 4.3. For a given graph G, the set Rel(G, G) of all relations satisfying G * R = G forms a monoid. Proof. Firstly, because G is a finite graph, the set Rel(G, G) is also finite. Furthermore, R, S G Rel(G, G) implies G * R = G and G * S = G and thus G * (R o S) = G, so that R o S G Rel(G, G). Finally, the identity relation IG is a left and right identity for relational composition: IG o R = R o IG = R. □ A relation R c VG x VG can be interpreted as a directed graph R with vertex set VG and a directed edge u ^ v if and only if (u, v) g R. Note that R may have loops. We say that v g Vg is recurrent if and only if there exists a walk (of length at least 1) from v to itself. Let SG be the set of all the recurrent vertices. Furthermore, we define an equivalence relation £ on SG by setting (u, v) G £ if there is a walk in R from u to v and vice versa. The equivalence classes w.r.t. £ are denoted by R/£ = {D1, D2, • • • , Dm}. We furthermore define a binary relation > over R/£ as follows: if there is a walk from a vertex u in Di to a vertex v in Dj, then we say u > v. It is easily seen that > is reflexive, antisymmetric, and transitive, hence (R/£, >) is a partially ordered set. W.l.o.g. we can assume {D1, D2,..., Ds} are the maximal elements w.r.t. >. Now let Gr = G[D1 U • • • U Ds] be the subgraph of G induced by these maximal elements. In the following we write R1 for the l-fold composition of R with itself. Lemma 4.4. For arbitrary x G VG, there exists I G N and a recurrent vertex v such that (v, x) G R1. Proof. Set x0 = x and choose xi G R-1(xi-1) for all i > 1. Since |VG| < to, there are indices j, k g N, j < k, xj = xk. Then xj is recurrent vertex. The lemma follows by setting I = j and v = xi. □ Lemma 4.5. For every v G VGr, R-1(v) C VGr. Proof. Suppose x G R-1(v) is not recurrent. Lemma 4.4 implies that there is I G N and a recurrent vertex w such that (w, x) G R1. Hence the definitions of E and > imply [w] > [v], where [v] denotes the equivalent class (w.r.t. E) containing the vertex v. Since [v] is maximal w.r.t. >, we have [v] = [w]. Consequently, there exists an index k G N such that (v, w) G Rk. On the other hand, we have (x, x) = (x, v) o (v, w) o (w, x) G Rk+1+1. Thus, x is recurrent, a contradiction. Therefore, every vertex x G R-1(v) is recurrent. Hence [x] > [v] together with the maximality of [v] gives [x] = [v], and thus x G VGr. □ Lemma 4.6. For every x G VG, there is I G N such that, for arbitrary i > l, there exists u G VGr satisfying (u, x) G Ri. Proof. From Lemma 4.4 and Lemma 4.5 we conclude that it is sufficient to show that for an arbitrary recurrent vertex v there is a k G N and w G VGr such that (w, v) G Rk. The lemma now follows easily from the finiteness of VG. □ From these three lemmata we can deduce Theorem 4.7. All solutions of G * R = G are automorphisms if and only if G has property N. Proof. Suppose there are distinct vertices x, y e VG such that NG (x) C NG (y). Then R = 1G U (x, y), which is not functional, satisfies G * R = G. Thus G * R = G is also solved by relations that are not automorphisms of G. This proves the "only if" part. Conversely, suppose G has property N. Claim: There is a k e N such that Rk n ( VGr x VGr ) = ^Gr . For each vj e VGr there is a walk of length sj > 1 from vj to itself. Hence (vj, vj) e Rsi. Let s be the least common multiple of the sj. Then (vj, vj) e Rs for all vj e VGr. Define Q := Rs n (VGr x VGr ). Thus IGr C Q and moreover Qj C Qj+1 for all j e N. Since VGr is finite there is an n e N such that Qn+1 = Qn, and hence Q2n = Qn. Let us write R-j(v) := {u e VG : (u, v) e Rj}. For v e VGr we have R-j(v) e VGr (from Lemma 4.5) and hence Q-n(v) = R-sn(v) for all v e VGr. If Qn = IGr, then there are two distinct vertices u, v e VGr, such that (u, v) e Qn. NG(u) ^ NG(v) and G = G * Rsn allows us to conclude that R-sn(u) £ R-sn(v) and R-sn(v) £ R-sn(u). Hence, there is a vertex w, such that (w, u) e Qn and (w, v) e Qn. From (u, v) e Qn and (w, u) e Qn we conclude (w, v) e Qn o Qn = Q2n, contradicting to Q2n = Qn. Therefore Qn = IGr. Setting k = sn now implies the claim. Finally, we show VGr = VG. For any v e VG \ VGr, Lemma 4.6 implies the existence of w e VGr and m e N such that (w, v) e Rmk. However, we have claimed R-k (w) = {w}, hence R-mk(w) = {w}. This, however, implies NG(w) C NG(v) and thus contradicts property N. Therefore, VG = VGr and moreover Rk = IG. This R is an automorphism. □ 5 R-retraction A particularly important special case of ordinary graph homomorphisms are homomor-phisms to subgraphs, and in particular so-called retractions: Let H be a subgraph of G, a retraction of G to H is a homomorphism r : VG ^ VH such that r(x) = x for all x e VH. We introduced the graph cores in section 3 as minimal representatives of the homo-morphism equivalence classes. The classical and equivalent definition is the following: A (graph) core is a graph that does not retract to a proper subgraph. Every graph G has a unique core H (up to isomorphism), hence one can speak of H as the core of G, see [7]. Here, we introduce a similar concept based on relations between graphs. Again to obtain a structure related to graph homomorphisms, in this section we require all relations to have full domain unless explicitly stated otherwise. Definition 5.1. Let H be a subgraph of G. An R-retraction of G to H is a relation R such that G * R = H and (x, x) e R for all x e VH .If there is an R-retraction of G to H we say that H is a retract of G. Lemma 5.2. If H is an R-retract of G and K is an R-retract of H, then K is an R-retract of G. Proof. Suppose T is an R-retraction of H to K and S is an R-retraction of G to H. Then (G * S) * T = G * (S o T) = K. Furthermore (x, x) e T for all x e VK C VH, and (u, u) e S for all u e VH, hence (x, x) e S o T for all x e Vk. Therefore S o T is an R-retraction from G to K. □ Hence, the following definition is meaningful. Definition 5.3. A graph is R-reduced if there is no R-retraction to a proper subgraph. Thus, we can also speak about "the R-reduced graph of a graph G" as the smallest subgraph on which it can be retracted. We shall see below that the R-reduced graph of a graph is always unique up to isomorphism. We shall remark that R-reduced graphs differs from R-cores introduced in section 3, thus we choose an alternative name used also in homomorphism setting (cores are also called reduced graphs). Lemma 5.4. Let G be a graph with loops and o a vertex of G with a loop on it. Then the R-reduced graph of G is the subgraph induced by {o}. Proof. Let O be the graph induced by {o}, and R = {(x, o)|x e VG}, then it is easily seen R is a R-retraction of G to O. Moreover, since O has only one vertex, thus there is no R-retraction to its subgraphs. So O is a R-reduced graph of G. Conversely, let H be a R-reduced graph of G and denote by R the R-retraction from G to H. Then a loop of G must generate a loop of H via R, denote it by O. Similarly to above, we see O is a R-retract of H, hence it is also a R-retract of G (by Lemma 5.2). Therefore the definition of R-reduced graph implies H = O. □ In the remainder of this section, therefore, we will only consider graphs without loops. Lemma 5.5. If G is R-reduced, then G has property N. Proof. Suppose there are two distinct vertices x,y e VG with NG(x) C NG(y) and consider the induced graph G/x := G\Vg \ {x}] obtained from G by deleting the vertex x and all edges incident with x. The relation R = {(z, z)|z e VG \ {x}} U {(x, y)} satisfies G * R = G/x: the first part is the identity on G/x and already generates all necessary edges in G/x. The second part transforms edges of the form (x, z) e EG to edges (y, z). Since R has full domain and contains the identity relation restricted to G/x, it is an R-retraction of graph G, and hence G is not R-reduced. □ Proposition 5.6. A graph G is R-reduced if and only if it has no relation to a proper subgraph. Proof. The "if" part is trivial. Now we suppose that H is a proper induced subgraph of graph G with the minimal number of vertices such that there is a relation R satisfying G * R = H. Then H does not have a relation to a proper subgraph of itself. We claim that H has property N; otherwise, one can find a vertex u e VH and construct a retraction from H to H/u as in Lemma 5.5, which causes a contradiction. Denote R = R n (VH x VH), then K = H * R is a subgraph of H. From our assumptions on H we obtain K = H .By virtue of Theorem 4.7, R is induced by an automorphism of H. Hence R o R+ is again a relation of G to H that contains the identity on H, i.e., it is an R-retraction. □ Since graph cores are induced subgraphs and retractions are surjective they also imply relations. Proposition 5.6 is also a consequence of this fact. We refer to [7] for a formal proof. We call R a minimal R-retraction if there is no R-retraction R' such that R D R' D IH. Lemma 5.7. Let H be an R-retract of G. Then any minimal R-retraction of G to H is functional. Figure 5: A graph G and its core. Proof. Suppose R is a minimal R-retraction of G to H. If R is not functional, then there exist distinct x, y G VH such that (u, x), (u, y) G R. Hence we could always pick a vertex from {x, y} which is different of u, w.l.o.g. suppose it is x. Then R/(u, x) is an R-retraction, which contradicts minimality. To see this, set R' = R/(u, x), then R D R' D IH and moreover H = G * IH C G * R' C G * R = H, and thus G * R' = H. □ Proposition 5.8. A graph is R-reduced if and only if it is a graph core. Proof. If H is R-reduced from G there is an R-retraction from G to H which can be chosen minimal and hence by Lemma 5.7 is functional and hence is a homomorphism retraction. Conversely, a homomorphism retraction is also an R-retraction. Hence the R-reduced graphs coincide with the graph cores. □ Proposition 5.9. Suppose H is the core of graph G. If H * R = K then there is a relation R' such that G * R' = K .If K * S = G, then there is a relation S' such that K * S ' = H. Proof. Since H is the core of graph G, there is a relation Ri such that G * Ri = H .If H * R = K we have G * R1 * R = K and R' = R1 o R satisfies G * R' = K .If K * S = G we have K * S * R1 = H and S' = S o R1 satisfies K o S' = G. □ 5.1 Cocores In the classical setting of maps between graphs, one can only consider retractions from a graph to its subgraphs, since graph homomorphisms of an induced subgraph to the original graph are just the identity maps. In the setting of relations between graphs, however, it appears natural to consider relations with identity restriction between a graph and an induced subgraph. This gives rise to notions of R-coretraction and R-cocore in analogy with R-retractions and R-reduced graphs. Definition 5.10. Let H be a subgraph of graph G. An R-coretraction of H to G is a relation R such that H * R = G and (x, x) G R for all x G VH. We say that H is an R-coretract of G. Lemma 5.11. If H is an R-coretract of graph G and K is an R-coretract of H, then K is an R-coretract of G. Proof. Suppose T is an R-coretraction of K to H and S is an R-coretraction of H to G. Then (K * T) * S = K * (T o S) = G. Furthermore (x, x) G T for all x G VK C VH, and (v, v) G S for all v G VH, hence (x, x) G T o S for all x G VK. Therefore T o S is an R-coretraction from K to G. □ Hence, the following definition is meaningful. Definition 5.12. An R-coretract H of a graph G is an R-cocore of G if H does not have a proper subgraph that is an R-coretract of H (and hence of G). G cocore(G) Figure 6: A graph and its cocore Clearly, the reference to G is irrelevant: A graph G is an R-cocore if there is no proper subgraph of G that is an R-coretract of G. Similarly, we call R to be a minimal R-coretraction of H to G if there exists no R-coretraction R', such that R' c R. Lemma 5.13. Let H be an R-coretract of graph G, and let R be a minimal R-coretraction of H to G. Then the restriction of R to H equals IH. Proof. Suppose R n (VH x VH) = IH and define R1 = R \ {(x, y) G R : x, y G VH, x = y}. Then H * R1 C H * R = G. We claim that H * R1 = H * R and thus R1 is an R-coretraction of H to R, contradicting the minimality of R. To prove this claim, it is sufficient to show that any edge e G EG is contained in H * R1 .If e is not incident with any vertex in VH or e G EH, the conclusion is trivial. So we only need to consider e = (z, u) with z G EH and u G VG \ VH. Since G = H * R, one can find x1;x2 G VH such that (x1,z), (x2,u) G R and (x1;x2) G EH. Because H C H * (IH U (x1; z)) C H * (R n (VH x VH)) = H, we get NH(x1) C NH(z). It follows that (z, x2) G Eh and hence e = (z, u) G G * R1. □ Like R-reduced graphs, R-cocores satisfy a stringent condition on their neighborhood structure. Definition 5.14. A graph G satisfies property N* if, for every vertex x G VG, there is no subset Ux C VG \ {x} such that Ng(X) = J NG(y) (5.1) yeUx In other words, no neighborhood can be represented as the union of neighborhoods of other vertices of graph G. Proposition 5.15. G is an R-cocore if and only if G has property N*. Proof. Consider a vertex set Ux as in Definition 5.14 and suppose that there is a vertex x G VG such that NG(x) = Uy£U^ NG(y). Then the relation R := I \ (x, x) U {(y, x) : y G Ux} is an R-coretraction from G/x to G. Thus G is not a R-cocore. Conversely, suppose that G is not an R-cocore, let H be a coretract of G, and denote by R a minimal R-coretraction of H to G. Then, by Lemma 5.13, R n (VH x VH) = IH. Consider a vertex v G VG \ VH and set R-1(v) = {x1; • • • , x^}. Then N (v) = U i N (xž), contradicting property N*. □ Proposition 5.16. The R-cocore of G is unique up to isomorphism. Proof. We denote by N the collection of all open neighborhoods of vertices in G, i.e., N = {NG(x1), NG(x2), • • • , NG(xk)}, where VG = {x1, x2, • • • , xk}. From the definition of the R-cocore we know that the subcollection M of N consisting of all the open neighborhoods of vertices in R-cocore is a basis of N, i.e., any set in N can be expressed by the union of some sets in M. W.l.o.g., we denote the vertex set in a R-cocore C of G is {x1,x2, ••• ,xm} where m < k, then M = {NG(x1), NG(x2), ••• , NG(xm)}. We claim that any element in {NG(x1), NG(x2), • • • , NG(xm)} cannot be expressed as the union of other elements, i.e., M is a minimal basis. Otherwise, w.l.o.g., suppose Ng(x1) = Uxk Ng(x&),xfc G {x2,...,xm}. For any 1 < k < m, Ng(x&) = NG (xfc ) or Ng(x&) = Nc(xfc) U {xj|(xj,xfc) G EG,m + 1 < i < n}, so either NG(x1) = UxkNc(xfc),xfc G {x2,... ,xm} or Nc(x1) = UxkNc(xfc)U{xž 1 (xi , xk ) G Eg, m +1 < i < n, xk G {x2,..., xm}}, the former contradicts to Proposition 5.15, which implies any element in {NC(x1), NC(x2), • • • , NC(xm)} cannot be expressed as the union of other elements, the latter is impossible because {xj|(xj,xfc) g EG,m +1 < i < n, xk G {x2,... ,xm}} ^ C. Now we prove that this minimal basis is unique. Note that in N we view any vertex with the same neighborhood as the same, since any vertex in R-cocore has different neighborhoods. Let us consider two minimal sub-collections A, B. Neither contains the other by their minimality. Since everything is finite, let A G A/B be an element of minimal size. Now A can be expressed as a union of elements of B, which all need to be of smaller cardinality than A (or same but A G B), but A then contains all of them, letting A be expressed by a union of elements of A contradicting the minimality of A. □ These results allow us to construct an algorithm that computes the cocore of given graph G in polynomial time. First observe that the cocore of a graph G that contains isolated vertices is the disjoint union of cocore of the graph G' obtained from G by removing isolated vertices and the graph consisting of a single isolated vertex. It is thus sufficient to compute cocores for graphs without isolated vertices in Algorithm 2. Proposition 5.17. Suppose H is a cocore of G. If K * R = H, then there is a relation R' such that K * R' = G. If G * S = K, then there is a relation S' such that H * S' = K. Proof. Since H is a cocore of G, there exists an R-coretraction R1 such that H * R1 = G. If K * R = H, then letting R' = R o R1 implies K * R' = G. If G * S = K, we have H * R1 * S = K .Let S ' = R1 o S, then H * S' = K. □ 6 Computational complexity In this section we briefly consider the complexity of computational problems related to graph homomorphisms. The homomorphism problem Hom(H) takes as input some finite G and asks whether there is a homomorphism from G to H. The computational complexity of the homomorphism problem is fully characterized. It is known that Hom(H) is NP-complete if and only if H has no loops and contains odd cycles. All the other cases are polynomial, see [7]. The analogous problem for relations between graphs can be phrased as follows: The full relation problem Ful-Rel(H) takes as input some finite G and asks whether there is a relation with full domain from G and asks whether there is a relation from G to H. We Algorithm 2 The cocore of a graph Input: Graph G with loops and without isolated vertices specified by its vertex set V and the neighborhoods NG(i), i e V. 1: for i e V do 2: W (i) = 0 3: for j e V \ {i} do 4: if N (j ) C N (i) then 5: W (i) := W (i) U N (j ) 6: end if 7: end for 8: if W(i) = N(i) then 9: delete i from V 10: N (i) = 0 11: end if 12: end for 13: return G[V], the cocore of G. show that this problem can be easily converted to a related problem on surjective homomorphisms. The surjective homomorphism problem Sur-Hom(H) takes as input some finite G and asks whether there is a surjective homomorphism from G to H. Let x(K3) = 3. Lemma 2.7 and Lemma 2.8 do not hold for weak relations. For example, if G is a graph consisting of a single isolated vertex, then P3 * R = G and C3 * R = G, but there are no walk in G. With respect to complete graphs, weak relational composition also behaves different from strong composition. If Kk * R = H then R(i) can contain more that one vertex in VH. Compared to Proposition 2.12, we also obtain a different result: Theorem 7.2. There is a relation R such that Kk * R = H if and only if every connected component of H is a complete graph, and the number of connected components of H containing at least 2 vertices is at most k. Proof. If every connected component of H is a complete graph, denoted the vertex sets of the connected components containing at least 2 vertices by H1;..., Hm, m < k and the vertices of KU by 1, • • • , k. Let R = {(i, u)|i = 1, • • • , k, u G VH.} U {(j, v) : 1 < j < k, v G VH \ U m 1 VHi}. One easily checks that Kk * R = H. Conversely, let R be a relation satisfying Kk * R = H. Consider the set Ui = {u g VH|R-1(u) = {i}}. Then u and v are not adjacent for arbitrary u, v G Ui, while u is adjacent to w for every w G VH \ Ui. Hence H(Ui) is a connected component of H, which is also a complete graph. Given w g VH Ui, R-1(w) must have at least 2 vertices in Kk, hence w is adjacent to every vertex in H except itself; in other words, w is an isolated vertex in H. Therefore the number of connected components of H containing at least 2 vertices is no more than k. □ The results in subsection 3.1 also remain true for weak relations. Acknowledgments We thank Rostislav Matveev for helpful discussions in the beginning of the project and pointing out the decomposition as in Lemma 2.3, and to Jaroslav Nesetril for pointing out the equivalence of some complexity problems and enlightening questions for further works. L.Y. is grateful to the Max Planck Institute for Mathematics in the Sciences in Leipzig for its hospitality and continuous support. This work was supported in part by the NSFC (to L.Y.), the VW Foundation (to J.J. and P.F.S.), the Czech Ministry of Education, and ERC-CZ LL-1201, and CE-ITI of GACR (to J.H). References [1] E. Babson and D. Kozlov, Complexes of graph homomorphisms, Israel J. Math.. 152 (2006), 285-312. [2] M. Bodirsky, J. Kara and B. Martin: The complexity of surjective homomorphism problems — a survey, Discr. Appl. Math. 160 (2012), 1680-1690. [3] S. N. Dorogovtsev and J. F. F. Mendes, Evolution of Networks: From Biological Nets to the Internet and WWW, Oxford Univ. Press, Oxford, UK, 2003. [4] T. Feder and P. Hell, On realizations of point determining graphs, and obstructions to full homomorphisms, Discrete Math. 308 (2008), 1639-1652. [5] P. A. Golovach, B. Lidicky, B. Martin and D. Paulusma: Finding vertex-surjective graph homomorphisms, Acta Informatics 49 (2012), 381-394. [6] R. Hammack, W. Imrich and S. KlavZar, Handbook of Product Graphs, Discrete Mathematics and Its Applications, CRC Press, Boca Raton, FL, 2011. [7] P. Hell and J. Nešetril, Graphs and homomorphisms, Oxford University Press, Oxford, UK, 2004. [8] J. Hubicka, J. Fiala and Y. Long: Constrained homomorphism orders, in preparation, 2012. [9] W. Imrich and S. KlavZar, Product Graphs: Structure and Recognition, Wiley, New York, 2000. [10] J. Matoušek, Using the Borsuk-Ulam Theorem, Springer, Berlin, DE, 2003. [11] M. E. J. Newman, Networks: An Introduction, Oxford Univ. Press, Oxford, UK, 2010. [12] A. Schrijver, Combinatorial optimization: B, Springer, Berlin, 2003. [13] D. P. Sumner, Point determination in graphs, Discrete Mathematics 5 (1973), 179-187. [14] S. Wuchty and E. Almaas, Evolutionary cores of domain co-occurrence networks, BMC Evol. Biol. 5 (2005), 24. [15] Y. Xiao, B. D. MacArthur, H. Wang, M. Xiong and W. Wang, Network quotients: Structural skeletons of complex systems, Phys. Rev. E 78 (2008), 046 102. [16] A. Zhang: Protein interaction networks: computational analysis. Cambridge University Press, Cambridge, UK, 2009. MFA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 351-364 ARS MATHEMATICA CONTEMPORANEA CI-groups with respect to ternary relational structures: new examples Edward Dobson * Department of Mathematics and Statistics Mississipi State University PO Drawer MA, Mississipi State, MS 39762 USA and UP IAM, University of Primorska Muzejski trg 2, 6000 Koper, Slovenia Pablo Spiga Dipartimento di Matematica Pura e Applicata Universita degli Studi di Milano-Bicocca Via Cozzi 53, 20126 Milano, Italy Received 22 February 2012, accepted 25 October 2012, published online 14 January 2013 Abstract We find a sufficient condition to establish that certain abelian groups are not CI-groups with respect to ternary relational structures, and then show that the groups Z3 x Z2, Z7 x Z2, and Z5 x Z2 satisfy this condition. Then we completely determine which groups Z2 x Zp, p a prime, are CI-groups with respect to color binary and ternary relational structures. Finally, we show that Z25 is not a CI-group with respect to ternary relational structures. Keywords: CI-group, ternary relation. Math. Subj. Class.: 05C15, 05C10 1 Introduction In recent years, there has been considerable interest in which groups G have the property that any two Cayley graphs of G are isomorphic if and only if they are isomorphic by a group automorphism of G. Such a group is a called a CI-group with respect to graphs, and this problem is often referred to as the Cayley isomorphism problem. The interested * Project sponsored by the National Security Agency under Grant Number H98230-11-1-0179. E-mail addresses: dobson@math.msstate.edu (Edward Dobson), pablo.spiga@unimib.it (Pablo Spiga) reader is referred to [11] for a survey on CI-groups with respect to graphs. Of course, the Cayley isomorphism problem can and has been considered for other types of objects (see, for example, [9,14,16] for work on this problem on codes and on designs). Before proceeding we give the relevant definitions. (There are several equivalent definitions of combinatorial object [1,15], here we follow [13].) Definition 1.1. A k-ary relational structure is an ordered pair X = (V, E), with V a set and E a subset of Vk. Furthermore, a color k-ary relational structure is an ordered pair X = (V, (Ei,..., Ec)), with V a set and Ei,..., Ec pairwise disjoint subsets of Vk. If k = 2,3, or 4, we simply say that X is a (color) binary, ternary, or quaternary relational structure. A combinatorial object is a pair X = (V, E), with V a set and E a subset of uc=i V \ The following two definitions are due to Babai [1]. Definition 1.2. For a group G, define gL : G ^ G by gL(h) = gh, and let GL = [gL : g G G}. Then GL is a permutation group on G, called the left regular representation of G. We will say that a (color) k-ary relational structure X is a Cayley (color) k-ary relational structure of G if GL < Aut(X ) (note that this implies V = G). In general, a combinatorial object X will be called a Cayley object of G if GL < Aut(X ). Definition 1.3. For a class C of Cayley objects of G, we say that G is a CI-group with respect to C if whenever X, Y G C, then X and Y are isomorphic if and only if they are isomorphic by a group automorphism of G. It is clear that if G is a CI-group with respect to color k-ary relational structures, then G is a CI-group with respect to k-ary relational structures. Perhaps the most significant result in this area is a well-known theorem of Palfy [15] which states that a group G of order n is a CI-group with respect to every class of combinatorial objects if and only if n = 4 or gcd(n, y(n)) = 1, where ^ is the Euler phi function. In fact, in proving this result, Palfy showed that if a group G is not a CI-group with respect to some class of combinatorial objects, then G is not a CI-group with respect to quaternary relational structures. As much work has been done on the case of binary relational structures (i.e., digraphs), until recently there was a "gap" in our knowledge of the Cayley isomorphism problem for k-ary relational structures with k = 3. As additional motivation to study this problem, we remark that a group G that is a CI-group with respect to ternary relational structures is necessarily a CI-group with respect to binary relational structures, see [5, page 227]. Although Babai [1] showed in 1977 that the dihedral group of order 2p is a CI-group with respect to ternary relational structures, no additional work was done on this problem until the first author considered the problem in 2003 [5]. Indeed, in [5] a relatively short list of groups is given and it is proved that every CI-group with respect to ternary relational structures lies in this list (although not every group in this list is necessarily a CI-group with respect to ternary relational structures). Additionally, several groups in the list were shown to be CI-groups with respect to ternary relational structures. Recently, the second author [17] has shown that two groups given in [5] are not CI-groups with respect to ternary relational structures, namely Z3 k Q8 and Z3 x Q8. In this paper, we give a sufficient condition to ensure that certain abelian groups are not CI-groups with respect to ternary relational structures (Theorem 2.1), and then show that 1?2 X Z3, Z 2 x Z7, and Z| x Z5 satisfy this condition in Corollary 2.4 (and so are not CI-groups with respect to ternary relational structures). We then show that Zf x Z5 is a CI-group with respect to ternary relational structures. As the first author has shown [6] that Zf x Zp is a CI-group with respect to ternary relational structures provided that p > 11, we then have a complete determination of which groups Z3 x Zp, p a prime, are CI-groups with respect to ternary relational structures. Theorem A. The group Zf x Zp is a CI-group with respect to color ternary relational structures if and only if p G {3,7}. We will show that both Z3 x Z3 and Z3 x Z7 are CI-groups with respect to color binary relational structures. As it is already known that Zf is a CI-group with respect to binary relational structures [11], we have the following result. Corollary A. The group Z3 x Zp is a CI-group with respect to color binary relational structures for all primes p. We are then left in the situation of knowing whether or not any subgroup of Zf x Zp is a CI-group with respect to color binary or ternary relational structures, with the exception of Zf x Z7 with respect to color ternary relational structures (as Zf x Z7 is a CI-group with respect to color binary relational structures [10]). We show that Zf x Z7 is a CI-group with respect to color ternary relational structures (which generalizes a special case of the main result of [10]) and we prove the following. Corollary B. The group Zf x Zp is a CI-group with respect to color ternary relational structures if and only if p = 3. Finally, using Magma [2] and GAP [8], we show that Zf is not a CI-group with respect to ternary relational structures. We conclude this introductory section by recalling the following. Definition 1.4. For g, h in G, we denote the commutator g-1h-1gh of g and h by [g, h]. 2 The main ingredient and Theorem A We start by proving the main ingredient for our proof of Theorem A. Theorem 2.1. Let G be an abelian group and p an odd prime. Assume that there exists an automorphism a of G of order p fixing only the zero element of G. Then Zp x G is not a CI-group with respect to color ternary relational structures. Moreover, if there exists a ternary relational structure Z on G with Aut(Z) = {GL, a), then Zp x G is not a CI-group with respect to ternary relational structures. Proof. Since a fixes only the zero element of G, we have |G| = 1 (mod p) and so gcd(p, |G|) = 1. For each g G G, define g : Zp x G ^ Zp x G by g(i,j) = (i,j + g). Additionally, define t, y, a : Zp x G ^ Zp x G by t (i, j) = (i + 1,j), 7 (i, j) = (i,ai(j)), and a(i,j) = (i, a(j)). Then (Zp x G)L = {t, g : g G G). Clearly, {GL, a) = GL x {a) is a subgroup of Sym(G) (where GL acts on G by left multiplication and a acts as an automorphism). Note that the stabilizer of 0 in {GL, a) is {a). As a fixes only 0, we conclude that for every g G G with g = 0, the point-wise stabilizer of 0 and g in (GL, a) is 1. Therefore, by [18, Theorem 5.12], there exists a color Cayley ternary relational structure Z of G such that Aut(Z) = (GL, a). If there exists also a ternary relational structure with automorphism group (GL, a), then we let Z be one such ternary relational structure. Let U = {((0zp,g), (0Zp,h)):(0G,g,h) G E(Z)}, and S = {([g,Y](1, 0g), [g, Y](2,0g)): g g G} U U and define a (color) ternary relational structure X by V(X) = Zp X G and E(X) = {k(0zpXG, si, s2) : (si, s2) G S, k G (Zp x G)L}. If Z is a color ternary relational structure, then we assign to the edge k(0ZpXG, s1, s2) the color of the edge (0G,g,h) in Z if (si,s2) G U and (si,s2) = ((0Zp,g), (0Zp, h)), and otherwise we assign a fixed color distinct from those used in Z. By definition of X we have (Zp X G)l < Aut(X) and so X is a (color) Cayley ternary relational structure of Zp x G. We claim that a G Aut(X ). As a is an automorphism of Zp x G, we see that a G Aut(X) if and only if a(S) = S and a preserves colors (if X is a color ternary relational structure). By definition of Z and U, we have a(U) = U and a preserves colors (if X is a color ternary relational structure). So, it suffices to consider the case s G S - U, i.e., s = ([g, y](1 , 0), [g, y](2, 0)) for some g G G. Note that now we need not consider colors as all the edges in S - U are of the same color. Then ag(i, j) = (i, a(j) + a(g)) = a(g)a(i, j). Thus ag = a(g)a. Similarly, ag-1 = a(g) a. Clearly a commutes with y, and so a[g, Y] = [a(g), y]a. As a fixes (1,0) and (2,0), we see that a(s) = a([g,Y](1,0), [g, Y](2, 0)) = (a[g,Y](1, 0),a[g,Y](2, 0)) = ([a(g), Y]a(1, 0), [a(g), Y]a(2, 0)) = ([a"(g), y](1 , 0), [a(g), y](2, 0)) G (S - U). Thus a (S) = S, a preserves colors (if X is a color ternary relational structure) and a G Aut(X ). We claim that Y-1(Zp x G)ly is a subgroup of Aut(X). We set t' = y-1ty and g' = Y-1gY, for g G G. Note that t' = t«-1. As a G Aut(X), we have t' G Aut(X). Therefore it remains to prove that (g' : g G G) is a subgroup of Aut(X). Let e G E(X) and g G G. Then e = k((0,0), s), where s G S and k = tal, for some a G Zp, l G G. We have to prove that g'(e) G E(X) and has the same color as e (if X is a color ternary relational structure). Assume that s G U .As g'(i, j ) = (i, j + a-i(g)), by definition of U, we have g'[k((0,0), s)] G E(X) and has the same color of e (if X is a color ternary relational structure). So, it remains to consider the case s g S - U, i.e., s = ([g, y](1 , 0), [x, y](2, 0)) for some x g G. As before, we need not concern ourselves with colors because all the edges in S - U are of the same color. Set m = ka-a(g). Since ag = a(g)a and a,Y commute, we get ag' = (a(g))'a. Also observe that as G is abelian, g' commutes with h for every g, h G G. Hence g'k 7-1 gjral = 7 gr a7aa1 = 7 -1t "(/«v a^ga^l = Ta (a-a(g))'l = t al(a-a(g))' ---1 — = t 7 = ka-a(g)a-a(g) Y-1a-a(g)Y = m[a-a(g),7] and g'[k((0,0), s)] = g'k((0,0), [x,y](1, 0), [x,y](2, 0)) = m[(y/—a(g), 7]((0, 0), [g, y](1, 0), [g, y](2, 0)) = m((0,0), [or^g), y][(, y](1, 0), Mg), 7][g,7](2, 0)) = m((0,0), [aTa^x,Y](l,0), [a^^Yp,0)) G E(X). This proves that g' G Aut(X ). Since g is an arbitrary element of G, we have y-1Gly C Aut(X). As claimed, 7-1(Zp x G)ly is a regular subgroup of Aut(X) conjugate in Sym(Zp x G) to (Zp x G)L. We now see that Y = y(X) is a Cayley (color) ternary relational structure of Zp x G as Aut(Y) = YAut(X)y-1. We will next show that Y = X. Assume by way of contradiction that Y = X. As 7(0, g) = (0, g) for every g G G, the permutation 7 must map edges of U to themselves, so that 7(5 - U) = S — U. We will show that 7(S — U) = S — U. Note that we need not concern ourselves with colors as all the edges derived from S — U via translations of (Zp x G)L have the same color. Observing that ([g,7](l, 0), [g,7](2, 0)) = (g-17-137(l,0),g-17-137(2,0)) = (g-17-1ff(i, 0),rS-1ff(2,0)) = (g-17-1(i,g),rS-1(2,g)) = (g-1(l, a-1(g), 3-1(2, a-2(g)) = ((l, a (g) — g), (2, a (g) — g)), we see that 7(S — U) = {((l, g — a(g)), (2, g — a2 (g))) : g G G}. Moreover, as S — U = {(l, a-1(g) — g), (2, a-2(g) — g) : g G G}, we conclude that for each g G G, there exists hg G G such that g — a(g) = a (hg) — hg and g — a (g) = a (hg) — hg. Setting i : G ^ G to be the identity permutation, we may rewrite the above equations as (i — a)(g) = (a-1 — i)( hg ) and (i — a2)(g) = (a-2 — i)(hg ). Computing in the endomorphism ring of the abelian group G, we see that (a-2 — i) = (a-1 + i)(a-1 — i). Applying the endomorphism (a-1 + i) to the first equation above, we then have (a-1 + i)(i — a)(g) = (a-1 + i)(a-1 — i)(hg ) = (a-2 — i)(hg ) = (i — a2)(g). Hence (a-1 + i)(i — a) = i — a2, and so 0 = (a-1 + i)(i — a) — (i — a2) = ((a-1 + i) — (i + a))(i — a) = (a-1 — a)(i — a), (here 0 is the endomorphism of G that maps each element of G to 0). As a fixes only 0, the endomorphism i — a is invertible, and so we see that a-1 — a = 0, and a = a-1. However, this implies that p = |a| = 2, a contradiction. Thus j(S — U) = S — U and so Y = X. We set T = y(S), so that ((0,0), t) G E(Y) for every t G T, where if X is a color ternary relational structure we assume that y preserves colors. Now suppose that there exists ß G Aut(Zp X G) such that ß(X) = Y. Since gcd(p, |G|) = 1, we obtain that Zp x 1 a and 1Zp x G are characteristic subgroups of Zp x G. Therefore ß(i, j) = (ß1 (i), ß2 (j)), where ß1 G Aut(Zp) and ß2 G Aut(G). As ß fixes (0,0), we must have ß(S) = T. Observe that every element of S and of T is of the form ((0, g), (0, h)) or ((1, g), (2, h)), for some g, h G G. In particular, we must have ß1(1) = 1 and hence ß1 = 1. As a G Aut(X) and X = Y, we have ß2 G (a). Now observe that ß(U) = U. Thus ß2 G Aut(Z) = (Gl, a). We conclude that ß2 G (a), a contradiction. Thus X, Y are not isomorphic by a group automorphism of Zp x G, and the result follows. □ The following two lemmas, which in our opinion are of independent interest, will be used (together with Theorem 2.1) in the proof of Corollary 2.4. Lemma 2.2. Let G be a transitive permutation group on O. If x G Q and Staba(x) in its action on Q — {x} is the automorphism group of a k-ary relational structure with vertex set Q — {x}, then G is the automorphism group of a (k + 1)-ary relational structure. Proof. Let Y be a k-ary relational structure with vertex set Q — {x} and automorphism group Staba(x) in its action on Q — {x}. Let W = {(x, v1,..., vk) : (v1,..., vk) G E(Y)}, and define a (k + 1)-ary relational structure X by V(X) = Q and E(X) = {g(w) : w G W and g G G}. We claim that Aut(X) = G. First, observe that Staba(x) maps W to W. Also, if e G E(X) and e = (x, v1,..., vk) for some v1,..., vk G Q, then there exists (x, u1,..., uk) G W and g G G with g(x, u1,..., uk) = (x, v1,..., vk). We conclude that g(x) = x and g(u1,..., uk) = (v1,..., vk). Hence g G Staba(x) and (v1,..., vk ) G E(Y). Then W is the set of all edges of X with first coordinate x. By construction, G < Aut(X). For the reverse inclusion, let h G Aut(X). As G is transitive, there exists g G G such that g-1h G StabAut(X)(x). Note that as g G G, the element g-1 h G G if and only if h G G. We may thus assume without loss of generality that h(x) = x. Then h stabilizes set-wise the set of all edges of X with first coordinate x, and so h(W) = W and h induces an automorphism of Y. As Aut(Y) = Staba(x) < G, the result follows. □ Lemma 2.3. Let m > 2 be an integer and p G Sym(Zms) be a semiregular element of order m with s orbits. Then there exists a digraph r with vertex set Zms and with Aut(r) = (p). Proof. For each i G Zs, set Pi = (0,1,...,m — 1) ••• (im, im +1,...,im + m — 1) and Vi = {im+j : j G Zm}. We inductively define a sequence of graphs r0,..., rs-1 = r such that the subgraph of r induced by Z(i+1)m is Tj, the indegree in r of a vertex in Vj is i + 1, and Aut(Tj) = (pi), for each i g Zs. We set r0 to be the directed cycle of length m with edges {(j, j + 1) : j g Zm} and with automorphism group (p0). Inductively assume that rs—2, with the above properties, has been constructed. We construct rs-1 as follows. First, the subgraph of rs-1 induced by Z(s-1)m is rs-2. Then we place the directed m cycle {((s - 1)m + j, (s - 1)m + j + 1) : j G Zm} whose automorphism group is (((s - 1)m, (s - 1)m +1,..., (s - 1)m + m -1)) on the vertices in Vs-1. Additionally, we declare the vertex (s - 1)m to be outadjacent to (s - 2)m and to every vertex that (s - 2)m is outadjacent to that is not contained in Vs-2. Finally, we add to rs-1 every image of one of these edges under an element of (ps-1). By construction, ps-1 is an automorphism of rs-1 and the subgraph of rs-1 induced by Z(s-1)m is rs-2. Then each vertex in rs-1 n Vi has indegree i + 1 for 0 < i < s - 2, while it is easy to see that each vertex of Vs-1 has indegree s. Finally, if S g Aut(rs-1), then S maps vertices of indegree i + 1 to vertices of indegree i + 1, and so S fixes setwise Vj, for every i G Zs. Additionally, the action induced by (S) on Vs-1 is necessarily (((s - 1)m, (s - 1)m +1,..., (s - 1)m + m -1)) as this is the automorphism group of the subgraph of rs-1 induced by Vs-1. Moreover, arguing by induction, we may assume that the action induced by S on V (rs-1)-Vs-1 is given by an element of (ps-2). If S G (ps-1), then Aut(rs-1) has order at least m2, and there is some element of Aut(rs-1) that is the identity on V(rs-2) but not on Vs-1 and vice versa. This however is not possible as each vertex of Vs-2 is inadjacent to exactly one vertex of Vs-1. Then Aut(rs-1) = (ps-1) and the result follows. □ Corollary 2.4. None of the groups Z3 x Z2, Z7 x Z3, or Z5 x Z| are CI-groups with respect to ternary relational structures. Proof. Observe that Z2 has an automorphism a3 of order 3 that fixes 0 and acts regularly on the remaining 3 elements, and similarly, Z3 has an automorphism a7 of order 7 that fixes 0 and acts regularly on the remaining 7 elements. As a regular cyclic group is the automorphism group of a directed cycle, we see that ((Z3 xZ2)L, a3) and ((Z7 xZ3)L, a7) are the automorphism groups of ternary relational structures by Lemma 2.2. The result then follows by Theorem 2.1. Now Z2 has an automorphism a5 of order 5 that fixes 0 and acts semiregularly on the remaining 15 points. Then (a5) in its action on Z| - {0} is the automorphism group of a binary relational structure by Lemma 2.3. By Lemma 2.2, there exists a ternary relational structure with automorphism group ((Z5 x Z2 )L, a5). The result then follows by Theorem 2.1. □ Before proceeding, we will need terms and notation concerning complete block systems. Let G < Sym(n) be a transitive permutation group (acting on Zn, say). A subset B C Zn is a block for G if g(B) = B or g(B) n B = 0 for every g G G. Clearly Zn and its singleton subsets are always blocks for G, and are called trivial blocks. If B is a block, then g(B) is a block for every g G G, and the set B = {g(B) : g G G} is called a complete block system for G, and we say that G admits B. A complete block system is nontrivial if its blocks are nontrivial. Observe that a complete block system is a partition of Zn, and any two blocks have the same size. If G admits B as a complete block system, then each g g G induces a permutation of B, which we denote by g/B. We set G/B = {g/B : g e G}. The kernel of the action of G on B, denoted by fixG(B), is then the subgroup of G which fixes each block of B set-wise. That is, fixG(B) = {g e G : g(B) = B for all B e B}. For fixed B e B, we denote the set-wise stabilizer of B in G by StabG(B). That is StabG(B) = {g e G : g(B) = B}. Note that fixG(B) = nBeBStabG(B). Finally, for g e StabG(B), we denote by g|B the action induced by g on B e B. Note that Corollary 2.4, together with the fact that Z3 x Zp, p > 11, is a CI-group with respect to color ternary relational structures [6], settles the question of which groups Z3 x Zp are CI-groups with respect to color ternary relational structures except for p = 5. Our next goal is to show that Z3 x Z5 is a CI-group with respect to color ternary relational structures. From a computational point of view, the number of points is too large to enable a computer to determine the answer without some additional information. Lemma 6.1 in [6] is the only result that uses the hypothesis p > 11. For convenience, we report [6, Lemma 6.1]. Lemma 2.5. Let p > 11 be a prime and write H = Z\ x Zp. For every $ e Sym(H), there exists S e (HL, $-1HL$) such that (HL, S-1$-1HL$S) admits a complete block system consisting of 8 blocks of size p. In particular, to prove that Z23 x Z5 is a CI-group with respect to color ternary relational structures, it suffices to prove that Lemma 2.5 holds true also for the prime p = 5. We begin with some intermediate results which accidentally will also help us to prove that Z3 x Z7 is a CI-group with respect to color binary relational structures. (Here we denote by Alt(X) the alternating group on the set X and by Alt(n) the alternating group on {1,..., n}.) Lemma 2.6. Let p be an arbitrary divisor of n with p = 1 and let P1 and P2 be partitions of Zn where each block in P1 and P2 has size p. Then there exists $ e Alt(Zn) such that $(Pl) = P2. Proof. Let Pi = {Ai,..., An/p} and P2 = {Q1,..., Qn/p}. As Alt(n) is (n - 2)-transitive, there exists $ e Alt(n) such that $(Aj) = for i e {1,... ,n/p - 1}. As both P1 and P2 are partitions, we see that $(An/p) = Qn/p as well. □ Lemma 2.7. Let p be a prime, let G = Z3 x Zp and let S e Sym(G). Suppose that (GL,S-1GLS) admits a complete block system C with p blocks of size 8 such that Alt(C ) < Stab^GLj^-iGLa)(C)|C, where C e C. Then there exists 7 e (GL, S-1 GLS) such that (Gl, y-1S-1GLSj) admits a complete block system B with 4p blocks of size 2. Proof. Write H = (GL, S-1GLS), N = fixH(C) and M = [N, N]. Clearly both GL and S-1GLS are regular, and so both fixGL (C) and fixÄ-iGLÄ(C) are semiregular of order 8. Moreover, as fixGL(C)|C and fix^-iGl$(C)|c have exponent 2, we see that they are both consist of even permutations and hence they are contained in Alt(C), for each C e C. From the previous paragraph, as Alt(8) is simple and 1 = N|C < Stab(G,Ä-iGÄ) (C)|C, we have Alt(C ) = M |C, for every C e C .In particular, M is isomorphic to a subgroup of Alt(8)p. Denote by M(C) the pointwise stabilizer of C e C. Define an equivalence relation = on C by C = C' if and only if M(C) = M(C'). Clearly, = is an H-invariant equivalence relation because M-1HL4>) such that (HL, S-1$-1HL$S) admits a complete block system with blocks of size p. Proof. Let p G H be of order p. Then HL admits a complete block system B of £ blocks of size p formed by the orbits of (p). Note that as £ < p, a Sylow p-subgroup of Sym(H) has order p£. In particular, (p|B : B G B) is a Sylow p-subgroup of Sym(H) isomorphic to Zp, an elementary abelian p-group of order pe. Let P and P1 be Sylow p-subgroups of (Hl, $-1Hl $) containing p and 4>-1p4>, respectively. Then there exists S G (Hl, $-1Hl$) such that S-1P1S = P. Now, every element of HL normalizes (p), and so normalizes (p|B : B G B). This then implies that HL normalizes P because P = (p|b : B gB)H(Hl,$-1Hl$). Let B' be the complete block system of S-1$-1HL$S formed by the orbits of the cyclic group S-1$-1(p)$S. Arguing as above, we see that S-1$-1HL$S normalizes M = ((S-1 $-1p$S)|b' : B' G B') n (Hl, S-1$-1Hl$S). However, M is the Sylow p-subgroup of (Hl, S-1$-1Hl$S) containing S-1$-1(p)S$, which is P. Thus we have P < (Hl ,S-11 Hl $S), and the orbits of P form the required complete block system. □ Lemma 2.9. Let p > 5, H = Zf x Zp, and $ G Sym(H). Then either there exists S G (Hl, $-1Hl$) such that (HL, S-1$-1HL$S) admits a complete block system with blocks of size p or (HL, $-1HL$) admits a complete block system B with blocks of size 8 and (B)|b is isomorphic to a primitive subgroup of AGL(3,2), for B G B. Proof. Set K = (HL, $-1HL$). As H has a cyclic Sylow p-subgroup, we have by [4, Theorem 3.5A] that K is doubly-transitive or imprimitive. If K is doubly-transitive, then by [12, Theorem 1.1] we have Alt(H) < K. Now Lemma 2.6 reduces this case to the imprimitive case. Thus we may assume that K is imprimitive with a complete block system C. Suppose that the blocks of C have size £p, where £ = 2 or 4. Notice that p > £. As H is abelian, fixHL (C ) is a semiregular group of order £p and fix0-iHL^(C ) is also a semiregular group of order £p. Then, for C G C, both fixHL(C)|C and fix0-iHL0(C)|C are regular groups of order £p. Let C G C .By Lemma 2.8, there exists S G (fixHL (C ), fix0-iHL^(C )) such that (fixHL (C), fixđ-i0-iHL0đ(C))|C admits a complete block system BC consisting of blocks of size p. Let C' G C with C' = C. Arguing as above, there exists S' G (fix^L(C),fixđ-i0-iHL0đ(C)) such that (fixÄ[(C), fix^-i^-i^-iH^M'(C))|c admits a complete block system BC' consisting of blocks of size p. Note that the restriction S' | C is in (fixHL(C),fixđ-i0-inL0đ(C))|c and so (fix^(C),fixđ'-iđ-i0-inL0đđ'(C))|c admits BC as a complete block system. Repeating this argument for every block in C, we find that there exists S G (fixfli (C), fix0-i hl^(C)) suchthat (fixfli (C ), fixÄ-i0-i (C))|c admits a complete block system BC consisting of blocks of size p. Let B = UCBC. We claim that B is a complete block system for (HL, S-V-1HL^S), which will complete the argument in this case. Let p G HL be of order p. By construction, p G fixHL (B). As H is abelian, fixHL (C) |C is abelian, for every C G C. Then BC is formed by the orbits of some subgroup of fixHL (C)|C of order p, and as (p)|C is the unique subgroup of fixHL (C)|C of order p, we obtain that BC is formed by the orbits of (p)| C. Then B is formed by the orbits of (p) < HL and B is a complete block system for HL. An analogous argument for S-1^-1 (p)^S gives that B is a complete block system for S-1^-1HL^S. Then B is a complete block system for (HL, S-1^-1Hl^S) with blocks of size p, as required. Suppose that the blocks of C have size 8. Now HL /C and ^-1HL^/C are cyclic of order p, and as Zp is a CI-group [1, Theorem 2.3], replacing by a suitable conjugate, we may assume that (HL, ^-1HL^)/C = HL/C. Then K/C is regular and StabK(C) = fixK (C), for every C G C. Suppose that StabK (C)|C is imprimitive, for C G C. By [4, Exercise 1.5.10], the group K admits a complete block system D with blocks of size 2 or 4. Then K/D has degree 2p or 4p and, by Lemma 2.8, there exists S G K such that (HL, S-1^-1HL^S)/D admits a complete block system B' with blocks of size p. In particular, B' induces a complete block system B'' for (HL, S-1^-1HL^S) with blocks of size 2p or 4p, and we conclude by the case previously considered applied with C = B''. Suppose that StabK (C)|C is primitive, for C G C. If StabK (C) | C > Alt(C), then the result follows by Lemma 2.7, and so we may assume this is not the case. By [12, Theorem 1.1], we see that StabK(C)|C < AGL(3, 2). The result now follows with B = C. □ Corollary 2.10. Let H = ZfxZ5 and ^ G Sym(H). Then there exists S G (HL, ^-1HL^) such that (Hl, S-1^-1Hl^S) admits a complete block system with blocks of size 5. Proof. Set K = (HL, ^-1HL^). By Lemma 2.9, we may assume that K admits a complete block system B with blocks of size 8 and with StabK (B)|b < AGL(3,2), for B G B. As |AGL(3,2)| = 8 • 7 • 6 • 4, we see that a Sylow 5-subgroup of K has order 5. Let (p) be the subgroup of HL of order 5. So (p) is a Sylow 5-subgroup of K. Then ^-1(p)^ is also a Sylow 5-subgroup of K, and by a Sylow theorem there exists S G K such that S-1^-1(p)^S = (p). We then see that (HL, S-1^-1HL^S) has a unique Sylow 5-subgroup, whose orbits form the required complete block system B. □ We are finally ready to prove Theorem A. Proof of Theorem A. If p is odd, then the paragraph following the proof of Corollary 2.4 shows that it suffices to prove that Lemma 2.5 holds for the prime p = 5. This is done in Corollary 2.10. If p = 2, then the result can be verified using GAP or Magma. □ 3 Proof of Corollaries A and B Before proceeding to our next result we will need the following definitions. Definition 3.1. Let G be a permutation group on Q and k > 1. A permutation a G Sym(Q) lies in the k-closure G(k) of G if for every k-tuple t G there exists gt G G (depending on t) such that a(t) = gt(t). We say that G is k-closed if the permutations lying in the k-closure of G are the elements of G, that is, G(k) = G. The group G is k-closed if and only if there exists a color k-ary relational structure X on Q with G = Aut(X), see [18]. Definition 3.2. For color digraphs r1 and r2, we define the wreath product of r1 and r2, denoted r1 l r2, to be the color digraph with vertex set V(r1) x V(r2) and edge set E1 U E2, where E1 = {((^1,^1), (x^)) : X1 G V(rO, (y1,ys) G E(r2)} and E2 = {((x1,y1), (x2,y2)) : (x^) G E(^),y1,y2 G V(r2)}. The edge ((x1, y1), (x1, y2)) G E1 is colored with the same color as (y1, y2) in r2 and the edge ((x1, y1), (x2, y2)) G E2 is colored with the same color as (x1, x2) in r1. Definition 3.3. Let G < Sym(X) and let H < Sym(Y). We define the wreath product of G and H, denoted by G l H, to be the semidirect product G k Hx , where HX is the direct product of | X| copies of H (labeled by the elements of X) and where G acts on HX as a group of automorphisms by permuting the coordinates according to its action on X. The group G l H has a natural faithful action on X x Y, where for (x, y) G X x Y the element g G G acts via (x, y) ^ (g(x),y) and the element (hz)zeX G HX acts via (x, y) ^ (x, hx(y)). We refer the reader to [4, page 46] for more details on this construction. The following very useful result (see [1, Lemma 3.1]) characterizes CI-groups with respect to a class of combinatorial objects. Lemma 3.4. Let H bea group and let K bea class of combinatorial objects. The following are equivalent. 1. H is a CI-group with respect to K, 2. whenever X is a Cayley object of H in K and ^ G Sym(H) such that ^-1HL^ < Aut(X), then HL and ^-1HL^ are conjugate in Aut(X). Proof of Corollary A. From Theorem A, it suffices to show that Z2 x Z3 and Z3 x Z7 are CI-groups with respect to color binary relational structures. As the transitive permutation groups of degree 24 are readily available in GAP or Magma, it can be shown using a computer that Z32 x Z3 is a CI-group with respect to color binary relational structures. It remains to consider H = Z3 x Z7. Fix ^ G Sym(H) and set K = {HL, ^-1HL^). Assume that there exists S G K such that {Hl, S-1^-1Hl^S) admits a complete block system with blocks of size 7. Now, it follows by [6] (see the two paragraphs following the proof of Corollary 2.4) that HL and S-V-1Hl^S are conjugate in {Hl, S-1^-1Hl^S)(3). Since {Hl,S-1^-1Hl^S)(3) < {Hl, S-1^-1Hl^S)(2), the corollary follows from Lemma 3.4 (and from Definition 3.1). Assume that there exists no S g K such that {HL, S-1^-1 HL^S) admits a complete block system with blocks of size 7. By Lemma 2.9, the group K admits a complete block system C with blocks of size 8 and fixK (C)|C is isomorphic to a primitive subgroup of AGL(3, 2), for C G C. Suppose that 7 and |fixK(C)| are relatively prime. So, a Sylow 7-subgroup of K has order 7. We are now in the position to apply the argument in the proof of Corollary 2.10. Let {p) be the subgroup of HL of order 7. Then (p)^ is a Sylow 7-subgroup of K, and by a Sylow theorem there exists S G K such that S-14>-1 {p) ^S = {p). We then see that {HL, S-1^-1HL^S) has a unique Sylow 7-subgroup, whose orbits form a complete block system with blocks of size 7, contradicting our hypothesis on K. We thus assume that 7 divides |fixK(C) | and so fixK(C) acts doubly-transitively on C, for C G C. Fix C G C and let L be the point-wise stabilizer of C in fixK(C). Assume that L = l. Now, we compute K(2) and we deduce that HL and are conjugate in from which the corollary will follow from Lemma 3.4. As L< fixK (C), we have L|c < fixK(C)|C, for every C' G C. As a nontrivial normal subgroup of a primitive group is transitive [19, Theorem 8.8], either L|c is transitive or L|c = l. Let r be a Cayley color digraph on H with K(2) = Aut(r). Let C = {Ci : i G Z7} where Ci = {(x1, x2, x3, i) : x1,x2,x3 G Z2}, and assume without loss of generality that C = C0. Suppose that there is an edge of color k from some vertex of Ci to some vertex of Cj, where i = j. Then there is an edge of color k from some vertex of C0 to some vertex of Cj-i. Additionally, j — i generates Z7, so there is a smallest integer s such that L|cs(j_i) = l while L|C(s+1)(j_i) is transitive. As there is an edge of color k from some vertex of Cs(j-i) to some vertex of C(s+1)(j-i), we conclude that there is an edge of color k from every vertex of Cs(j-i) to every vertex of C(s+1)(j-i). This implies that there is an edge of color k from every vertex of Ci to every vertex of C j, and then r is the wreath product of a Cayley color digraph r1 on Z7 and a Cayley color digraph r2 on Z2. Since fixK(C) is doubly-transitive on C, we have Aut(r2) = Sym(8). Therefore K(2) = Aut(r1) i Aut(r2) = Aut(r1) i Sym(8). By [7, Corollary 6.8] and Lemma 3.4 Hl and are conjugate in K(2). We henceforth assume that L = l, that is, fixK(C) acts faithfully on C, for each C G C. Define an equivalence relation on H by h = k if and only if it holds StabfiXK (C) (h) = StabfiXK(C)(k). The equivalence classes of = form a complete block system D for K. As fixK (C)|c is primitive and not regular, each equivalence class of = contains at most one element from each block of C. We conclude that D either consists of 8 blocks of size 7 or each block is a singleton. Since we are assuming that K has no block system with blocks of size 7, we see that each block of D is a singleton. Fix C and D in C with C = D and h G C. Now, StabfiXK(C)(h) is isomorphic to a subgroup of GL(3,2) and acts with no fixed points on D. From [4, Appendix B]), we see that AGL(3,2) is the only doubly-transitive permutation group of degree 8 whose point stabilizer admits a fixed-point-free action of degree 8. Therefore fixK(C) = AGL(3, 2). Additionally, StabfiXK(C)(h)|D is transitive on D. Suppose that r is a color digraph with K(2) = Aut(r) and suppose that there is an edge of color k from h to l G E, with E G C and E = D. Then StabfiXK(C)(h)|E is transitive, and so there is an edge of color k from h to every vertex of E. As fixK (C) is transitive on both C and E, we see that there is an edge of color k from every vertex of C to every vertex of D. We conclude that r is a wreath product of two color digraphs r1 and r2, where r1 is a Cayley color digraph on Z7 and r2 is either complete or the complement of a complete graph, and K(2) = Aut(r1) i Sym(8). The result then follows by the same arguments as above. □ Proof of Corollary B. From Corollary 2.4 and Theorem A, it suffices to show that Z2 x Z7 is a CI-group with respect to color ternary relational structures. As the transitive permutation groups of degree 28 are readily available in GAP or Magma, it can be shown using a computer that Z2 x Z7 is a CI-group with respect to color ternary relational structures. (We note that a detailed analysis similar to the proof of Corollary A for the group Z2 x Z7 also gives a proof of this theorem.) □ 4 Concluding remarks In the rest of this paper, we discuss the relevance of Theorem A to the study of CI-groups with respect to ternary relational structures. Using the software packages [2] and [8], we have determined that Z2 is not a CI-group with respect to ternary relational structures. Here we report an example witnessing this fact: the group G has order 2048, V and W are two nonconjugate elementary abelian regular subgroups of G, and X = ({1,..., 32}, E) is a ternary relational structure with G = Aut(X ). The group V is generated by (1.2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32), (1.3)(2,4)(5,7)(6,8)(9,11)(10,12)(13, 15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32), the group W is generated by (1.2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32), (1.3)(2,4)(5,7)(6,8)(9,11)(10,12)(13, 15)(14,16)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31), (1,5)(2,6)(3,7)(4,8)(9,14)(10,13)(11,16)(12,15)(17,22)(18,21)(19,24)(20, 23)(25,29)(26,30)(27,31)(28,32), (1,9)(2,10)(3,11)(4,12)(5,14)(6,13)(7,16)(8,15)(17,27)(18,28)(19,25)(20,26)(21,32)(22,31)(23,30)(24,29), (1,17)(2,18)(3,20)(4,19)(5,22)(6,21)(7,23)(8,24)(9,27)(10,28)(11,26)(12,25)(13,32)(14,31)(15,29)(16,30), the group G is generated by V, W, (25,26)(27,28)(29,30)(31,32),(1,11)(2,12)(3,9)(4,10)(5,13)(6, 14)(7,15)(8,16)(17,19)(18,20)(25,27)(26,28), the set E is defined by {g((1, 3, 9)),g((1, 5, 25)) : g G G}. Definition 4.1. For a cyclic group M = (g) of order m and a cyclic group (z) of order 2d, d > 1, we denote by D(m, 2d) the group (z) k M with gz = g-1. Combining Theorem A with [5, Theorem 9], [5, Lemma 6], the construction given in [17] and the previous paragraph, we have the following result which lists every group that can be a CI-group with respect to ternary relational structures (although not every group on the list needs to be a CI-group with respect to ternary relational structures). Theorem 4.2. If G is a CI-group with respect to ternary relational structures, then all Sy-low subgroups of G are of prime order or isomorphic to Z4, Zd, 1 < d < 4, or Qg. Moreover, G = U X V, where gcd(|U |, |V |) = 1, U is cyclic of order n, with gcd(n, y(n)) = 1, and V is one of the following: 1. Zd, 1 < d < 4, D(m, 2), or D(m, 4), where m is odd and gcd(nm, y(nm)) = 1, 2. Z4, Qg. Furthermore, (a) if V = Z4, Qg, or D(m, 4) and p | n is prime, then 4 / (p - 1), (b) if V = Zd, d > 2, or Qg, then 3 / n, (c) if V = Zd, d > 3, then 7 / n, (d) if V = Z|, then 5 / n. References [1] L. Babai, Isomorphism problem for a class of point-symmetric structures, Acta Math. Acad. Sci. Hungar. 29 (1977), 329-336. [2] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), 235-265, Comput. algebra and number theory (London, 1993). [3] P. J. Cameron, Finite permutation groups and finite simple groups, Bull. London Math. Soc. 13 (1981), 1-22. [4] J. D. Dixon and B. Mortimer, Permutation groups, Graduate Texts in Mathematics, vol. 163, Springer-Verlag, New York, 1996. [5] E. Dobson, On the Cayley isomorphism problem for ternary relational structures, J. Combin. Theory Ser. A 101 (2003), 225-248. [6] E. Dobson, The isomorphism problem for Cayley ternary relational structures for some abelian groups of order 8p, Discrete Math. 310 (2010), 2895-2909. [7] E. Dobson and J. Morris, Automorphism groups of wreath product digraphs, Electron. J. Com-bin. 16 (2009), R#17, 30 p. [8] The GAP Group, Gap - groups, algorithms, and programming, version 4.4, (2005), http: //www.gap-system.org. [9] W. C. Huffman, V. Job and V. Pless, Multipliers and generalized multipliers of cyclic objects and cyclic codes, J. Combin. Theory Ser. A, 62, (1993), 183-215. [10] I. Kovacs and M. Muzychuk, The group Zp x Zq is a CI-group,Comm. Algebra 37 (2009), 3500-3515. [11] C.H. Li, On isomorphisms of finite Cayley graphs—a survey, Discrete Math. 256 (2002), 301334. [12] C.H. Li, The finite primitive permutation groups containing an abelian regular subgroup, Proc. London Math Soc. (3) 87 (2003), 725-747. [13] M. Muzychuk, On the isomorphism problem for cyclic combinatorial objects, Discrete Math. 197 (1999), 589-606. [14] M. Muzychuk, A solution of an equivalence problem for semisimple cyclic codes, ArXiv:1105.4320v1. [15] P. P. Palfy, Isomorphism problem for relational structures with a cyclic automorphism, European J. Combin. 8 (1987), 35-43. [16] K. T. Phelps, Isomorphism problems for cyclic block designs, Combinatorial design theory, North-Holland, 1987, 149, 385-391 [17] P. Spiga, On the Cayley isomorphism problem for a digraph with 24 vertices, Ars Math. Contemp. 1 (2008), 38-43. [18] H. Wielandt, Permutation groups through invariant relations and invariant functions, lectures given at The Ohio State University, Columbus, Ohio, 1969. [19] H. Wielandt, Finite permutation groups, Translated from the German by R. Bercov, Academic Press, New York, 1964. d MFA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 365-388 On the rank two geometries of the groups PSL(2, q): part II* Francis Buekenhout, Julie De Saedeleer, Dimitri Leemans f Université Libre de Bruxelles, Departement de Mathématiques - C.P.216 Boulevard du Triomphe, B-1050 Bruxelles Received 30 November 2010, accepted 25 January 2013, published online 15 March 2013 Abstract We determine all firm and residually connected rank 2 geometries on which PSL(2, q) acts flag-transitively, residually weakly primitively and locally two-transitively, in which one of the maximal parabolic subgroups is isomorphic to A4, S4, A5, PSL(2, q') or PGL(2, q' ), where q' divides q, for some prime-power q. Keywords: Projective special linear groups, coset geometries, locally s-arc-transitive graphs. Math. Subj. Class.: 51E24, 05C25 1 Introduction In [5], we started the classification of the residually weakly primitive and locally two-transitive coset geometries of rank two for the groups PSL(2, q). The aim of this paper is to finish this classification. It remains to focus on the cases in which one of the maximal parabolic subgroups is isomorphic to A4, S4, A5, PSL(2, q') or PGL(2, q') where q' divides q. For motivation, basic definitions, notations and context of the work we refer to [5]. In Section 3, we sketch the proof of our main result: Theorem 1.1. Let G = PSL(2, q) andr(G; {G0, G1 ,G0nGi}) be a locally two-transitive RWPRI coset geometry of rank two. If G0 is isomorphic to one of A4, S4, A5, PSL(2, q') or PGL(2, q'), where q' divides q, then r is isomorphic to one of the geometries appearing in Table 1, Table 2, Table 3, Table 4, Table 5, and Table 6. *This paper is a part of SIGMAP'10 special issue Ars Math. Contemp. vol. 5, no. 2. t Supported by the "Communaute Francaise de Belgique - Actions de Recherche Concertees" E-mail addresses: fbueken@ulb.ac.be (Francis Buekenhout), judesaed@ulb.ac.be (Julie De Saedeleer), dleemans@ulb.ac.be (Dimitri Leemans) Go = A5 q = 4r with r prime Go n Gi Gi J Geom. J Geom. Extra conditions loc. (G, s)- up to conj. up to isom. on q arc-trans. g. ri Dio Dso 1 1 3H odd s = 3 r2 A4 Ei6 : 3 1 1 q = 16 s = 3 rs A4 Ei6 : 3 5 2 q = 64 s = 3 r,, AA P . „ 4r-1-1 2(4r-2-i) + s.2r-2 s 3 s sr Go = A5 q = p = ±1(5) with p odd prime Go n Gi Gi J Geom. J Geom. Extra conditions loc. (G, s)- up to conj. up to isom. on q arc-trans. g. r5 Dio D2o 2 1 q = ±1(20) s = 3 r6 Dio Dso 2 1 q = ±1(30) s = 3 r7 Dio A5 2 1 211 even s = 2 rs Dio A5 1 1 21 odd s = 2 rg A4 S4 2 1 q = ±1(40) or q = ±9(40) s = 3 rio A4 A5 2 1 q = ±1(40) or q = ±9(40) s = 2 rii A4 A5 1 1 q = ±11(40) or q = ±19(40) s = 2 Go = A5 q = p2 = -1(5) with p odd prime Go n Gi Gi J Geom. J Geom. Extra conditions loc. (G, s)- up to conj. up to isom. on q arc-trans. g. ri2 Dio D2o 2 1 q = -1(20) s = 3 ris Dio Dso 2 1 q = -1(30) s = 3 ri4 Dio A5 2 1 2+1 even s = 2 ri5 Dio A5 1 1 odd io s = 2 ri6 A4 S4 2 1 q = -1(40) or q = 9(40) s = 3 ri7 A4 A5 2 1 q = -1(40) or q = 9(40) s = 2 ris A4 A5 1 1 q = = -11(40) or q = 19(40) s = 2 Table 1: The RWPRI and (2T)1 geometries with G0 = A5. Go = A4 q = p > 3 and q = 3,13, 27, 37(40) or q = 5 Go n Gi Gi J Geom. J Geom. Extra conditions locally(G, s)-arc- up to conj. up to isom. on q transitive graphs ri r2 3 3 Z6 D6 1 q+i 6 1 q+1 + 1 6 +1 2 q = 13, 37, 83,107(120) ^ odd s = 3 s = 3 rs r4 3 3 D6 D6 q-i 2+1 6 q-1 +1 6 +1 6+1 12 ^ odd 41 even 6 s = 3 s = 3 r5 3 D6 q-i 6 q-i 12 2-1 even 6 s = 3 r6 3 A4 q+i 1 s 1 q+i 6 3 | q + 1 s = 2 r7 3 A4 2-1 _ 1 s 1 q-i 6 3 | q - 1 s = 2 Table 2: The RWPRI and (2T)i geometries with Go = A4 Go = S4 q = p> 2 and q = ±1(8) Go n Gi Gi J Geom. J Geom. Extra conditions locally(G, s)-arc- up to conj. up to isom. on q transitive graphs ri Da D12 2 1 q = ±1(24) s = 3 r2 Da Dis 2 1 q = ±1(72) or q = ±17(72) s = 3 Ts Da S4 2 1 ^ even s = 2 r4 Da S4 1 1 2±i odd a s = 2 r5 Ds Dia 2 1 q = ±1(16) s = 7 ra Ds D24 2 1 q = ±1(24) s = 3 r7 Ds S4 2 1 ^ even s = 4 Ts Ds S4 1 1 sii odd s = 4 Tg A4 A5 2 1 q = ±1(40) or q = ±9(40) s = 3 Table 3: The RWPRI and (2T)i geometries with Go = S4. Go = PSL(2, 2n) q = 2nm, with m prime Go n Gi Gi J Geom. J Geom. Extra conditions loc.(G, s)-arc- up to up to on q trans. conj. isom. graphs ri E2n : (2n - 1) E2mn : (2n — 1) 1 1 m = 2, n = 1 s = 3 r2 2 Da 1 1 q = 4; n = 1, m = 2 s = 2 rs 2 22 1 1 q = 4; n = 1, m = 2 s = 3 r4 3 A4 1 1 q = 4; n = 1, m = 2 s = 3 r5 Dio Dso 1 1 q = 4m ; n = 2; ^odd s = 3 Table 4: The RWPRI and (2T)i geometries with Go = PSL(2, 2n). Go = PSL(2,pn) q = pnm, p and m odd primes Go n Gi Gi J Geom. up to conj. J Geom. up to isom. Extra conditions on q locally(G, s)-arc-transitive graphs ri r2 3 3 A4 A4 3m-i — 1 8 3m-i-i 2m 2 q = 3m; n = 1, m = 3 q = 27; n = 1, m = 3 s = 2 s = 2 Table 5: The RWPRI and (2T)i geometries with Go = PSL(2, q'), q' odd. Go = PGL(2,pn) q = p2n, with p odd prime Go n Gi Gi J Geom. J Geom. Extra conditions loc. (G, s)-arc- up to conj. up to isom. on q transitive graphs ri Epn : (pn — 1) Ep2n : (pn — 1) 2 1 none s = 3 r2 PSL(2,pn) A5 2 1 q = 9 s = 3 rs Ds PGL(2, 3) 1 1 q = 9 s = 4 Table 6: The RWPRI and (2T)1 geometries with G0 = PGL(2, q'). Observe that, geometry r5 in Table 4 is exactly geometry ri in Table 1. In Section 4, we recall the subgroup lattice of PSL(2, q), and we give the two-transitive representations of the maximal subgroups. In Section 5, we prove Theorem 1.1, which is based on the proof of Propositions 5.5, 5.6, 5.10, 5.12, 5.16 and 5.21. For that purpose, we determine the rank two RWPRI and (2T)1 geometries of PSL(2, q) and their number, up to isomorphism and up to conjugacy. The existence of such geometries is equivalent to the existence of a locally 2-arc transitive bipartite graph for which the action of G is primitive on one of the bipartite halves (see [8]). Our result is also a part of the program initiated in [8]. These graphs are interesting in their own right because of the numerous connections they have with other fields of mathematics (see [8] for more details). We also refer to the classification of these graphs for almost simple groups with socle a Ree simple group Ree(q) (see [7]). In terms of locally 2-arc-transitive graphs, we obtain here the classification of these graphs with one vertex-stabilizer maximal in PSL(2, q) and isomorphic to A4, S4, A5, PSL(2, q') or PGL(2, q'). The last column of Table 1, Table 2, Table 3, Table 4, Table 5 and Table 6 gives, for each geometry r, the value of s such that r is a locally s-arc-transitive but not a locally (s + 1)-arc-transitive graph. In section 6, we determine the exact value of s in all cases that are not current by the method of Leemans. In Tables 1, 2, 3, 4, 5, 6 and 9 most values are s = 2 or s = 3, but there are some spectacular examples with larger values of s. Indeed we obtain a locally 4-arc transitive graph and a locally 7-arc transitive graph. As one of the referees pointed out, the (G, 2)-arc transitive graphs with L2(q) < G < PrL2(q) were classified by Hassani, Nochefranca and Praeger in [9]. Therefore, they already classified the geometries of Theorem 1.1 in which G0 n G1 is of index two in one of G0 or G1. Our proof of Theorem 1.1 uses a completely different approach. In cases where our work overlaps with [9], the results are the same. Also, in Table 3, geometry r5 is due to Wong in [22] and geometries r7 and r8 are the Biggs-Hoare graphs in [1] (see also [14], Table 1). 1.1 Aknowledgement The authors warmly thank the referees for many corrections and improvements to the initial text. 2 Definitions and notation For basic notions on coset geometries and locally s-arc-transitive graphs needed to understand this paper we shall freely use the definitions from Section 2 in [5]. Let us nevertheless recall concepts related to isomorphism. Let G be a group and Aut(G) be its automorphism group. The coset geometries r(G; {G0,G1}) and r(G; {G0, G1}) are conjugate (resp. isomorphic) provided there exists an element g G G (resp. g G Aut(G)) such that {G0,G1} = {G0,G1} (resp. {g(G0), g(G^} = {G0,G1}). We classify geometries up to conjugacy and up to isomorphism. That is, for each triple {G0, G1, G0 n G1}, we give the number of corresponding classes of geometries with respect to conjugacy and isomorphism. 3 Sketch of the proof of Theorem 1.1 Let G = PSL(2, q). Let G0 and G1 be subgroups of G and let G01 = G0 n G1. The RWPRI condition in rank two requires that either G0 or G1 is a maximal subgroup of G and that G01 is a maximal subgroup of G0 and G1. The (2T) 1 condition requires that both G0 and G1 act two-transitively on the respective cosets of G01. We break down the task by classifying those geometries with a fixed subgroup G0. Since we may assume without loss of generality that G0 is maximal in G, we follow Tables 7 and 8 that give all maximal subgroups of PSL(2, q). The number of RWPRI and (2T)1 geometries of rank 2 depends on the value of q = pn. More precisely, it usually depends on whether p = 2 or p = 2. Knowing that q = pn with p a prime, the two cases are q = 2n or q odd. The way we work to determine the RWPRI and (2T)1 geometries of rank two always follows the same path. To achieve our goal we first choose a subgroup G0, which is a maximal subgroup of G = PSL(2, q). Then, using the results obtained in Proposition 4.6, we determine the possibilities for G01 := G0 n G1. They are the two-transitive pairs (G0, G01 ). At last, in Section 5 we determine the possible subgroups G1 of PSL(2, q) such that (G1, G01) is a two-transitive pair. Finally, we determine for each triple (G0, G1, G01) the number of geometries it gives rise to, up to conjugacy and up to isomorphism. 4 Structure of subgroups of PSL(2, q) To follow the approach described above, we first recall the list of subgroups of the pro-jective special linear groups PSL(2, q). We then give the list of maximal subgroups of PSL(2, q). Finally we determine the two-transitive representations of the maximal subgroups of PSL(2, q) in order to be able to check the (2T)1 property easily. 4.1 The subgroups of PSL(2, q) We recall the complete subgroup structure of PSL(2, q) for which we refer to Dickson [6], Moore [15], Huppert [10] and Suzuki [16]. In the statement of Lemma 4.1, we make use of the phrasing due to O. H. King [11]. Lemma 4.1. [Dickson-Moore] The group PSL(2, q) of order , where q = pn (p prime), contains exactly the following subgroups: 1. The identity subgroup. 2. A single class of q +1 conjugate elementary abelian subgroups of order q, denoted by Eq. 3. A single class of q(q+1) conjugate cyclic subgroups of order d, denoted by either Zd or d; for every divisor d of q — 1 for q even and 2—1 for q odd, with d > 1. 4. A single class of q(q—1) conjugate cyclic subgroups of order d,denoted by either Zd or d; for every divisor d of q + 1 for q even and for q odd, with d > 1. 5. • For q odd, a single class of q(q4—1) dihedral groups of order 2d, denoted by Did, for every divisor d of 2—1 with 2—1 odd, with d > 1; • For q odd, two classes each of q(q8d 1) dihedral groups of order 2d, denoted by D2d,for every divisor d > 2 of with qd even; • For q even, a single class of dihedral groups of order 2d, denoted by D2d, for every divisor d of q — 1, with d > 1; • For q odd, a single class of dihedral groups of order 2d, denoted by D2d,for every divisor d of ^j1 with odd, with d > 1; • For q odd, two classes each of dihedral groups of order 2d, denoted by D2d,for every divisor d> 2 of with even; • For q even, a single class of g(g2—1) dihedral groups of order 2d, denoted by D2d, for every divisor d of q + 1, with d > 1. 6. • A single class of g(g24-1) conjugate dihedral groups of order 4 denoted by 22 when q = ±3(8); • Two classes each of g(g4—1) conjugate dihedral groups of order 4 denoted by 22 when q = ±1(8); • When q is even, the groups 22 are in the case 7. 2 — 1 7. A number of classes of (21 1)(pk- 1) conjugate elementary abelian subgroups of order pm, denoted by Epm for every natural number m, such that 1 < m < n — 1, where k is a common divisor of n and m and (2,1,1) is equal to 2 (resp. 1, 1) if p > 2 and n is even (resp. p > 2 and n is odd, p = 2). ( q2 1)pn — m 8. A number of classes of ((2g1~1)(pfc_ 1) conjugate subgroups Epm : d which are semidi-rect products of an elementary abelian group Epm and a cyclic group of order d, d > 1, for every natural number m such that 1 < m < n and every natural number d dividing (^yyy, where k is a common divisor of n and m and (1,2,1) is one of • 1 for p > 2 and n is even • 2 for p > 2 and n is odd • 1 for p = 2 These subgroups are Frobenius groups. 9. • Two classes each of g(g48~1) conjugates of A4 when q = ±1(8); • A single class of g(g24-1) conjugates of A4 when q = ±3(8); • A single class of ^^ conjugates of A4 when q is an even power of 2. 10. Two classes each of g(g48~1) conjugates of S4 when q = ±1(8). 11. Two classes each of q(q1201) conjugate alternating groups A5 when q = ±1(10). 12. • Two classes each of ^(y^!) groups PSL(2, q'), where q is an even power of q', for q odd; A single class of g<^(q,2l1-i>) groups PSL(2, q'), where q is an odd power of q', for q odd; • A single class of groups PSL(2, q'), where q is a power of q', for q even. 13. Two classes each of groups PGL(2, q'), where q is an even power of q', for q odd. 14. PSL(2, q) itself. Remark 4.2. Subgroups A5 are given either by case 11 (when q = ±1(5) ) or by case 12 (when q = 0(5) and q = 4m) of Lemma 4.1. Also, if q is even, the PGL(2, q') are given by case 12, since PGL(2, q') = PSL(2, q') provided q is even. Remark 4.3. Let us mention that in the cases 7 and 8 of Lemma 4.1, the number of con-jugacy classes is not given. The number of conjugacy classes of the elementary abelian subgroups Epm given by Dickson (see [6], §260) is incorrect. For an example we refer to [5] Remark 7. Notice that Dickson does not give the number of conjugacy classes of the subgroups Epm : d, except in the particular case where m = n and d = (Pq- j). There are q +1 subgroups Eq : (2q--j1), all conjugate. 4.2 Maximal subgroups of PSL(2, q) In this section, we list the maximal subgroups of PSL(2, q). As the classification of geometries usually depends on whether q is even or odd, we give in Table 7 and Table 8 the maximal subgroups of PSL(2, q) in these two cases. We borrowed this result from Suzuki [16], page 417. Notice that the subgroups A5 appear both as A5 and PSL(2, q') for q' = 5. Let us mention that a little error in Suzuki [16] was detected and corrected by Patricia Vanden Cruyce [19] in her thesis: Indeed the subgroup A5 is maximal if r is an odd prime. Because if r = 2 we have that A5 < PGL(2,5) < PSL(2,25). However there remains a missing case in Suzuki [16] because, A4 is maximal if q = 5. We include it in Table 8. 4.3 Two-transitive representations of the maximal subgroups of PSL(2, q) The first lemma is obvious but used often in the next section as a necessary condition to have a two-transitive action. Lemma 4.4. Let G be a group and let H be a subgroup of G. If G acts two-transitively on the cosets of H in G, then |G| must be divisible by [G : H]([G : H] — 1). A group G is said to act regularly on a set Q if G is transitive on Q and the stabilizer in G of a point x G Q is the identity. Lemma 4.5. [21] Let (G, Q) be a permutation group which is transitive over Q and let G be abelian. Then G is regular. Moreover, if G is two-transitive then |Q| = 2. In order to simplify notation used throughout this section and the following one, we need another basic definition (borrowed from [2]). In a group G, an ordered pair of subgroups (A, B) is called two-transitive provided that B is a maximal subgroup of A and that the action of A on the left cosets of B is two-transitive. Structure Order Index Eq :(q - 1) q(q -1) q +1 D2(q+1) q = 2 2(q + 1) q(q-i) 2 D2(q-1) 2(q - 1) q(q+i) 2 A5 q = 4r r is prime 60 q(q2-i) 60 PSL(2, q') = PGL(2, q') q' > 4, q = q'm, m is prime or q' = 2, q = q'2 q'(q'2 -1) q(q2-1) q'(q'2-1) Table 7: The maximal subgroups of PSL(2, q), for q even Structure Order Index e : q-1 Eq: 2 q(q-1) 2 q +1 D(q+1) q = 7, 9 q +1 q(q-1) 2 D(q-1) q = 3, 5, 7, 9, 11 q - 1 q(q+1) 2 A4 if q = P > 3 and q = 3, 13, 27, 37(40) or q = 5 12 q(q2-1) 12x2 S4 if q = P > 2 and q = ±1(8) 24 q(q2-1) 24x2 A5 f q = 5r r odd prime or if < p = q = ±1(5) p prime or 1 q = p2 = -1(5) p prime or 60 q(q2-1) 60x2 PSL(2, q') q' = 5, q = q'm m odd prime q'(q'2-1) 2 q(q2-1) q'(q'2-1) PGL(2, q') 2 q = q q'(q'2 -1) q(q2-1) q'(q'2-1) Table 8: The maximal subgroups of PSL(2, q), for q odd We now provide the classification (existence and uniqueness) of all two-transitive representations of every maximal subgroup of PSL(2, q), a result borrowed from [2]. For the time being, let U be a group acting 2-transitively on a set 0. Let Ker U be the kernel of the representation, namely, the set of all u G U such that u(x) = x for every x G 0. Let U0 be the stabilizer in U of some element 0 in 0. Proposition 4.6. [2] Let G = PSL(2, q) for some power q of a prime p. Let (U, U0) be a 2-transitive pair of subgroups of G with U maximal in G. Then one of the following holds: 1. U = Eq : , q = 1(4), Ker U is the unique subgroup of index 2 of U, |0| = 2, U0 = Ker U (unique up to conjugacy); 2. U = Eq : (q — 1), q even, |0| = q, Ker U = 1, U0 is a cyclic subgroup of order (q - 1) (unique up to conjugacy); 3. U = PSL(2,2) = S3, |0| = 2, Ker U = Z3 = U0 (unique up to conjugacy); 4. U = PSL(2,2) = S3, |0| = 3, Ker U = 1, U0 = Z2 (unique up to conjugacy); 5. U = PSL(2,3) = A4, |0| = 4, Ker U =1, U0 = Z3 (unique up to conjugacy); 6. U = A5 = PSL(2,5) = PSL(2,4), p = 2, p = 5, either q = p = ±1(5) or q = p2 = —1(5). Here |0| = 5, Ker U = 1, U0 = A4; (two such representations, up to conjugacy; they are fused in PGL(2, q)); or |0| = 6, Ker U = 1, U0 = D10. 7. U = PSL(2,11), |0| = 11, Ker U = 1, U0 = A5 (two such representations, up to conjugacy; they are fused in PGL(2,11) = Aut(U )); 8. U = PSL(2,9) = A6, |0| = 6, Ker U = 1, U0 = A5 (two such representations, up to conjugacy; they are fused in PGL(2, 9)); 9. U = PSL(2, 7) = PSL(3,2), |0| = 7, Ker U = 1, U0 = S4 (two such representations, up to conjugacy; they are fused in PGL(2, 7)); 10. U = PSL(2, r) for every r = ps, s > 1, r > 3 with q = rm and m prime. Moreover, for p > 2 we also require m > 2. Here |0| = r +1, Ker U = 1, U0 = Er : (2^rT11) (unique up to conjugacy for given r); 11. U = PGL(2, r), r odd, r = ps, q = r2, |0| = 2, Ker U = U0 = PSL(2, r) (unique up to conjugacy); 12. U = PGL(2, r), r odd, r = ps, s > 1 with q = r2. Here |0| = r + 1, Ker U =1, U0 = Er : (r — 1) (unique up to conjugacy); 13. U = PGL(2, 3) = S4, q = ±1(8), |0| = 3, Ker U = E4, U/Ker U = S3, U0 = D8 (two such representations, up to conjugacy; they are fused in PGL(2, q)); 14. U is dihedral of order 2(q — 1) or 2(q + 1), q even. |0| = 2, Ker U = U+ = U0 where U + is the cyclic subgroup of index 2 of U, (unique up to conjugacy for each of the two possible values of |U |); 15. U is dihedral of order (q — 1) or (q + 1), q odd. |0| = 2, Ker U = U+ = U0 where U + is the cyclic subgroup of index 2 of U, (unique up to conjugacy for each of the two possible values of |U|). In the particular case where q = 3, the case of (q + 1) provides U = E4 , |0| =2. Then U0 is one of the three subgroups of order 2 in U (unique up to conjugacy); 16. U is dihedral of order either 2 (q — 1) or 2 (q + 1), q even, and 3 | | U |Q| = 3, Ker U is the unique cyclic subgroup of index 6 in U. Then U0 is one of the three dihedral subgroups of index 3 in U, U /Ker U = S3 (unique up to conjugacy); 17. U is dihedral of order either (q — 1) or (q + 1), q odd, and 3 | |U |. Here |0| = 3, Ker U is the unique cyclic subgroup of index 6 in U. Then U0 is one of the three dihedral subgroups of index 3 in U, U /Ker U = S3 (unique up to conjugacy); 18. U is dihedral of order either (q — 1) or (q + 1), q odd, q > 5, and 4 | |U |. Here |0| = 2, Ker U = U0 is one of the two dihedral subgroups of index 2 in U (two such representations, up to conjugacy; they are fused in PGL(2, q)); Ker U is dihedral, U0 is dihedral of index 2; 19. U is dihedral of order 4, q is one of 3,5; |0| = 2, Ker U = U0 is one of the three dihedral subgroups of index 2 in U (unique up to conjugacy); 20. U = PGL(2,5) = S5, |Q| = 5, Ker U =1, U0 = S4 (unique up to conjugacy). 4.4 Some other useful results An observation used in our proofs is that PGL(2, q) can be viewed as a subgroup of PSL(2, q2) and also that PGL(2, q) has a unique subgroup isomorphic to PSL(2, q). This lets us extract the list of subgroups of PGL(2, q) from the list of subgroups of PSL(2, q2). Therefore we require the properties of the subgroup lattice of PGL(2, q) for which we refer to [4] (see also [13] and [15]). The next lemma is often used to count the geometries up to isomorphism. Lemma 4.7. • Assume that is even. In this case both conjugacy classes of D2d for every d > 2 dividing (2q±11) fuse in PGL(2, q) and thus also in PTL(2, q). • Assume that q = ±1(8). In this case both conjugacy classes of S4 and A4 fuse in PGL(2, q) and thus also in PFL(2, q). • Assume that q = ±1(5). In this case both conjugacy classes of A5 fuse in PGL(2, q) and thus also in PTL(2, q). • Assume that q = p2n is odd. In this case both conjugacy classes of PGL(2,pn) fuse in PGL(2,p2n) and thus also in PTL(2, q). 5 Proof of Theorem 1.1 In this section, we prove the Classification Theorem 1.1 by a case analysis. We determine the rank 2 RWPRI and (2T)i geometries of PSL(2, q). In order to structure this work we introduce a subsection for each type of G0. There are 5 such subsections left to consider, which are the different types of maximal subgroups of G = PSL(2, q), listed in section 4.2. The cases Eq : i9-1^, D0 (g-i) and D0 (q+i) have (2'q 1) 2 (2 ,q — i) 2 (2 ,q — i) been treated in [5]. The various cases for the two-transitive pairs (G0, G01) with G0 maximal in G are provided by Proposition 4.6. Those situations are analysed in order to detect the admissible G1 in a series of Lemmas. During this analysis, candidates for G1 are represented by the symbol H. They become G1 only when they resist the analysis. 5.1 The case where G0 = A5 Recall that following Table 7 and Table 8, the subgroup A5 is maximal in PSL(2, q) if q = 5r r odd prime or q = 4r r prime or q = p = ±1(5) p odd prime or q = p2 = —1(5) p odd prime . In this section we assume these conditions on q. Observe that if q = 0(5) the group A5 is isomorphic to PSL(2, 5) which is a particular case of the family PSL(2,5n) with q = 5nm for m an odd prime. In this section we treat this particular situation. The general situation is treated in Proposition 5.16. If q = 0(4) the group A5 is isomorphic to PSL(2,4) which is a particular case of the family PSL(2,4n) with q = 4nm for m prime. In this section we analyse this particular situation. The general situation is treated in Proposition 5.12. In view of (6) in Proposition 4.6 there are two cases for G01, namely the case of D10 and A4. For each of these G01 we look for the various possible groups H in one of the two following Lemmas. Remember that H is any subgroup of G such that (H, G01) is a two-transitive pair. In order to determine all H candidates we scan the list of maximal subgroups. For each maximal subgroup we analyse its subgroup lattice. Lemma 5.1. Let G = PSL(2, q) with q as required in this section. If H is a subgroup of G such that (H, D10 ) is a two-transitive pair then one of the three following statements holds: • H = D20 provided 10 | (2q±11) ; • H = D30 provided 15 | (2qq—11) ; • H = PSL(2,5) = A5. Proof. Left to the reader. See Appendix pg 1. (The Appendix contains details for this and several other results to follow.) □ Lemma 5.2. Let G = PSL(2, q) with q as required in this section. If H is a subgroup of G such that (H, A4) is a two-transitive pair then one of the five following statements holds: • H = E16 :3 provided q = 4r ; • H = PSL(2,4) = A5 provided q = 4r; • H = PSL(2,5) provided q = 5r; • H = S4 provided q = ±1(5) and q = ±1(8); • H = As. Proof. Left to the reader. See Appendix pg 2. □ In Remark 4.3 of section 4.1. we mention that the number of conjugacy classes of cases 7 and 8 are not given in Lemma 4.1. To prove the following Proposition we need the number of conjugacy classes of a particular situation, treated in the next two Lemmas. Lemma 5.3. The number of conjugacy classes of E16 :3 in PSL(2,4r ), for an odd prime r, is equal to 4 15 -1. Proof. Step 1: We must count the number of conjugacy classes of subgroups E16 : 3 in PSL(2,4r ). Therefore we first count the total number of subgroups E16 : 3 in PSL(2,4r ) and divide this number by the length of the conjugacy classes. We shall indeed see that this number is constant. Step 2: We consider G = PSL(2,4r) as a permutation group acting on the projective line PG(1,4r). This group is sharply 3-transitive on 4r + 1 points. Given a point to, its stabilizer is E4r : 4r — 1 = AGL(1,4r). The latter contains our E16 : 3. Let H be any subgroup E4 : 3 = A4 = AGL(1,4). It is contained in a subgroup K := PGL(2,4) = A5 which has an orbit of length five namely PG(1,4). Step 3: Let us see AG(1,4r ) = PG(1,4r)\{to} as an affine space V of dimension r over the field GF(4). The subgroup H stabilizes a line l of V namely AG(1,4). Hence, l contains the points 0 and 1. The space V endowed with the point 0 is a vector space of dimension r on GF (4). Observe that H fixes a unique point namely to. In A5 there are four conjugate subgroups E4 : 3 say X1,X2,X3, X4 other than H, each fixing a unique point which belongs to l. Moreover, H stabilizes no other line l' in V since otherwise l' U {to} is an orbit of length five of A5 and so each of X1, X2, X3, X4 fixes a point on l' while this point is on l implying l = l'. Therefore, H stabilizes a unique line of V which is l. 4V /4V _ 1) Step 4: Observe that AG(1,4r ) is transitive on the lines of V. There are —^—- lines in V and, taking the point to into account, we see that the conjugacy class of H in G 4V /4V 1) consists of —L12—subgroups E16 : 3. Step 5: Coming back to the beginning of Step 3, the multiplicative group of GF(4) is a cyclic group Z3 which is a subgroup of H and so also a subgroup of A5 namely E16 : 3. Step 6: The group Z3 stabilizes the point 0 and every line on 0 in the space V. Therefore, it also stabilizes every plane on 0 in this space, in particular every plane containing l. There are 46—4 = 4V 3-1 such planes on l. Step 7: Let n be a plane of V containing l. It is invariant under 16 translations and Z3. Thus n is invariant under a subgroup E16 : 3 containing H. Conversely, every E16: 3, say L, containing H also contains Z3 which fixes the point 0. The orbit of 0 under L is its orbit under E16. And Z3 acts on this orbit, hence this orbit is a plane. In conclusion, the subgroups E16 : 3 containing H and the planes containing l are in one-to-one correspondence. Step 8: Combining Steps 3, 6 and 7 we see that the number of conjugacy classes of subgroups E16: 3 containing H and fixing to is 4V 3-1.5 as required. □ For the particular situation of Lemma 5.3, we count the number of geometries up to conjugacy and up to isomorphism in the following Lemma. Lemma 5.4. Let r be an odd prime. Let ac (r) (resp. aI (r) ) be the number of geometries of type r(PSL(2,4r),A5,A4,E16 : 3) up to conjugacy (resp. isomorphism). Then the following hold: 1. ac (3) = 5; 2. a/(3) = 2; 3. if r > 3, then ac(r) = 4V 3-1 ; 4. if r > 3, then a/ (r) = —-—. Proof. Step 1: Lemma 5.3 gives the number of conjugacy classes of E16 : 3 for a given PSL(2,4r ). Every E16 : 3 has five conjugacy classes of subgroups E4 : 3. Moreover, each E4 : 3 is contained in a unique A5. Therefore, we get the number of triples consisting of a representative G1 of every conjugacy class of E16 : 3, a representative G01 of every conjugacy class of E4 : 3 in G1 and the unique subgroup G0 = A5 containing G01. Hence aC (r) = 4r 15-1 • 5 • 1. In particular aC (3) = 5. This is proving respectively (3) and (1). Step 2: Let to, V, H and l be defined as in the proof of Lemma 5.3, Steps 2 and 3. Recall that l contains 0 and 1. To get a/(r), we still have to figure out how NPrL(2,4r )(H) acts on the subgroups E16 : 3 containing H. In other words, how does NPrL(2,4r)(H) act on the planes of V containing l ? Step 3: To answer the question of Step 2 we shall show that NPrL(2,4r)(H) = H : K, where K is the group of field automorphisms of GF(4r ). Recall the fact that the group PrL(2,4r ) is PSL(2,4r ) : K. Recall also that K is a cyclic group of order 2r. The group K leaves every subfield of GF (4r ) invariant. Hence K leaves GF (4) invariant, thus also the line l, and it normalizes H. Applying Lemma 4.1 we see that NPSL(2,4r)(H) = H in view of the fact that H = A4 and of the restrictions on the values taken by q. We get that NPrL(2,4r )(H) is a group of order H.K.e and we want now to show that e =1. Let N1 (resp. N2) be the number of conjugate subgroups of H in G (resp. PrL(2,4r)). Then N1 < N2, N1 = jfj, N2 = '^[Kj.? = \§\~e and so e = 1. Therefore NprL(2,4r)(H) = H : K. Step 4: In our count of triples, we may assume that G0 and G01 are fixed because, up to isomorphism, the chain of subgroups PSL(2,4r ) - A5 - A4 is unique. Moreover, without loss of generality, we suppose that G01 is H. Step 5: We consider G = PSL(2,4r) and H in it. We recall the 4r 15-1 conjugacy classes of subgroups E16 : 3 containing H as found in Lemma 5.3. Let Q be the set of these 4r 15-1 conjugacy classes. Recall that NPSL(24r)(H) = H and so the action of H on Q is the identity. Next we consider the action of K on Q which is also the action of H : K. The number of orbits of this K-action on Q is the number a/ (r) we have to determine. Step 6: As in the proof of Lemma 5.3, Step 2 we consider G = PSL(2,4r ) as a triply transitive permutation group acting on the projective line PG(1,4r ). For every t dividing 2r there is a subfield GF(24) of GF(4r). It fixes 24 + 1 points on PG(1,4r). This set of points is called a circle as well as all of its transforms under G. Every triple of distinct points on PG(1,4r ) is contained in one and only circle of 2l + 1 points. Step 7: Given three points to, 0 and 1, there is a unique circle C5 of five points, namely PG(1,4) = {to} U l and there is a unique circle of 2r + 1 points C2r+1, namely PG(1,2r ). The involution ß G K fixes all the points of C2r+1. On C5, it fixes to, 0 and 1, and it permutes the remaining 2 points that we call a and ß(a). The group induced on C5 by the stabilizer of C5 in PSL(2,4r ) is A5 = PSL(2,4) extended by ß, that is, S5. The unique subgroup K + of K, of order r fixes all points of C5 and splits the remaining points of PG(1,4r ) in orbits of length r. Therefore, (22r + 1) — 5 must be divisible by r. Indeed, (22r + 1) — 5 = 4r — 4 = 4(4r-1 — 1), the latter being divisible by r thanks to Fermat's little theorem. The subgroup H = E4 : 3 fixes to. Every cyclic subgroup of order 3 of H fixes two points of C5. This gives ten conjugate subgroups of order 3 in A5. The group K + fixes C5 point-wise. Suppose K + stabilizes a plane n of V containing l. Then it must decompose the 16 — 4 = 12 points of n\l in orbits of length r. If r = 3, this may occur and K + indeed stabilizes two of the five planes containing l, hence it normalizes two of the E16 : 3 containing H. Moreover, it fuses the other three. The two E16 : 3 normalized by K + are swapped by ß, giving a/(3) = 2. This is proving (2). If r > 3, no plane of V that contains l can be stabilized by K +. Hence K + fuses the 4r 3-1 subgroups E16: 3 in 4r 3r -1 orbits of length r. Step 8: It remains to look at the action of ß on these orbits. In GF(4r), there are three proper subfields, namely GF(2), GF(4) and GF(2r ). The involution ß fixes all the elements of GF (2r ). Let us show that ß stabilizes planes containing l. Given an element x G GF(2r ), the plane n containing 0, 1 and x is stabilized since 0, 1 and x are fixed by ß. Moreover, n contains the point x + 1 G GF(2r). Hence, there are at least four points fixed in n by ß. If there are more, there must be at least 8 points fixed and the whole plane n is fixed point-wise, a contradiction with the fact that a G n and a is not fixed by ß. Therefore, the elements of GF(2r ) give distinct planes that are stabilized by ß. Step 9: We claim that the remaining planes of V that contain l are fused in pairs by ß. Indeed, suppose that there exists a plane n containing l and no other element of GF (2r ) in V, and such that ß(n) = n. In n, the only fixed points are thus 0 and 1. For every x G n\C5, the line xß(x) is stabilized by ß. It is either secant or parallel to l. Suppose first that it is secant. Then, it intersects l in either 0 or 1 and the fourth point of xß(x) must be fixed, a contradiction. Suppose then that it is parallel. The other two points of xß(x) may be written as y and ß(y) . Let us recall that we denote the points of l as 0, 1, a and ß(a). The lines ax and ß(a)ß(x) are swapped and parallel. One of the lines 1y or 1ß(y) must also be parallel to ax. Its image by ß is not parallel to ax. This is a contradiction. Therefore, no other plane of V containing l can be stabilized by ß. Step 10: In conclusion, we gety r-1 sets of r isomorphic geometries and 27 (4r 3-1 — (2r -1 — 1)) sets of 2r isomorphic geometries. Finally, we obtain a/(r) = 2r 7-1 + 27 (3-1 — (2r-1 — 1)) and the formula given in the Lemma is obtained by a straightforward simplification. This is proving (4). □ Proposition 5.5. Let G = PSL(2, q) with q as required in this section. Every RWPRI and (2T)1 geometry of rank two r(G; Go, G1, Go n G1) in which G0 = A5 is isomorphic to one of the geometries appearing in Table 1. Proof. Let Go = A5. We subdivide our discussion in two cases, namely the two G01-candidates in view of (6) in Proposition 4.6 which are: D10 and A4. In each of these two cases we review all possibilities for G1 given in the previous Lemmas 5.1 and 5.2, as well as the number of classes of geometries with respect to conjugacy (resp. isomorphism). Subcase 1: G01 = G0 n G1 = D10. This is dealt with in the appendix, (pg 2-6). Subcase 2: G01 = G0 n G1 = A4. By Lemma 5.2 the possibilities for G1 are E16 : 3 if q = 4r, PSL(2,4) = A5 if q = 4r, PSL(2, 5) = A5 if q = 5r, S4 if q = ±1(5) as well as q = ±1(8) and A5. 2.1 We consider the case where G1 = E16: 3. The condition on q is q = 4r with r prime. In this situation there is only one conjugacy class of A5 and one of A4 in PSL(2, q). Notice that there are 5 conjugacy classes of A4 in E16: 3. Since PSL(2,16) is simple and A5 maximal, A5 is self-normalized. Moreover, A4 is self-normalized in PSL(2,4r). The normalizer of E16 : 3 depends on whether r = 2 or not. We distinguish three cases namely: r = 2, r = 3 and r > 3. In the latter two, notice that since r = 2, E16: 3 is self-normalized in PSL(2,4r). • Let us first consider the particular case where r = 2. In this situation there exists only one conjugacy class of E16 : 3 in PSL(2,16). We also have that NPSL(2,16)(E16 : 3) = E16 : 15. Therefore the number of subgroups E16 : 3 containing a given subgroup A4 in PSL(2,16) is equal to | PSL(2,16) | | E16 :3 | 5 | A4 | | E16: 15 | ^ | A4 | ^ •l PSL(2,16) | . Thus the RWPRI and (2T)1 geometry r2 = r (PSL(2,16); A5,E16 :3,A4) exists and is unique up to conjugacy and also up to isomorphism. • In view of Lemma 5.3 and Lemma 5.4 we know that if r = 3 there exist up to conjugacy exactly five RWPRI and (2T) 1 geometries r := r(PSL(2, 64),A5,A4, E16: 3) and exactly two up to isomorphism. • In view of Lemma 5.3 and Lemma 5.4 we know that if r > 3 there exist up to conjugacy exactly 4r—3-1 RWPRI and (2T)1 geometries r4 := r(PSL(2, q), A5, A4, E16 : 2(4^ — 2_1)13 2r — 2 3) and exactly 2-3^+--up to isomorphism. This geometry is new and the number of classes up to conjugacy (resp. isomorphism) is confirmed by Magma for q = 16,64. For q = 16, it is also confirmed by [20]. 2.2 We consider the case where G1 = S4. The conditions given on q are q = ±1(5) and q = ±1(8). They imply that there are two conjugacy classes of S4, two of A5 and also two of A4 in PSL(2, q). Therefore we consider two situations: either q = p = ±1(5) or q = p2 = -1(5), with p an odd prime. We distinguish these two cases in the discussion below. • Assume q = p = ±1(5) with p prime. All conditions given on q imply that either q = ±1(40) or q = ±9(40). In both situations we know that S4 is a maximal subgroup of PSL(2, q). Therefore NPSL(2,q)(A4) = S4 = NSi(A4) and NA5 (A4) = A4. Now all A4 in an S4 are conjugate and this is also the case for all A4 in an A5. The number of subgroups A5 containing a given subgroup A4 in PSL(2, q) is equal to l PSL(2, q) | | A5 | | S4 | =2 | A5 | | A4 | | PSL(2, q) | . To count the geometries up to conjugacy we need to know whether the S4 normalizes each of the A5. This is not the case because |NPSL(2,q) (A4) n NPSL(2,q)(S4)| = |S4| = 2|A41. Hence, there exist exactly two RWPRI and (2T)1 geometries r'9 = r(PSL(2, q); A5, S4, A4) up to conjugacy, provided q = ±1(40) or q = ±9(40). Let us deal with the fusion of non-conjugate classes. Following Lemma 4.7 the two classes of S4, A4 and A5 are fused under the action of PGL(2, q) and thus also under the action of PrL(2, q). Therefore, there exists exactly one RWPRI and (2T)1 geometry r9 = r (PSL(2, q); A5, S4, A4) up to isomorphism provided q = ±1(40) or q = ±9(40). • Assume q = p2 = —1(5) with p prime. All conditions given on q imply that either q = —1(40) or q = 9(40). All A4 in an S4 are conjugate and NPSL(2,q)(A4) = S4 = NSi(A4) and NA5(A4) = A4. We also know that NPSL(2,q)(S4) = S4. Therefore the number of S4 containing a given A4 is one. To count the geometries up to conjugacy we need to know whether the S4 normalizes each of the A5. This is not the case because |NPSL(2,q)(A4) n NPSL(2,q)(S4)| = |S4| = 2|A41. Therefore, up to conjugacy there exist exactly two RWPRI and (2T)1 geometries r16 = r (PSL(2, q); As, S4, A4) provided either q = —1(40) or q = 9(40), with q = p2. Let us deal with the fusion of non-conjugate classes. Following Lemma 4.7 the two classes of A4, S4 and A5 are fused under the action of PGL(2, q) and thus also under the action of PrL(2, q). Therefore, there exists exactly one RWPRI and (2T)1 geometry r16 = r (PSL(2, q); A5, S4, A4) up to isomorphism, provided either q = —1(40) or q = 9(40), with q = p2. This geometry is new and the number of classes up to conjugacy (resp. isomorphism) is confirmed by Magma for q = 9, 31,41,49. For q = 9, it is also confirmed by [3]. 2.3 Consider the case where G0 = G1 = A5. With the given conditions on q there are three cases to consider: • If q = 4r with r prime, there is only one conjugacy class of A5 and also one of A4. Since every A4 is contained in only one A5, there is no such geometry. • Assume q = 5r with r an odd prime. The number of conjugacy classes of A4 in PSL(2, q) depends on whether q = ±1(8) or q = ±3(8). If q = ±1(8) there is a contradiction with r odd in q = 5r. Now q = ±3(8) implies that there is one conjugacy class of A4 and also one of A5. Since every A4 is contained in only one A5, there exists no such geometry. • Assume q = p = ±1(5) or q = p2 = —1(5) with p an odd prime. There are two conjugacy classes of A5 in PSL(2, q). The number of conjugacy classes of A4 in PSL(2, q) depends on whether q = ±1(8) or q = ±3(8). We distinguish these two cases. If q = ±1(8) there are two classes of A4, all A4 in an A5 are conjugate, and the normalizer of A4 in PSL(2, q) is S4. All conditions on q imply that if q = p = ±1(5) either q = ±1(40) or q = ±9(40); and if q = p2 = —1(5) either q = —1(40) or q = +9(40). The number of subgroups A5 containing a given subgroup A4 in PSL(2, q) is equal to I PSL(2,q) | | As | | S4 | =2 | As | | A4 | | PSL(2, q) | . Therefore, there exist exactly two RWPRI and (2T)1 geometries ru = r(PSL(2, q); A5, A5, A4) up to conjugacy, provided either q = ±1(40) or q = ±9(40), with q prime, one for each class of A5. Also, there exist exactly two RWPRI and (2T) 1 geometries r17 = r (PSL(2, q); As, As, A4) up to conjugacy, provided either q = —1(40) or q = +9(40), with q = p2, one for each class of As. Let us deal with the fusion of non-conjugate classes. Following Lemma 4.7 the two classes of As are fused under the action of PGL(2, q) and thus also under the action of PrL(2, q). Therefore there exists exactly one RWPRI and (2T)1 geometry r10 = r (PSL(2, q); As, As, A4) up to isomorphism provided either q = ±1(40) or q = ±9(40). Also, there exists exactly one RWPRI and (2T)1 geometry r^ = r (PSL(2, q); A5, A5, A4) up to isomorphism provided either q = —1(40) or q = +9(40). If q = ±3(8), there is one conjugacy class of A4 in PSL(2, q). All conditions on q imply that if q = p = ±1(5) either q = ±11(40) or q = ±19(40); and if q = p2 = —1(5) either q = —11(40) or q = +19(40). Every A4 is contained in exactly one A5, and there are two conjugacy classes of A5 in PSL(2, q). Hence, there exists exactly one RWPRI and (2T)1 geometry rn = r(PSL(2, q); A5, A5, A4) up to conjugacy and thus also exactly one up to isomorphism provided either q = ±11(40) or q = ±19(40), with q prime. This geometry is new and the number of classes up to conjugacy (resp. isomorphism) is confirmed by Magma for q = 11,19,29,31,41,61. For q =11,19, it is also confirmed by [20]. Also, there exists exactly one RWPRI and (2T)1 geometry r18 = r(PSL(2, q); A5, A5, A4) up to conjugacy and thus also exactly one up to isomorphism provided either q = — 11(40) or q = +19(40), with q = p2. This geometry is new and the number of classes up to conjugacy (resp. isomorphism) is confirmed by MAGMA for q = 9,49. □ 5.2 The case where G0 = A4 Recall that following Table 8, the subgroup A4 is maximal in PSL(2, q) provided q is prime, q > 3 and either q = 3,13,27, 37(40) or q = 5. Therefore q = ±3(8) and there exists only one conjugacy class of subgroups isomorphic to A4. In view of (5) in Proposition 4.6 there is only one case for G01, namely the cyclic subgroup of order 3. The proof of all following propositions are very similar to that of Proposition 5.5. Therefore we do not give the details and we refer to the Appendix. The proof of proposition 5.6 may be found in the Appendix (pg. 6-9). Proposition 5.6. Let G = PSL(2, q) with qprime, q > 3 and either q = 3,13, 27, 37(40) or q = 5. Every RWPRI and (2T )1 geometry of rank two r(G; G0, G1, G0 n G1) in which G0 = A4 is isomorphic to one of the geometries appearing in Table 2. 5.3 The case where G0 = S4 Recall that following Table 7 and Table 8, the subgroup S4 is maximal in PSL(2, q) if q is an odd prime and q = ±1(8). In this section we assume these conditions on q. Moreover, there are two conjugacy classes of subgroups isomorphic to S4 in G. In view of (11), (12) and (13) in Proposition 4.6 there are three cases for G01, namely the case of D6 = E3 : 2, the case of D8 and the case of A4. For each of these G01 we look for the various possible groups H in one of the three following Lemmas, whose proofs are left to the reader. The proof of proposition 5.10 may be found in the Appendix (pg. 8-12). Lemma 5.7. Let G = PSL(2, q) with q an odd prime and q = ±1(8) as required in this section. If H is a subgroup of G such that (H, D6) is a two-transitive pair then one of the three following statements holds: H = D12 provided 6 | q±±1 ; H = D18 provided 9 | q±±1 ; or H = S4. Lemma 5.8. Let G = PSL(2, q) with q an odd prime and q = ±1(8) as required in this section. Then the following statement holds: If H is a subgroup of G such that (H, D8) is a two-transitive pair then H = D16 provided 8 | q±±1, H = D24 provided 12 | q±±i; orH = S4. Lemma 5.9. Let G = PSL(2, q) with q an odd prime and q = ±1(8) as required in this section. Then the following statement holds: If H is a subgroup of G such that (H, A4) is a two-transitive pair then H = S4; or H = A5 provided q = ±1(5). Proposition 5.10. Let G = PSL(2, q) with q an odd prime and q = ±1(8). Every RWPRI and (2T )1 geometry of rank two r(G; G0, G1, G0 n G1) in which G0 = S4 is isomorphic to one of the geometries appearing in Table 3. 5.4 The case where Go = PSL(2, q') In this section we make a distinction between the cases q odd and q even with q = pnm. The subgroups PSL(2, q') and PGL(2, q') with q' = pn are isomorphic provided q is even and they are distinct provided q is odd. 5.4.1 The case q even Since q is even, PSL(2, q') = PGL(2, q'). Recall that following Table 7, the subgroup PSL(2, q') = PGL(2, q') is maximal in PSL(2, q) provided q' = 2n and q = q'm = 2n m for m prime; moreover for n = 1 we need m = 2. In this section we assume these conditions on q. In view of (3), (4), (6) and (10) in Proposition 4.6 there are three cases for G01, namely: the case of the cyclic subgroup of order 3 provided q' = 2, the case of D10 provided q' = 4 and the case of E2 n : (2n — 1). For each of these G01 we look for the various possible groups H; the case of E2 : (2n — 1) is treated in the following Lemma, whose proof is left to the reader. The proof of proposition 5.12 may be found in the Appendix (pg. 13-14). Lemma 5.11. Assume q = 2nm with m prime and n = 1 and let G = PSL(2, q). If H is a subgroup of G such that (H, E2n : 2n — 1) is a two-transitive pair then one of the two following statements holds: H = E22n : 2n — 1 provided m = 2 or H = PSL(2,2n). Notice that if n = 2, PSL(2,2n) = A5. Proposition 5.12. Assume q' = 2n and q = q'm = 2nm for m prime ; moreover for n =1 we need m = 2. Let G = PSL(2,2n m). Every RWPRI and (2T)1 geometry of rank two r(G; G0, G1, G0 n G1) in which G0 = PSL(2, q') is isomorphic to one of the geometries appearing in Table 4. 5.4.2 The case q odd Since q is odd we need to consider two distinct maximal subgroups which are PSL(2,pn) provided q = pmn where m and p are odd primes and PGL(2,pn) provided q = p2n where p is an odd prime. The latter will be treated in section 5.5. Recall that following Table 8, the subgroup PSL(2,pn) is maximal in PSL(2, q) provided q = pmn with m and p odd primes. In this section we assume these conditions on q. In view of (5)-(10) in Proposition 4.6 there are four possibilities for G01, namely: A4 provided q' = 5, S4 provided q' = 7, A5 provided q' = 9,11 and Eq- : q-—1. For each of these G01 we look for the various possible groups H in the three following Lemmas, whose proofs are left to the reader. The proof of proposition 5.16 may be found in the Appendix (pg. 14-17). The case of A4 provided q' = 5, will be treated directly in the proof of the Proposition. Lemma 5.13. Assume q odd, q = pnm with m prime and let G = PSL(2, q); then the following statement holds: If H is a subgroup of G such that (^H, Epn : p 2-1 j is a two-transitive pair then H = PSL(2, pn). Notice that if pn = 3, PSL(2,pn) = A4 and if pn = 5, PSL(2,pn) = A5. They are particular cases of PSL(2,pn). Lemma 5.14. Assume q is either 11m or 9m, with m an odd prime and let G = PSL(2, q). Then the following statement holds: If H is a subgroup of G such that (H, A5) is a two-transitive pair then H = PSL(2, q') provided q' = 9 or 11. Lemma 5.15. Assume q = 7m, with m odd prime and let G = PSL(2,7m). Then the following statement holds: If H is a subgroup of G such that (H, S4) is a two-transitive pair then H = PSL(2,7). Proposition 5.16. Assume q = pnm with p and m odd primes and let G = PSL(2, q). Every RWPRI and (2T)1 geometry of rank two r(G; G0, G1,G0 n G1) in which G0 = PSL(2,pn) is isomorphic to one of the geometries appearing in Table 5. 5.5 The case where Go = PGL(2, q') If q is even, PGL(2, q') = PSL(2, q') and this situation has been treated in Section 5.4. Therefore, we assume in this section that q is odd. Recall that following Table 8, the subgroup PGL(2, q') is maximal in PSL(2, q) provided q' = pn and q = q'2 = p2n with p an odd prime. In view of (11), (12), (13) and (20) in Proposition 4.6 there are four cases for G01, namely the case of Epn : (pn - 1), the case of PSL(2, q'), the case of D8 provided q = 32 and the case of S4 provided q = 52. For each of these four G01 we look for the various possible groups H in one of the four following Lemmas, whose proofs are left to the reader. The proof of proposition 5.21 may be found in the Appendix (pg. 17-19). Lemma 5.17. Let G = PSL(2,32). Then the following statement holds: If H is a subgroup of G such that (H, D8) is a two-transitive pair then H = PGL(2, 3). Lemma 5.18. Assume q is odd and let G = PSL(2,p2n). Then the following statement holds: If H is a subgroup of G such that (H, Epn : (pn — 1)) is a two-transitive pair then H = Ep2n : (pn — 1) or H = PGL(2,pn). Lemma 5.19. Assume q is odd and let G = PSL(2,p2n). Then the following statement holds: If H is a subgroup of G such that (H, PSL(2,pn )) is a two-transitive pair then H = A5 providedpn = 3; or H = PGL(2,pn). Notice that if pn = 3, PGL(2,pn) = S4. Lemma 5.20. Let G = PSL(2,52). Then the following statement holds: If H is a subgroup of G such that (H, S4) is a two-transitive pair then H = PGL(2,5). Proposition 5.21. Assume q' = pn and q = q'2 = p2n with p an odd prime. Let G = PSL(2, q). Every RWPRI and (2T)1 geometry of rank two r(G; Go, Gu Go n G1) in which G0 = PGL(2, q') is isomorphic to one of the geometries appearing in Table 6. The proof of Theorem 1.1 readily follows from Propositions 5.6, 5.10, 5.5, 5.12, 5.16 and 5.21. The main Theorem of [5] and Theorem 1.1 complete the classification of rank two resid-ually weakly primitive and locally two-transitive coset geometries for the groups PSL(2, q). We also give the number of classes of all such geometries with respect to conjugacy and isomorphism. This classification includes infinite classes of geometries up to conjugacy and up to isomorphism. This number is dependent on the prime power q = pn; it is a function of n and p. 6 Locally s-arc-transitive graphs The construction of the (G, 2)-arc-transitive graphs, using Tits' Theorem, is studied in full detail in Leemans [12]. This construction shows that the rank two incidence structures are also locally-2-arc-transitive graphs in the sense of [8]. All the RWPRI and (2T)1 geometries we have obtained are bipartite graphs and also locally 2-arc-transitive graphs. Now we want the value of s such that the incidence graph of r is a locally s-arc-transitive but not a locally (s + 1)-arc-transitive graph. We mainly use the method of D. Leemans [12] (Lemma 5.1). This provides the value of s in all cases given in Tables 1, 2, 3, 4, 5 and 6 (in the introduction) except those listed in Table 9. We don't give the details in the cases for which the Leemans' method works. We now discuss the nine cases left over in Table 9. In every case if p is a vertex of the graph, we write p1 for the set of neighbours of p which is also the residue of p. We give the details for four of them, the other five are dealt with in the Appendix (pg. 1920). Case of Table 1, geometry r1, case of Table 1, geometries F and r13 and case of Table 4, geometry r5. We know that s > 2. Consider a path (a, b, c) such that a is of type 0, b is of type 1, c is of type 0. Here, Gabc = Z5. This acts on the five 1-elements d1,..., d5 other than b in c1. The action is transitive since otherwise Z5 would be in the kernel of the action of Gc on c1 contradicting the simplicity of G0 = A5 = Gc. This provides s > 3 for paths starting at a 0 — element. Next consider a path (h, i,j) such that h is of type 0, i is of type 1, j is of type 0. Here, Ghij = Z2. This acts on the two 0-elements k1, k2 other than i in j1. The action is transitive since otherwise Z2 would be in the kernel of the action of G j on j1. This kernel for the action of D30 on the cosets of D10 is a group Z5, a contradiction. Hence s > 3. Applying Leemans' method we get s = 2 or 3. Thus s = 3. Case of Table 3, geometry r6. We know that s > 2. Consider a path (a, b, c) as in the preceding case. Here, Gabc = Z4. This acts on the two 1-elements d1, d2 other than b in c1. The action is transitive since otherwise Z4 would be in the kernel of the action of Gc on c1. This kernel for the action G0 = A5 G01 G1 D10 D10 D30 D30 Table 1, r Table 1, re and r13 G0 = S4 G01 G1 De Dg Dg Dg D 1g D16 D24 S4 Table 3, r2 Table 3, r5 Table 3, re Table 3, rr and Tg G0 = PSL(2,2n) G01 G1 E2n : (2n — 1) D10 E2mn : (2n — 1) D30 Table 4, r Table 4, r5 G0 = PGL(2,pn) G01 G1 Epn :(pn — 1) Ep2n :(pn — 1) Table 6, r Table 9: Cases in which s cannot be decided by Leemans' method. of S4 on the cosets of D8 is 22, a contradiction. This provides s > 3 for paths starting at a 0 — element. Next consider a path (h, i, j) as in the preceding case. Here, Ghij = 22. This acts on the two 0-elements k1, k2 other than i in ja. The action is transitive since otherwise 22 would be in the kernel of the action of Gj on . This kernel for the action D18 on the cosets of D8 is a group Z4, a contradiction. Hence s > 3. Applying Leemans' method we get that s equals 3 or 4. We now prove that s cannot be equal to 4 thanks to the following argument due to an unknown referee: Given the path (a, b, c) starting at a 0-element we have shown that Gabc = Z4 and that this is transitive on the two elements adjacent to c other than b. Thus G abed = Z2 = (x), where x is the square of an element of order 4 in Gabc < Ga = S4. Thus x lies in the normal subgroup of Ga of order 4 and so acts trivially on the set of neighbours of a. Thus Gdcba is not transitive on the set of 4-arcs starting with (d, c, b, a) and so the graph is not locally 4-arc transitive. Hence s = 3. Let us make some observations on the results: In Tables 1, 2, 3, 4, 5, 6 and 9 most values are s = 2 or s = 3. There are some spectacular examples with larger values of s. Indeed we obtain a locally 4-arc transitive graph and a locally 7-arc transitive graph which are respectively r (PSL(2, q); S4, S4, D8) due to Biggs-Hoare [1] and r (PSL(2, q); D16, S4, D8) due to Wong [22] These examples also appear in Li [14]. However, let us pay more attention to the case q = 9. Here we are dealing with a geometry whose Buekenhout diagram is given by o-4-o 2 2 B = D 15 15 RPRI S4 S4 (2T)1, s = 4 This is the smallest thick generalised quadrangle. Its origin is the symplectic group Sp4(2); in that context it is known at least from [17]. It is also famous as Tutte's 8-cage [18]. Its incidence graph admits an automorphism group four times as big as group PSL(2, 9) which is PrL(2,9). Under the action of this group we check that the graph is actually 5-arc-transitive and this is also provided by Tutte. Moreover, for the cases in which q = 17,23,31,41,47, 71, 73, 79, 89 the full automorphism group of the incidence graph is the group PGL(2, q). This group has a unique conjugacy class of subgroups S4, according to E.H. Moore as we see in [4]. Thus PGL(2, q) fuses the two classes of S4 in PSL(2, q) and so it cannot provide 5-arc-transitivity. Finally, for the case r (PSL(2, q); D16, S4, for q = 17,31,79,97, there are two classes of S4 in PSL(2, q) that are fused in PGL(2, q). There are two such geometries for each value of q and so the full automorphism group of r is PSL(2, q). (see Proposition 5.10). 7 Appendix The Appendix contains details for several results of this paper, except the proofs of Lemmas 5.7, 5.8, 5.9, 5.11, 5.13, 5.14, 5.15, 5.17, 5.18, 5.19, 5.20 which are left to the reader. Appendix is available on-line at: http://amc-journal.eu/index.php/ amc/issue/view/17. References [1] N. L. Biggs and M. J. Hoare, The sextet construction for cubic graphs, Combinatorica 3 (1983), 153-165. [2] F. Buekenhout, J. De Saedeleer and D. Leemans, Two-transitive pairs in PSL(2, q), in preparation. [3] F. Buekenhout, M. Dehon and D. Leemans, An Atlas of residually weakly primitive geometries for small groups, Mem. Cl. Sci., Coll. 8, Ser. 3, Tome XIV. Acad. Roy. Belgique, 1999. [4] P. J. Cameron, G. R. Omidi and B. Tayfeh-Rezaie, 3-designs from PGL(2,q), Electron. J. Combin. 13 (2006), R#50, 11 pp. [5] J. De Saedeleer and D. Leemans, On the rank two geometries of the groups PSL(2, q): part I, Ars Math. Contemp. 3 (2010), 177-192. [6] L. E. Dickson, Linear groups: With an exposition of the Galois field theory, with an introduction by W. Magnus. Dover Publications Inc., New York, 1901. [7] X. G. Fang, C. H. Li and C. E. Praeger, The locally 2-arc transitive graphs admitting a Ree simple group, J. Algebra (2004), 638-666. [8] M. Giudici, C. H. Li and C. E. Praeger, Analysing finite locally s-arc transitive graphs, Trans. Amer. Math. Soc. 356 (2004), 291-317. [9] A. Hassani, L. R. Nochefranca and C. E. Praeger, Two-arc transitive graphs admitting a two-dimensional projective linear group, J. Group Theory 2 (1999), 335-353. [10] B. Huppert, Endliche Gruppen. I, Die Grundlehren der Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967. [11] O. H. King, The subgroup structure of finite classical groups in terms of geometric configurations, in: Surveys in combinatorics 2005, volume 327 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 2005, pages 29-56. [12] D. Leemans, Locally s-arc-transitive graphs related to sporadic simple groups, J. Algebra 322 (2009), 882-892. [13] D. Leemans and E. Schulte, Polytopes with groups of type PGL(2, q), Ars Math. Contemp. 2 (2009), 163-171. [14] C. H, Li, The finite vertex-primitive and vertex-biprimitive s-transitive graphs for s > 4, Trans. Amer. Math. Soc. 353 (2001), 3511-3529. [15] E. H. Moore, The subgroups of the generalized finite modular group, Decennial Publications of the University of Chicago 9 (1904), 141-190. [16] M. Suzuki, Group theory. I, volume 247 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1982, translated from the Japanese by the author. [17] J. Tits, Sur la trialite et certains groupes qui s'en deduisent, Publ. Math. I.H.E.S. 2 (1959), 14-60. [18] W. T. Tutte, A family of cubical graphs, Proc. Cambridge Philos. 43 (1947), 459-474. [19] P. Vanden Cruyce, Géométries des groupes PSL(2,q), PhD thesis, Universite Libre de Bruxelles, 1985. [20] K. Vanmeerbeek, Rwpri meetkunden voor kleine bijna enkelvoudige groepen met als sokkel PSL(2,q). deel ii: de atlas, Master's thesis, Vrije Universiteit Brussel, 1999. [21] H. Wielandt, Finite permutation groups, translated from the German by R. Bercov, Academic Press, New York, 1964. [22] W. J. Wong, Determination of a Class of Primitive Permutation Groups, Math. Zeitschr. 99 (1967), 235-246. d MFA Also available at http://amc.imfm.si ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 389-392 A note on a conjecture on consistent cycles Stefko MiklaviC * University of Primorska, Andrej Marusic Institute, Muzejski trg 2, 6000 Koper, Slovenia Received 28 December 2011, accepted 9 July 2012, published online 17 April 2013 Abstract Let r denote a finite digraph and let G be a subgroup of its automorphism group. A directed cycle C of r is called G-consistent whenever there is an element of G whose restriction to C is the 1-step rotation of C. In this short note we prove a conjecture on G-consistent directed cycles stated by Steve Wilson. Keywords: Digraphs, consistent directed cycles. Math. Subj. Class.: 05C20, 05C38, 05E18 1 Introduction Let r denote a finite digraph (without loops and multiple arcs). By a directed cycle in r we mean a cyclically ordered set C = {v0, v1, v2,..., vr-1}, r > 3, of pairwise distinct vertices of r such that (v®, vi+1) is an arc of r for every i e Zr (the addition being mod r). Let G be a subgroup of the automorphism group of r. Directed cycle C is called G- consistent, if there exists g e G such that vg = vi+1 for each i e Zr .In this case g is called a shunt for C. Clearly, G acts on the set of G-consistent directed cycles: for h e G, Ch = {vh, vh, vh,..., vh_1} is G-consistent with a shunt h-1gh. Consistent cycles in finite arc-transitive graphs were introduced by J. H. Conway in one of his public lectures [3]. Since then a number of papers on consistent cycles and their applications appeared, see [1, 2, 4, 5, 6, 7, 8, 9, 10, 11]. Observe that if (u, v) is an arc of r and g e G is such that u9 = v, then the orbit of u under g induces a G-consistent directed cycle {u, v = u9,u9 ,...} (provided that u9 = u). Steve Wilson [12] stated the following conjecture on consistent cycles. *This work is supported in part by "Agencija za raziskovalno dejavnost Republike Slovenije", research program P1-0285 and research project J1-4010. E-mail address: stefko.miklavic@upr.si (Stefko Miklavic) Conjecture 1.1. Let r denote a finite digraph (without loops and multiple arcs) and let G be an arc-transitive subgroup of its automorphism group. Pick vertices u, v of r, such that (u, v) is an arc of r. For a G-orbit A of G-consistent directed cycles, let Ba denote the set of all automorphisms g G G, such that ug = v, and the orbit of u under g is in A. Let G(uv) denote the G-stabilizer of the arc (u, v). Then the number of elements in Ba is independent of A, and is equal to the order of G(u,v). In this short note we prove the above conjecture. 2 Proof of the conjecture In this section we prove Conjecture 1.1. We prove Conjecture 1.1 in two steps. In Proposition 2.1 we prove that \G(uv)\ < \Ba \, and in Proposition 2.2 we prove that \Ba\ < \G(u,v) \. Proposition 2.1. With the notation of Conjecture 1.1, we have \G(uv)\ < \Ba\. Proof. Since G is arc-transitive, there exists a G-consistent directed cycle C in A, which contains the arc (u, v). Let g denote a shunt for C. Let G g denote the pointwise stabiliser of C and let k be the index of G g in G(u v). Let gi,..., gk be representatives of cosets of GC in G(u,v). Observe that for each 1 < i < k and each h G Gc, the automorphism g-ighgi sends u to v. Furthermore, the orbit of u under g-1ghgi is the directed cycle Cgi. Namely, since g is a shunt for C and h G Gc, the image of vgj 9i under g-1ghgi is vgj+1gi. Moreover, C9i is clearly in A. Therefore, g-1ghgi G BA. We claim that if either i = j or h1 = h2 (h1,h2 G G g ), then a = g-1gh1gi and ß = g-1 gh2gj are distinct. Indeed, assume first that i = j. Note that Cgi = Cgj since gi and g j are from different cosets of G g in G(uv). Moreover, a is a shunt for Cgi and ß is a shunt for Cgj. Since Cgi = Cgj (and since Cgi and Cgj have at least the arc (u, v) in common), it follows that also a = ß. On the other hand, if i = j and a = ß, then h1 = h2. Therefore, if h1 = h2 and i = j, then a = ß. This proves the claim. It follows that |Ba| > k|GC\ = \G(u,v)\. □ Proposition 2.2. With the notation of Conjecture 1.1, we have \BA\ < \G(u,v)\. Proof. Let X denote the set of all G-consistent directed cycles in A, containing the arc (u,v). Clearly, BA is exactly the set of all shunts of directed cycles from X. Since all directed cycles from X have the arc (u, v) in common, every element of BA is a shunt for exactly one directed cycle from X. Note also that X is nonempty as G is arc-transitive. We now define a mapping ^ from BA to G(u v) as follows. Fix C G X and a shunt g g of C. For each D G X there exists an element of G which sends D to C. Pick such an element and denote it by h(D). Composing h(D) with an appropriate power of g g, we could assume that h(D ) G G(uv). For each g G Ba, let D (g) denote the unique directed cycle in X, for which g is a shunt (see Figure 1). For g G Ba define ^(g) = gh(D(g))g-1 and note that ^(g) G G(u,v). We now show that ^ is an injection. Pick g1, g2 G Ba and assume that ^(g1) = ^(g2). Let D (g1) = {u, v, v1, v2,..., vn-1} and D (g2) = {u,v,w1,w2,... ,wn-1}. We first D = D (g) Figure 1: Directed consistent cycles C and D. show that D(gl) = D(g2). Since tf(gi) = gih(D(g^)g_ 1 = g2h(D(g2))ffđ 1 = ^(ffa), we have g_1g1 = h(D(g2))h(D(g1))-1. This implies that g_1g1 is in G(u,v). We claim that = w„_j for i = 0,1,... n — 1, where vn = wn = u. We prove our claim using induction on i. Note that our claim is true for i = 0. Assume that our claim is true for i = 0,1,..., t, where 0 < t < n — 2. Note that h(D(g2))h(D(g1))_ fixes the arc (vn_t, vn_t+1,... vn_1,u, v), and therefore also g_1g1 fixes this arc. But since gi g2 gi vn_t_1 = vn_t = vn_t = wn1-^^ we have vn_t_1 = wn_t_1, verifying the claim. It follows that D(g1) = D(g2). But since D(g1) = D(g2), also h(D(g1)) = h(D(g2)). As g1h(D(g1))g_1 = g2h(D(gO)g_1, it follows that g1 = g2. Therefore ^ is an injection and so |BaI < |G(u,v) |. □ Corollary 2.3. With the notation of Conjecture 1.1, we have |BaI = |G(„jV) |. Proof. Immediately from Propositions 2.1 and 2.2. □ References [1] M. Boben, S. Miklavic and P. Potocnik, Consistent cycles in half-arc-transitive graphs, Electron. J. Combin. 16 (2009), R5. [2] M. Boben, S. Miklavic and P. Potocnik, Rotary polygons in configurations, Electron. J. Combin. 18 (2011), P119. [3] J. H. Conway, Talk given at the Second British Combinatorial Conference at Royal Holloway College, Egham, 1971. [4] H. H. Glover, K. Kutnar, A. Malnic and D. Marusic, Hamilton cycles in (2,odd,3)-Cayley graphs, J. London Math. Soc. 104 (2012), 1171-1197. [5] H. H. Glover, K. Kutnar and D. Marušic, Hamiltonian cycles in cubic Cayley graphs: the (2,4k, 3) case, J. Algebraic Combin. 30 (2009), 447-475. [6] W. M. Kantor, Cycles in graphs and groups, Amer. Math. Monthly 115 (2008), 559-562. [7] I. Kovacs, K. Kutnar and J. Ruff, Rose window graphs underlying rotary maps, Discrete Math. 310 (2010), 1802-1811. [8] I. Kovacs, K. Kutnar and D. Marušic, Classification of edge-transitive rose window graphs, J. Graph Theory 65 (2010), 216-231. [9] K. Kutnar and D. Marušic, A complete clasification of cubic symmetric graphs of girth 6, J. Combin. Theory Ser. B 99 (2009), 162-184. [10] S. Miklavic, P. Potocnik and S. Wilson, Consistent cycles in graphs and digraphs, Graphs Combin. 23 (2007), 205-216. [11] S. Miklavic, P. Potocnik and S. Wilson, Overlap in consistent cycles, J. Graph Theory 55 (2007), 55-71. [12] S. Wilson, Personal communication (2009). d MFA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 393-408 Sectional split extensions arising from lifts of groups Rok Požar Institut za matematiko, fiziko in mehaniko, Jadranska 19, 1000 Ljubljana, Slovenia and Fakulteta za matematiko, naravoslovje in informacijske tehnolgije, Univerza na Primorskem, Glagoljaška 8, 6000 Koper, Slovenia Received 3 September 2012, accepted 4 April 2013, published online 9 September 2013 Abstract Covering techniques have recently emerged as an effective tool used for classification of several infinite families of connected symmetric graphs. One commonly encountered technique is based on the concept of lifting groups of automorphisms along regular covering projections p : X ^ X. Efficient computational methods are known for regular covers with cyclic or elementary abelian group of covering transformations CT(p). In this paper we consider the lifting problem with an additional condition on how a group should lift: given a connected graph X and a group G of its automorphisms, find all connected regular covering projections p : X ^ X along which G lifts as a sectional split extension. By this we mean that there exists a complement G of CT(p) within the lifted group G such that G has an orbit intersecting each fibre in at most one vertex. As an application, all connected elementary abelian regular coverings of the complete graph K4 along which a cyclic group of order 4 lifts as a sectional split extension are constructed. Keywords: Covering projection, graph, group extension, lifting automorphisms, voltage assignment. Math. Subj. Class.: 05C50, 05E18, 20B40, 20B25, 20K35, 57M10 1 Introduction Graph covers play a significant role when symmetry properties of graphs are investigated. One of the commonly used techniques is based on the concept of lifting automorphisms along regular covering projections. Applications of this technique have been used to classify families of graphs with given structural properties (see for instance [2, 11, 12, 19,20]). E-mail address: pozar.rok@gmail.com (Rok Požar) In its most general form the problem of lifting automorphisms is well understood. Much attention has been devoted to finding the necessary and sufficient lifting conditions in combinatorial terms, see [15, 16,26,27]. Nevertheless, these general results are rather hopeless to apply when concrete examples and more detailed questions related to symmetry properties of graphs are considered. In a more specific setting of regular covers in which the group of covering transformations is either cyclic or elementary abelian, the situation changes. For such covers, efficient computational methods are known. For example, in the case of elementary abelian regular covers, the idea behind the approach developed in [19] is to reduce the general lifting problem to that of finding invariant subspaces of matrix groups over prime fields, linearly representing the action of automorphisms on the first homology group of the graph. Applying this method to a number of symmetric graphs - including the complete graphs K4 [20] and K5 [13], the Mobius-Kantor graph [18], the complete bipartite graph K33 [20], the Petersen graph [21], the Pappus graph [25], the octahedron graph [14], and the Hea-wood graph [19] - has resulted in the classification of connected elementary abelian regular covers admitting various types of subgroups of automorphisms. A similar approach, also based on linear criteria for lifting automorphisms, was proposed in [3], and has been used in order to find connected regular coverings with cyclic or elementary abelian group of covering transformation for the complete graph K4 [6], the 3-dimensional cube graph Q3 [7], the complete bipartite graph K3,3 [4], and the Petersen graph [5]. Assuming that a group G of automorphisms of X lifts along a regular covering projection p : X ^ X, the lifted group G is an extension of the group of covering transformations CT(p) by G. Specific types of extensions have usually a strong impact on structural properties of the covering graph X. In this context, the following two cases deserve special attention: (i) G is a split extension of CT(p) by G, and in particular, (ii) G is a direct split extension of CT(p) by G. For short we say that G lifts as a split extension or as a direct split extension, respectively. In the former case there exists, by definition, a complement G of CT(p) within G, and a normal complement G of CT(p) in the latter. This allows us to compare actions of two isomorphic groups, G on X and G on X, where G projects isomorphically onto G along p. However, it can happen that the complement is not unique, and what is more, different complements can exhibit different actions on X. Therefore, the analysis can be quite complicated. Certain algorithmic aspects related to the question of how difficult is to test conditions (i) and (ii) are considered in [22]. According to particular kinds of actions that can arise from complements, two extremal cases seem to stand out: (iii) there exists a complement G that acts transitively on the covering graph X, and (iv), there exists a complement G that is sectional. By this we mean that there is a section of X - a set of vertices containing at most one vertex from each fibre - invariant under the action of G. For short we say that G lifts as a transitive split extension or as a sectional split extension, respectively. Clearly, one might further restrict conditions (iii) and (iv) to normal complements. Certain particular questions along these lines have been addressed in [1, 8, 16, 17]. Motivated by the above discussion, the following problem is of interest. Given a connected graph X and a group G of its automorphisms, find all connected regular covering projections p : X ^ X along which G lifts in a prescribed way. In this paper we restrict to case (iv) - we introduce a method for finding regular coverings along which G lifts as a sectional split extension. The basic idea behind our approach is the following. First, we take the cone X over the graph X obtained by adding a^ew vertex * joined to every vertex of X, together with the group of automorphisms G of X that fixes * and acts on X as the group G. Next, the condition for lifting G as a sectional split extension is reduced to the general lifting problem of finding regular coverings of X admitting the lift of G. Consequently, the original problem can be solved as soon as the general lifting problem can be solved. Our approach is illustrated on a concrete example: we construct all connected elementary abelian regular coverings of the complete graph K4 along which a cyclic group of order 4 lifts as a sectional split extension. The rest of the paper is organized as follows. In Section 2 we review some preliminary concepts about regular graph covers and lifting automorphisms. In Section 3 we devise a method for constructing connected regular covering projections along which G lifts as a sectional split extension. A detailed example is provided in Section 4. 2 Preliminaries A graph is an ordered quadruple X = (D, V; beg,-1 ), where DX = D and VX = V are disjoint sets of darts and vertices, respectively, beg is a mapping that assigns to each dart x its initial vertex beg(x), and -1 is an involution interchanging every dart x and its inverse dart x-1. For a dart x, its terminal vertex is the vertex end(x) = beg(x-1). The orbits of -1 are called edges. An edge e = {x, x-1} is called a link whenever beg(x) = end(x). If beg(x) = end(x), then the respective edge is either a loop or a semi-edge, depending on whether x = x-1 or x = x-1, respectively. All graphs in this paper are assumed to be finite, meaning that the sets of vertices and darts are finite. A graph homomorphism f : Y ^ X is an adjacency preserving mapping taking darts to darts and vertices to vertices, or more precisely, f (beg(x)) = beg(f (x)) and f (x-1) = f (x)-1. An isomorphism is a bijective homomorphism. An isomorphism of a graph onto itself is an automorphism. All automorphisms of a graph X together with composition of automorphisms constitute the automorphism group Aut(X). A surjective homomorphism p : X ^ X is called a regular covering projection if there exists a semi-regular subgroup Sp of Aut(X) such that its vertex orbits and dart orbits coincide with vertex fibres p-1(v), v G VX, and dart fibres p-1(x), x G DX, respectively. Two regular covering projections p : X ^ X and p' : XX' ^ X are isomorphic if there exist an automorphism g of X and an isomorphism g : XX ^ XX' such that the following diagram X ——■ XX ' 4 Jy X -> X g commutes. In particular, if g = id then p and p' are equivalent. If, in the above setting, XX = XX' and p = p', then we say that g lifts along p or that g is a lift of g along p. A group G < Aut(X ) lifts if all g G G lift. The collection of all lifts of all elements in G forms a subgroup G < Aut(X), the lift of G. In particular, the lift of the trivial group is known as the group of covering transformations and denoted by CT(p). Observe that G is an extension of CT(p) by G. Furthermore, if G lifts along a given projection p, then it lifts along any covering projection equivalent to p. This allows us to study lifts of automorphisms combinatorially in terms of voltage assignments, a concept that we are going to describe now. Let X be a graph and let N be an (abstract) group, called the voltage group. Assign to each dart x of X a voltage Zx G N in such a way that Zx-i = Z—1. Such a function Z : DX ^ N is called a voltage assignment on X. Further, construct the derived graph Cov(Z) with vertex set VX x N and dart set DX x N, where beg(x, n) = (beg(x), n) and (x, n) —1 = (x—1,n Zx). The projection onto the first coordinate pz : Cov(Z) ^ X is then the derived regular covering projection, where the required semi-regular subgroup SP( of Aut(Cov(Z)) arises from the action of N on the second coordinate by left multiplication on itself. Conversely, any regular covering projection p : X ^ X can be reconstructed by a voltage assignment Z on X such that the projection pz derived from Z is equivalent to p. Moreover, one can assume that the voltage assignment Z is T-reduced for some arbitrarily chosen spanning tree T of X, meaning that Zx = 1 for all darts x in T, see [9] for more details. Consider now a regular covering projection p of connected graphs. Then we say that p is connected. Further, the semi-regular group Sp is equal to CT(p), and the voltage assignment Z that reconstructs the projection p is valued in the voltage group N = CT(p) (viewed as an abstract group). Such a voltage assignment Z is also called connected. It is well known that Z is connected if and only if each element of N appears as the voltage of some closed walk. Furthermore, by the basic lifting lemma [15, 16], an automorphism g of X lifts along pz if and only if each closed walk with trivial voltage is mapped by g to a walk with trivial voltage. Two assignments Z and Z' on X are equivalent whenever the respective derived regular covering projections pz and pz' are equivalent. Assuming that both assignments are connected and valued in N, then they are equivalent if and only if there exists an automorphism of N mapping Zw to ZW for each closed walk W at u0 [27]. For a given connected graph X and subgroup G < Aut(X ), the problem of finding regular covering projections p along which G lifts is very difficult in general. However, in the case of elementary abelian regular coverings p - that is, when CT(p) is isomorphic to an elementary abelian group - the necessary and sufficient lifting condition can be stated combinatorially by means of voltages as follows, see [19]. Let p be a prime. The first homology group H1(X ; Zp) is generated by the (directed) cycles of X and is isomorphic to the elementary abelian group Zp, where r is the Betti number of the graph X. The group H1(X ; Zp) is usually viewed as a vector space over Zp of dimension r. Since each automorphism a G Aut(X) maps a cycle in X to a cycle in X, there is a natural action of a on H1(X; Zp) which induces a linear transformation a# of H1(X ; Zp). Choose a spanning tree T of X and exactly one dart from each edge {x, x—1} that is not contained in T. Then the sequence x1, x2,..., xr G DX\Dt of all such darts naturally defines an (ordered) basis bt = {C1, C2,..., Cr} of H1(X; Zp), where C is the cycle arising from the spanning tree T and the dart x,. Next, denote the matrix representation of a# with respect to the basis bt by Ma G Zp,r. Thus, a subgroup G < Aut(X) induces a subgroup MG = {Ms | g G g} < GL(r, Zp). By Mfa we denote the dual group consisting of all transposes of matrices in MG. Theorem 2.1. ([19, Proposition 6.3, Corollary 6.5]) With the notation above, let Z : DX ^ Zd1 be a T-reduced voltage assignment on X, and let Z G Zdr be the matrix with columns Cxi , Zx2 , . . . , Zxr . If Z has rank d, then the derived graph Cov(Z ) is connected and the following hold: (i) A group G < Aut(X ) lifts along pz : Cov(Z ) ^ X if and only if the columns of ZG form a basis of a MlG-invariant d-dimensional subspace S (Z ) of Zp1 = H1(X ; Zp). (ii) If Z' : DX ^ Zp'1 is another voltage assignment on X satisfying the above conditions, then pz' is equivalent to pz if and only if S (Z ') = S (Z ). Moreover, pz' is isomorphic to pz if and only if there exists an automorphism a G Aut(X ) such that the matrix Ml0 maps S (Z ') onto S (Z ). By Theorem 2.1, we can find all pairwise nonequivalent connected elementary abelian regular coverings of X along which G lifts - in terms of voltages - as follows. First find a basis {u1,ut,..., ud} for each MtG-invariant subspace U of Zp'1. Next, for each basis {u1, ut,..., ud} consider a matrix Z with rows ut1,ut2,..., ufd, and then define the voltage assignment ZU : DX ^ Zp'1, mapping dart Xj to the i-th column of Z, i = 1, 2,... ,r, and mapping all darts of T to the trivial voltage. Observe that the choice of a spanning tree together with a sequence x1,xt,... ,xr as well as choosing a basis for an invariant subspace is irrelevant as long as we consider regular coverings up to equivalence. Thus, the problem of finding connected elementary abelian regular coverings along which a given group of automorphisms lifts translates to a purely algebraic question of finding invariant subspaces of finite linear groups. In this context, let A G Znn be an n x n matrix over a field Zp, acting as a linear transformation on the column vector space Zn1. Next, letka(x) = f1 (x)ni ft(x)n2 ■ ■ ■ fk(x)nk be the characteristic polynomial and mA(x) = f1(x)si ft(x)s2 ■ ■ ■ fk(x)sk the minimal polynomial of A where polynomials f are pairwise distinct and irreducible over Zp. Then Zpn,1 can be written as a direct sum of the A-invariant subspaces Fn'1 = Kerf1(A)S1 e Kerft(A)S2 e ■ ■ ■ e Kerffc(A)sk. Moreover, all A-invariant subspaces appear as direct sums of some A-invariant subspaces of Kerfj (A)Si. As for finding common invariant subspaces of a finite linear group, we can often exploit Maschke's theorem which states that if the characteristic of the field does not divide the order of the group, then the representation is completely reducible. In this case one essentially needs to find just the minimal common invariant subspaces. In particular, if the order of the matrix A is not divisible by p, each A-invariant subspace of Zn1 is a direct sum of the minimal ones. For a more detailed description of finding invariant subspaces we refer the reader to [10]. 3 Sectional split extensions We start by giving a more precise definition of a sectional split extension mentioned in the Introduction. Let p : X ^ X be a regular covering projection of connected graphs, and let Q be a nonempty set of vertices of X. A section over Q is a set of vertices & of X containing exactly one vertex from each vertex fibre over Q. Further, let G be a group of automorphisms of X. Assuming that Q is invariant under the action of G, we say that G lifts along p to G as a sectional split extension over Q if the following two conditions are met: (a) G lifts along p and (b) there exist a complement G to CT(p) within G and a section Q over Q that is invariant under the action of G. Such a complement is called sectional over Q. The necessary and sufficient conditions for G to lift as a sectional split extension over Q in terms of voltages were given by Malnic et al. This is summarized in the following theorem. Theorem 3.1. ([16, Theorem 9.1, Theorem 9.3]) With the notation and assumptions above, a group G lifts along p as a sectional split extension over Q if and only if p can be reconstructed by a voltage assignment Z on X such that the following condition Zw = 1 ^ ZgW = 1 (3.1) holds for each automorphism g G G and each walk W in X with both its endpoints in Q . Firstly, note that this theorem is an extended version of an old result of Biggs [1], retold in a different language. Secondly, Malnic took this result further in [17], and used it to sketch a method for testing whether G lifts along p as a sectional split extension over Q. The approach is based on introducing a new vertex joined to every vertex of Q, and then converting condition (3.1) to the general lifting problem (but no proof is given). In order to exploit this idea in another direction (see below), we introduce the following notation. The cone X (Q) over the graph X is the graph obtained by adding a new vertex * joined to every vertex of Q. Assuming that Q is invariant under the action of G, we denote by G the group of automorphisms of X(Q) that fixes * and acts on X as the group G. Also, for any voltage assignment Z on X, we extend Z to a voltage assignment Z on X(Q) by assigning the trivial voltage to the extra darts. More precisely, C f Zx, x G Dx ; i1, x G DX(q)\Dx. Conversely, for a voltage assignment Z on X(Q) being trivial on the set of extra darts we denote by C the restriction of Z to X. Clearly, if Z is not trivial on the set of extra darts, then we can always find an equivalent assignment that is. For example, we may choose a spanning tree T * of X (Q) such that all extra darts are included in T *, and then take an equivalent T *-reduced voltage assignment. Moreover, the following holds. Proposition 3.2. Let Z and Z' be two equivalent connected voltage assignments on X(Q), that are trivial on the set of extra darts D^(Q)\DX. Then their restrictions C and Z' to X are also equivalent. Hence they are either both connected or both disconnected. Proof. By definition of equivalence, there exists an isomorphism g from the derived graph Cov(Z) to the derived graph Cov(Z') such that pz = gpz>. Clearly, g maps the vertex fibre p-1(*) to the vertex fibre p-1(*). Therefore, when restricting to X, the isomorphism g induces an isomorphism from the derived graph Cov(C) to Cov(Z') that gives rise to an equivalence of C and Z'. It is then obvious that isomorphic graphs are either both connected or both disconnected, as required. □ We are now ready to forge a link between connected regular coverings of X along which G lifts as a sectional split extension over Q, and connected regular coverings of X(Q) admitting the lift of G. For completeness, we explicitly record the following theorem and provide the proof. Theorem 3.3. Let p : X ^ X be a regular covering projection of connected graphs, and let G be a group of automorphisms of X. Suppose that a nonempty subset Q of vertices of X is invariant under the action of G. Then the group G lifts along p as a sectional split extension over Q if and only if p can be reconstructed by a voltage assignment Z on X such that G lifts along the derived regular covering projection pc: Cov(Z ) ^ X(Q). Proof. Suppose that G lifts along p as a sectional split extension over Q. By Theorem 3.1, there exists a voltage assignment Z on X that reconstructs p and satisfies condition (3.1). Extend Z to a voltage assignment Z. We will show that G lifts along the projection p^ derived from Let W * be a closed walk at * in X (Q) with Zw * = 1, and let g* e G. In view of the basic lifting lemma we need to show that Zg*W* = 1. Write W* as a concatenation W* = W*W2*... W* of closed walks at * such that W* = PiWiQ where Wi : ui ^ vi is a walk in X with both its endpoints ui and vi in Q, while Pi : * ^ ui and Qi : * ^ vi are walks of length 1, for i = 1, 2,..., k. Observe that Zw Zw ... Zw = 1. Now choose a vertex u0 e Q. Let Ri : u0 ^ ui and Si : u0 ^ vi be walks with ZRi = Zs = 1, for i = 1,2,..., k (note that such walks always exist). Then the product of walks W = f]*=1 RiWiSi-1 is a closed walk at u0 with Zw = Zw ZW2 ... Zw = 1. By condition (3.1) we have that ZgW = 1 as well as ZgRj = ZgSj = 1, for i = 1, 2,..., k. Thus Zgw ZgW2 ... ZgWk = 1 implies that Zg*W* = 1, as required. Conversely, suppose that p is reconstructed by a voltage assignment Z on X such that G lifts along the covering projection p^. By Theorem 3.1, it is sufficient to prove that Z satisfies condition (3.1). Consider a walk W : u ^ v in X with both its endpoints u and v in Q such that ZW = 1. Let P : * ^ u and Q : * ^ v be the (unique) walks of length 1 in X(Q). Then the closed walk W* = PWQ-1 at * has voltage Cw* = 1. By the basic lifting lemma we have Zg*W* = 1 for any automorphism g* e G. Hence ZgW = 1, completing the proof. □ Coming back to methods for testing whether G lifts along p as a sectional split extension over Q, one possibility would be to use the latter theorem. However, from computational point of view that would be inefficient, since one has to seek for an appropriate voltage assignment that reconstructs the cover. For a more adequate approach to this problem we refer the reader to [23]. As already mentioned, Theorem 3.3 can be efficiently exploited in another direction: given a connected graph X, a group G of its automorphisms, and a nonempty subset Q C Vx invariant under the action of G, find, up to equivalence, all connected regular coverings p : X ^ X along which G lifts as a sectional split extension over Q. As a first step towards this aim we need to find, in view of Proposition 3.1 and Theorem 3.3, all pairwise nonequivalent connected regular coverings of X(Q) along which the group G lifts - combinatorially reconstructed in terms of voltage assignments Z being trivial on the set of extra darts. Although each Z is connected - as it reconstructs a connected cover - its restriction £ to X, however, might be disconnected. Thus, additional testing whether C is connected is required. These remarks are formally gathered in the following theorem. Theorem 3.4. Let X be a connected graph and Q a nonempty subset of vertices of X that is invariant under the action of a group of automorphisms G < Aut(X). Further, let Z be a voltage assignment on X(Q) that is trivial on the set of extra darts Dx(fi)\DX and gives rise to a connected regular covering projection along which the group G lifts. If the restriction £ to X is connected, then G lifts along the derived regular covering projection p^ as a sectional split extension over Q. Moreover, any connected regular covering of X along which G lifts as a sectional split extension over Q arises in this way. Remark 3.5. Even if Z and Z' are two nonequivalent connected assignments on X (Q) such that their restrictions C and C' to X are connected, it still might happened that C and C' are equivalent. Thus, additional testing is needed. Now we can more precisely summarize our approach. First, construct all voltage assignments Z on X(Q) giving rise to pairwise nonequivalent connected regular covering projections along which G lifts. Next, consider their restrictions Z to X and remove the disconnected ones. Finally, do further reduction to obtain all voltage assignments on X giving rise to pairwise nonequivalent connected regular covering projections along which G lifts as a sectional split extension over Q. 4 Elementary abelian regular covers of K4 In light of the discussion in Section 3 we now give an example to illustrate our approach. Let X = K4 be the complete graph on the vertex set VX = {1, 2, 3,4}, and let Q = VX. Further, denote by g = (1234) G Aut(X) the automorphism of X. We compute all voltage assignments on X giving rise to pairwise nonequivalent connected elementary abelian regular covering projections along which the cyclic group G = (g) lifts as a sectional split extension over Q. To start with, we need to find all voltage assignments on X(Q) giving rise to pairwise nonequivalent connected elementary abelian regular coverings along which the group G = (g*) lifts. Let T* be the spanning tree of X(Q) consisting of all extra darts, and let x1 = (1, 2), x2 = (2, 3), x3 = (3,4), x4 = (4,1), x5 = (2,4), x6 = (3,1) denote the six cotree darts of XX(Q). Denote by Bj* = {xj 11 < i < 6} the ordered basis of the vector space H1(XX (Q); Zp), where x is the cycle arising from the spanning tree T * and the dart xj. Next, in view of the remarks given in Preliminaries, let (g*)# be the linear transformation of H1(XX(Q); Zp) induced by the natural action of g* on H1(XX(Q); Zp), and let Mg* G Z6'6 be its matrix representation with respect to the basis Bj*. By computation we obtain that ^0 1 0 0 0 (f 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 0 By Theorem 2.1, we need to find A-invariant subspaces of Zp'1. However, note that every elementary abelian regular Zd1 -cover of X is disconnected if the dimension d is higher that the Betti number of X. Since the Betti number of X is three, it is therefore enough to find all A-invariant subspaces of dimension at most three. These subspaces define T *-reduced voltage assignments Z : Dx(n) ^ Zd'1, d =1, 2, 3 on X(Q) that give rise to pairwise nonequivalent connected regular coverings of X(Q) along which G lifts. In addition, as already explained in the previous section, their restrictions C to X might still be disconnected as well as connected but equivalent. In order to test whether the restriction Z to X stays connected, let T be the spanning tree of X consisting of the edges {1,2}, {1,3} and {1,4}. Denote by C1, C2 and C3 cycles a = mg* = arising from the spanning tree T and darts x2, x3 and x5, respectively. The connectedness condition, relative to the ordered basis Bt = {C1,C2,C3} of #i(X; Zp), translates to the requirement that the voltages Cci = Cci = Cxi + Cx2 + Cx6 , Cc2 Cc2 Cx3 + Cx4 Cx6, Cc3 = Cc3 = Cxi + Cx4 + Cx5 generate the voltage group Z^'1. As for the test of equivalence, let Z and Z' be two T*-reduced voltage assignments on X(Q) arising from two different d-dimensional subspaces U and U' of Zp'1, respectively. Suppose that their restrictions C and Z' are connected. Then C and C' are equivalent, in view of [27], if and only if there exists an automorphism of Z^'1 mapping Zci ^ ZC1, Zc2 ^ ZC2, and Zc3 ^ ZC3. For the purpose of finding A-invariant subspaces, note that ka(x) = (x4 - 1)(x2 + 1) is the characteristic polynomial of A, while m^(x) = x4 — 1 is its minimal polynomial. Further, observe that the factorization of mA(x) into irreducible factors over Zp depends on the congruence class of p modulo 4, namely {(x — 1)(x + 1)(x2 + 1), p = 3 (mod4); (x — 1)(x + 1)(x — i)(x + i), p = 1 (mod4), i2 = —1; (x — 1)4, p = 2. Therefore the analysis splits into three cases. Case p = 3 (mod 4). In this case the representation of the group (A) is completely reducible, by Maschke's theorem. The eigenvalues are 1 and —1, both of multiplicity 1. The respective eigenspaces are (1) = (v1) and LA( —1) = (v2), where v1 = (1,1,1,1,0,0)4 and v2 = (1, —1,1, —1,0,0)4. The whole space splits into a direct sum of A-invariant subspaces Zp'1 = la(1) e la( —1) ® Ker(A2 + I). It is obvious that the 1-dimensional A-invariant subspaces are LA(1) and LA( —1). The respective lists of voltages for the base homology cycles C1, C2, C3 in X are 2, 2,2 for the one arising from LA(1), and 0,0,0 for the one arising from LA( — 1). Thus, only LA(1) gives rise to a connected cover of X, while LA(—1) does not. Since the 2-dimensional A-invariant subspace arising from the direct sum LA(1) e (—1) does not give a connected cover of X, all others are necessarily contained in Ker(A2 + I). These subspaces are of the form (v, Av), for v G Ker(A2 + I). There are p2 + 1 distinct subspaces. To check which of these give rise to connected covers of X, choose a basis of Ker(A2 +1), for instance b1 =(1,0,-1, 0,0,0)4, b2 =(0,1,0, -1,0,0)4, b3 =(0,0,0, 0,1,0)4, b4 =(0,0,0, 0, 0,1)4. An arbitrary vector v e Ker(A2 +1) is then of the form v = (a, b, —a, -b, c, d)4, for some a, b,c,d e Zp, while Av = (b, —a, —b, a, d, — c)4. For convenience we denote Wa,b}c}d = ((a, b, —a, —b, c, df, (b, —a, —b, a, d, —c)4). Checking for connectedness gives that (a, b)4 + (b, —a)4 + (d, —c)4, (—a, —b)4 + (—b, a)4 — (d, —c)4 and (a, b)4 + (—b, a)4 + (c, d)4 should generate Zp'1. The condition is reduced to requiring that (a + b + d, —a + b — c)4 and (a — b + c, a + b + d)4 are linearly independent in Z^'1. Let x = a + b + d and y = a — b + c. The vectors (x, y)4 and (—y, x)4 are linearly dependent if and only if x2 + y2 = 0 (modp) Since p = 3 (mod 4), wemusthave x = 0 (mod p) and y = 0 (mod p). Thus a disconnected cover of X is obtained if and only if c = —a + b and d = —a — b; in this case Wa,b,c,d is generated by va b = a(1,0, —1,0, —1, —1)4 + b(0, —1,0,1, —1,1)4 and Ava,b. Observe that any va,b is contained in (v1,0, Av1,0). Hence (va,b, Ava,b) = (v1,0, Av1,0) for all a, b e Zp. This is therefore the only A-invariant 2-dimensional subspace giving rise to a disconnected cover of X. As for the remaining subspaces, these are Wa,b,c,d where (c, d) = (—a+b, —a—b). Furthermore, these subspaces all give rise to equivalent coverings of X. Indeed. Choose one of these subspaces, say W1'1'0'0 = ((1,1, —1, —1,0, 0)4, (1, —1, —1,1,0,0)4). Let Z and Z' be two assignments arising from Wa,b,c,d and W1,1,0,0, respectively. The base homology cycles C1, C2, C3 in X have the following voltages Zc = (a + b + d, —a + b — c)4, ZC, = (2, 0)4, Zc2 = (—a — b — d, a — b + c)4, ZC2 = (—2, 0)4, Zc3 = (a — b + c, a + b + d)4, ZC3 =(0, 2)4. By computation one can check that there exists amatrix in GL(2, Zp) = Aut(Zp'1) taking Zci, Zc2 , Zc3 to ZC !, ZC2, ZC3, respectively, if and only if (c, d) = (—a + b, —a — b), and the claim is proved. As a representative of the above 2-dimensional subspaces we take WM'0'0. Any 3-dimensional A-invariant subspace giving rise to a connected cover of X is equivalent to the homological cover of X. So it is enough to find one such a subspace, if it exists. For instance, we may take the subspace LA(1) © W110 0, as the reader can easily check. Case p = 1 (mod 4). The representation of the group (A) is again completely reducible, by Maschke's theorem. The matrix A is diagonalizable, having the diagonal form diagA(1, -l,i,i, -i, -i). Clearly, the 1-dimensional eigenspaces LA( 1) and LA(-1) are the same as before, where only LA(1) gives rise to a connected cover of X. As for the eigenvalues i and —i satisfying i2 = -1 (modp), the respective eigenspaces LA(i) = (ui,vi) and LA(-i) = (u-i, v-i) are 2-dimensional, where Uj = (1, i, -1, -i, 1, i)*, u-i = (1, -i, -1, i, 1, -i)*, vi =(1,i, -1, -i, 0, 0)*, v-i = (1, -i,-1,i, 0,0)*. The 1-dimensional subspaces in LA(i) can be conveniently parametrized as W«,(i) = (ui), Ws(i) = (sui + vi) = ((s + 1, (s + 1)i, (s + 1), (s + 1)i, s, si)*), s G Z p, while those in LA(-i) can be parametrized as WTO(-i) = (u-i), Ws(-i) = (su-i + v-i) = ((s + 1, -(s + 1)i, -(s + 1), (s + 1)i, s, -si)*), s G Zp. The conditions for connectedness of covers of X arising from (i), Ws (i), WTO(-i) and Ws(-i) become i - 2 = 0 (modp), s(i - 2) = 1 - i (modp), -i - 2 = 0 (modp) and s(—i - 2) = 1 + i (modp), respectively. We need to consider subcases p =5 and p = 5 separately. Let p = 5. Then i, -i = 2, and there are (2p + 1) 1-dimensional subspaces giving rise to connected covers of X, namely the set Wi = |Ws(i) | s G (Zp\{(1 - i)(i - 2)-1}) U of p subspaces in LA(i), the set W-i = {Ws(-i) | s G (Zp\{(1 + i)(-i - 2)-1}) U {TO}} of p subspaces in LA(-i), and the subspace LA(1). However, all subspaces in Wi give rise to equivalent coverings of X. To show this, let Z and Z' be two assignments arising from Ws(i) and WTO(i), respectively. By computation we have Zci =(s + 1)(1 + i) + si, ZC ! =1 + 2i, Zc2 = - (s +1)(1 + i) - si = -ZC!, ZC 1 = - 1 - 2i = -ZC!, ZC3 =(s + 1)(1 - i) + s = -iZC!, ZC3 =2 - i = -iZC 1 . Clearly, there exists an automorphism of Zp taking ZCl, ZC2, ZC3 to ZC, ZC2, ZC3, respectively, if and only if ZCl = 0. In fact, we do have ZCl = 0 since s = (1 - i)(i - 2)-1. Similarly, all subspaces in W-i give rise to equivalent coverings of X. As a representative in Wi we choose W0(i), while in W-i we choose W0(-i). In fact, there are exactly three pairwise nonequivalent connected coverings of X, namely the one arising from LA(1), and the two coverings arising from W0(i) and W0(-i). The respective lists of voltages for the base homology cycles Ci, C2, C3 in X are 2,2,2 for the one arising from LA(1), while 1 + i, -1 - i, 1 - i and 1 - i, -1 + i, 1 + i for the other two covers. The reader may check that there is no automorphism of Zp taking any of these triples to any other. Let p = 5. Then for each s g Z5 the subspace Ws(2) gives rise to a connected cover of X, while the subspace WTO(2) does not. On the other hand, for each s = 3 we obtain a connected cover of X arising from Ws(3), and one connected cover of X arising from WTO(3). Together with the cover of X arising from LA(1) we therefore have 2p +1 = 11 connected covers. If Z denotes an assignment arising from Ws (2), then the base homology cycles in X have voltages ZCi = 3(s + 1) + 2s = 3, ZC2 = -3(s + 1) - 2s = - ZCi, ZC3 = -(s + 1) + s = -2ZCl. It is obvious that the subspaces Ws(2), s g Z5, give rise to equivalent coverings of X. As a representative we take W0 (2). Let now Z be an assignment arising from Ws(3), where s g Zp and s = 3. Further, let Z' denote an assignment arising from Wto(3). Then we have Zci = 4(s + 1)+3s = 2s - 1, Zc2 = - 4(s +1) - 3s = -Zci, Zc3 = - 2(s +1) + s = -3Zci, ZCi = 2, ZC 2 = - 2 ZC 3 = -1 - ZC1, -3ZC i. Clearly, multiplication by s + 2 takes ZCi, ZC2, ZC3 to ZC, ZC2, ZC3, respectively. As a representative we take W0(3). The reader may check that LA(1), W0(2) and W0(3) give rise to pairwise nonequivalent coverings of X. Let us now consider the 2-dimensional subspaces. We shall need the following lemma. Lemma 4.1. Let T* be a spanning tree of X(Q) such that all extra darts are included in T*, and let the sequence xi, x2,..., xn contain exactly one dart from each edge not contained in T *. Further, let U, U ', W, W ' be subspaces of Zn 1 such that U n W = {0} = U ' n W ', and let ZU, ZU', ZW, ZW', ZU®W, ZU'®W' denote T*-reduced voltage assignments on X(Q), where the voltages of darts xj arise from U, U', W, W', U © W, U' © W', respectively. Suppose that all their restrictions to X are connected. If the restrictions of ZU and ZU' are equivalent and the restrictions of ZW and ZW' are equivalent, then the restrictions of ZU®W and Z U'®W' are also equivalent. Proof. Since U n W = {0} = U' n W' we may assume, up to equivalence of regular covering projections, that Z U0W = Zx and ZX U'eW ' = x= for all darts x in X(Q). Let r be the Betti number of X, and let C1, C2,..., Cr be an ordered basis of H1 (X, Zp). Since the restrictions of ZU and ZU' are equivalent, there exists an invertible matrix A mapping voltages ZU to voltages ZC', i = 1, 2,..., r. Similarly, there exists an invertible matrix B mapping voltages Z [1] p = 2 3 [1] [i] [-1] -i] [0] [0] p = 1 (mod 4), i2 = -1 4 [1] M [-1] [i] [0] [0] p = 1 (mod 4), i2 = -1 5 0 0 "0" 0 0 0 0 0 1" 1 1" 0 p=2 6 Wi, 1,0,0 1" 1 1 -1 -1 -1 -1 1 0" 0 "0" 0 p = 3 (mod 4) 7 1" 1 "1" i 1 -1 1 -i 0" 0 "0" 0 p = 1 (mod 4), i2 = -1 8 1" 1 1 -i 1 -1 "1" i 0" 0 "0" 0 p = 1 (mod 4), i2 = -1 9 1 1 1 -1 -1 -1 -1 1 0 0 0 0 p = 3 (mod 4) 11 1" 1 1 i 1 -1 1 -i 0" 0 0 0 p = 1 (mod 4), i2 = -1 1 -i -1 i 0 0 Acknowledgement. The author would like to thank Aleksander Malnic for enlightening discussions, and both referees for their helpful suggestions that improved the presentation. References [1] N. L. Biggs, Algebraic Graph Theory, Cambridge Univ. Press, Cambridge, 1974. [2] S. J. Curran, D. W. Morris and J. Morris, Cayley graphs of order 16p are hamiltonian, Ars Math. Contemp. 5 (2012), 189-215. [3] S. F. Du, J. H. Kwak and M. Y. Xu, Lifting of automorphisms on the elementary abelian regular coverings, Lin. Alg. Appl., 373 (2003), 101-119. [4] Y. Q. Feng and J. H. Kwak, s-regular cubic graphs as coverings of the complete bipartite graph K3,3, J. Graph Theory 45 (2004), 101-112. [5] Y. Q. Feng and J. H. Kwak, Classifying cubic symmetric graphs of order 10p or 10p2, Science China Ser. A: Math 49 (2006), 300-319. [6] Y. Q. Feng and J. H. Kwak, Cubic symmetric graphs of order a small number times a prime or a prime square, J. Combin. Theory Ser. B 97 (2007), 627-646. [7] Y. Q. Feng, J. H. Kwak and K. Wang, Classifying cubic symmetric graphs of order 8p or 8p2, Europ. J. Combin. 26 (2005), 1033-1052. [8] Y. Q. Feng, A. Malnic, D. Marušic and K. Kutnar, On 2-fold covers of graphs, J. Combin. Theory Ser. B 98 (2008), 324-341. [9] J. L. Gross and T. W. Tucker, Topological Graph Theory, Wiley - Interscience, New York, 1987. [10] N. Jacobson, Lectures in Abstract Algebra, II. Linear Algebra, Springer, New York, 1953. [11] M. Klin and C. Pech, A new construction of antipodal distance regular covers of complete graphs through the use of Godsil-Hensel matrices, Ars Math. Contemp. 4 (2011), 205-243. [12] K. Kutnar, D. Marušic, D. W. Morris, J. Morris and P. Sparl, Hamiltonian cycles in Cayley graphs whose order has few prime factors, Ars Math. Contemp. 5 (2012), 27-71. [13] B. Kuzman, Arc-transitive elementary abelian covers of the complete graph K5, Linear Algebra Appl. 433 (2010), 1909-1921. [14] J. H. Kwak and J. M. Oh, Arc-transitive elementary abelian covers of the octahedron graph, Linear Algebra Appl. 429 (2009), 2180-2198. [15] A. Malnic, Group actions, coverings and lifts of automorphisms, Discrete Math. 182 (1998), 203-218. [16] A. Malnic, R. Nedela and M. Skoviera, Lifting graph automorphisms by voltage assignments, European J. Combin. 21 (2000), 927-947. [17] A. Malnicš, Action graphs and coverings, Discrete Math. 224 (2002), 299-322. [18] A. Malnicš, D. Marusšicš, S. Miklavicš and P. Potocšnik, Semisymmetric elementary abelian covers of the Möbius-Kantor graph, Discrete Math. 307 (2007), 2156-2175. [19] A. Malnic, D. Marušic and P. Potocnik, Elementary abelian covers of graphs, J. Alg. Combin. 20 (2004), 71-96. [20] A. Malnic, D. Marušic and P. Potocnik, On cubic graphs admitting an edge-transitive solvable group, J. Algebraic Combin. 20 (2004), 99-113. [21] A. Malnicš and P. Potocšnik, Invariant subspaces, duality, and covers of the Petersen graph, European J. Combin. 27 (2006), 971-989. [22] A. Malnic and R. PoZar, On the Split Structure of Lifted Groups, submitted. [23] A. Malnic and R. PoZar, On the Split Liftings with Sectional Complements, in preparation. [24] J. M. Oh, A classification of cubic s-regular graphs of order 14p, Discrete Math. 309 (2009), 2721-2726. [25] J. M. Oh, Arc-transitive elementary abelian covers of the Pappus graph, Discrete Math. 309 (2009), 6590-6611. [26] J. Siran, Coverings of graphs and maps, ortogonality, and eigenvectors, J. Algebraic Combin. 14 (2001), 57-72. [27] M. Skoviera, A contribution to the theory of voltage graphs, Discrete Math. 61 (1986), 281292. d MFA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 409-417 Augmented down-up algebras and uniform posets Paul Terwilliger *, Chalermpong Worawannotai Department of Mathematics, University of Wisconsin 480 Lincoln Drive, Madison, WI 53706-1388 USA Received 9 July 2013, accepted 31 August 2013, published online 9 September 2013 Abstract Motivated by the structure of the uniform posets we introduce the notion of an augmented down-up (or ADU) algebra. We discuss how ADU algebras are related to the down-up algebras defined by Benkart and Roby. For each ADU algebra we give two presentations by generators and relations. We also display a Z-grading and a linear basis. In addition we show that the center is isomorphic to a polynomial algebra in two variables. We display seven families of uniform posets and show that each gives an ADU algebra module in a natural way. The main inspiration for the ADU algebra concept comes from the second author's thesis concerning a type of uniform poset constructed using a dual polar graph. Keywords: Uniform poset, dual polar space, dual polar graph, down-up algebra. Math. Subj. Class.: 06A07, 05E10, 17B37 1 Introduction In [10] the first author introduced the notion of a uniform poset, and constructed eleven families of examples from the classical geometries. Among the examples are the polar spaces Polarb(N, e) and the attenuated spaces Ab(N, M), as well as the posets Altb(N), Herq(N), and Quadb(N) associated with the alternating, Hermitean, and quadratic forms. Another example is Hemmeter's poset Hemb(N). In [12, Proposition 26.4] the second author constructed a new family of uniform posets using the dual polar graphs. We denote these posets by PolarJb°p(N, e) and describe them in Section 5 below. In [2] Benkart and Roby introduced the down-up algebras, and obtained modules for these algebras using Altb(N), Herq(N), Quadb(N), and Hemb(N). A down-up algebra module is obtained from Polarb°p(N, e) in a similar way. However, it appears that the down-up algebra concept is not sufficiently robust to handle Polarb(N, e) or Ab(N, M). The * Corresponding author E-mail addresses: terwilli@math.wisc.edu (Paul Terwilliger), worawann@math.wisc.edu (Chalermpong Worawannotai) same can be said for the generalized down-up algebras [5]. In the present paper we introduce a family of algebras called augmented down-up algebras, or ADU algebras for short. These algebras seem well suited to handle uniform posets. Indeed, we show that each of the uniform posets Polarb(N, e), Ab(N, M), Altb(N), Herq(N), Quadb(N), Hemb(N), Polarb°p(N, e) gives an ADU algebra module in a natural way. The ADU algebras are related to the down-up algebras as follows. Given scalars a, ß, 7 the corresponding down-up algebra A(a, ß, 7) is defined by generators e, f and relations e2f = aefe + ßfe2 + Ye, ef2 = afef + ßf 2e + f. See [2, p. 308]. To turn this into an ADU algebra we make three adjustments as follows. Let q denote a nonzero scalar that is not a root of unity. We first require a = q-2s + q-2t, ß = -q-2s-2t where s, t are distinct integers. Secondly, we add two generators such that ke = q2ek and kf = q-2fk. Finally we reinterpret 7 as a Laurent polynomial in k for which the coefficients of ks,kt are zero. From the above description the ADU algebras are reminiscent of the quantum univeral enveloping algebra Uq(sl2). To illuminate the difference between these algebras, consider their center. By [6, p. 27] the center of Uq (sl2) is isomorphic to a polynomial algebra in one variable. As we will see, the center of an ADU algebra is isomorphic to a polynomial algebra in two variables. The results of the present paper are summarized as follows. We define two algebras by generators and relations, and show that they are isomorphic. We call the common resulting algebra an ADU algebra. For each ADU algebra we display a Z-grading and a linear basis. We also show that the center is isomorphic to a polynomial algebra in two variables. We obtain ADU algebra modules from each of the above seven examples of uniform posets. We have a remark about the place of down-up algebras and ADU algebras in ring theory. A down-up algebra can be viewed as an ambiskew polynomial ring [7, Section 3], which in turn can be viewed as a generalized Weyl algebra [1], [7, Prop. 2.1]. By a comment in [8, p. 48] that cites a preprint version of the present paper, an ADU algebra can also be viewed in this way. Hoping to keep our paper accessible to nonexperts in ring theory, we will avoid this point of view and use only linear algebra. Recall the natural numbers N = {0,1,2,...} and integers Z = {0, ±1, ±2,...}. 2 Augmented down-up algebras Our conventions for the paper are as follows. An algebra is meant to be associative and have a 1. A subalgebra has the same 1 as the parent algebra. Let F denote a field. Let A denote an indeterminate. Let F[A, A-1] denote the F-algebra of Laurent polynomials in A that have all coefficients in F. Pick ^ g F[A, A-1] and write ^ = J2®ez ajA®. By the support of ^ we mean the set {i g Z|a® =0}. This set is finite. Fix distinct s, t g Z. Define F[A, A-1]s,t = Span{Aj|i G Z, i = s, i = t}. Note that F[A, A-1] = F[A, A-1]m + FAs + FA4 (direct sum). For ' G F[A, A-1] the following are equivalent: (i) ' G F[A, A-1]s,t; (ii) the integers s, t are not in the support of Fix a nonzero q G F that is not a root of unity. Definition 2.1. For p G F[A, A-1]s,t the F-algebra A = Aq(s,t, p) has generators e, f, k±1 and relations kk-1 = 1, k-1k = 1, ke = q2ek, kf = q-2 fk, e2f - (q-2s + q-2t)efe + q-2s-2tfe2 = ep(k), (2.1) ef2 - (q-2s + q-2t)fef + q-2s-2tf2e = p(k)f. (2.2) Remark 2.2. Referring to Definition 2.1, consider the special case in which p G F. Then the relations (2.1), (2.2) become the defining relations for the down-up algebra A(q-2s + q-2t, -q-2s-2t, p). Definition 2.3. For $ G F [A, A-1]s,t the F-algebra B = Bq (s, t, 4>) has generators Cs, Cu E, F, K±1 and relations Cs , Ct are central, KK-1 = 1, K-1K = 1, KE = q2EK, KF = q-2FK, FE = C s qs Ks + CtqtKt + (qK ), (2.3) EF = Csq-sKs + Ctq-tKt + ^(q-1K ). (2.4) Next we describe how the algebras in Definition 2.1 and Definition 2.3 are related. Definition 2.4. We define an F-linear map F[A, A-1] ^ F[A, A-1], ' ^ 's,t as follows. For ' G F[A,A-1], 's,t(A) = '(q-1A) - (q-2s + q-2t)'(qA) + q-2s-2t'(q3 A). Recall the basis {Ai}ieZ for F[A, A-1]. Lemma 2.5. Consider the map ' ^ 's,t from Definition 2.4. For i G Z the vector Ai is an eigenvector for the map. The corresponding eigenvalue is q3i (q-2i — q-2s)(q-2i — q-2t). This eigenvalue is zero if and only if i G {s, t}. Proof. Use Definition 2.4. □ The following two lemmas are routine consequences of Lemma 2.5. Lemma 2.6. For the map ' ^ 's,t from Definition 2.4 the image is F[A, A-1]s,t and the kernel is FAs + Fa4. Lemma 2.7. For the map ^ ^ ^s,t from Definition 2.4 the restriction to F[A, A 1]s,t is invertible. Let G F[A, A-1]s,t such that f = ^s,t. We are going to show that the algebras Aq(s, t, f) and Bq(s, t, <) are isomorphic. Lemma 2.8. For < G F [A, A-1 ]s,t the following hold in Bq (s, t, <): C = q-tFE — qtEF + qt<(q-1K) - q-t <(qK) R_s 2 qs-t — qt-s ' = q-sFE — qsEF + qs<(q-1K) — q-s<(qK) K- 2 qt-s — qs-t Moreover the algebra Bq (s, t, <) is generated by E, F, K±1. Proof. We first verify (2.5). In the expression on the right in (2.5), eliminate FE and E F using (2.3) and (2.4). After a routine simplification (2.5) is verified. The equation (2.6) is similarly verified. The last assertion follows from (2.5), (2.6). □ Lemma 2.9. For < G F [A, A-1 ]s,t the following hold in Bq (s, t, <): E2F — (q-2s + q-2t)EFE + q-2s-2t F E2 = E((K ), (2.7) EF2 — (q-2s + q-2t)FEF + q-2s-2t F2 E = (f(K)F. (2.8) In the above lines ( = < s,t. Proof. We first verify (2.7). In the expression on the left in (2.7), view E2F = E (EF ), EFE = E(FE), FE2 = (FE)E and eliminate each parenthetical expression using (2.3) and (2.4). Simplify the result using K E = q2EK along with f = 1, the F-vector space Bn has a basis FnKhCsCj h G Z, i, j G N. (3.2) (iii) For n > 1, the F-vector space B-n has a basis EnKhCiCj h G Z, i, j G N. (3.3) Moreover the union of (3.1)-(3.3) is a basis for the F-vector space B. Proof. Routinely applying the Bergman diamond lemma [3, Theorem 1.2] one finds that the union of (3.1)-(3.3) is a basis for the F-vector space B. Let B0 denote the subspace of B spanned by (3.1). For n > 1 let Bn and B-n denote the subspaces of B spanned by (3.2) and (3.3), respectively. We show that {Bn}neZ is a Z-grading of B. By construction the sum B = J2neZBn is direct. By construction and since Cs, Ct are central we have CsBn C Bn and CtBn C Bn for n G Z. Using KE = q2EK and KF = q-2FK we find K±xBn C Bn for n G Z. Using (2.3) and (2.4) we find EBn C Bn-1 and FBn C Bn+1 for n G Z. By these comments and the construction we see that BmBn C Bm+n for all m, n G Z. Therefore {Bn}eZ is a Z-grading of B. The result follows. □ We emphasize a few points from Theorem 3.1. Corollary 3.2. With respect to the above Z-grading ofB, the generators Cs,Ct, E, F, K±1 are homogeneous with the following degrees: v Cs Ct E F K±1 degree of v 0 0 -11 0 Corollary 3.3. The homogeneous component B0 is the subalgebra of B generated by Cs, Ct, K±1. The algebra B0 is commutative. Let {Aj}2=0 denote mutually commuting indeterminates. Corollary 3.4. There exists an F-algebra isomorphism B0 ^ F[A±1, A1, A2] that sends K±1 ^ A^1, Cs ^ A1, Ct ^ A2. The Z-grading {Bn}nei has the following interpretation. Lemma 3.5. Consider the F-linear map B ^ B, £ ^ K-1£K. For n G Z the n-homogeneous component Bn is an eigenspace of this map. The corresponding eigenvalue is q2n. Proof. Use the basis for Bn given in Theorem 3.1, along with the relations KE = q2 E K and K F = q-2FK. □ Corollary 3.6. The homogeneous component B0 consists of the elements in B that commute with K. Proof. Immediate from Lemma 3.5. □ 4 The center of Bq (s, t, 0) Recall the algebra B = Bq(s, t, from Definition 2.3. In this section we describe the center Z(B). Theorem 4.1. The following is a basis for the F-vector space Z (B): CS Ci i,j e N. (4.1) Proof. By Theorem 3.1 the elements (4.1) are linearly independent over F, so they form a basis for a subspace of B which we denote by Z'. We show Z' = Z(B). The elements Cs, Ct are central in B so Z ' C Z (B). To obtain the reverse inclusion, pick £ e Z (B). The element £ commutes with K, so £ e B0 by Corollary 3.6. Recall the basis (3.1) for B0. Writing £ in this basis, we find £ = J2 he z K h£h where £h e Z ' for h e Z. Using KE = q2EK and £E = E£ we obtain 0 = E^heZ Kh£h(q2h - 1). Combining this with Theorem 3.1 we find £h = 0 for all nonzero h e Z. Therefore £ = £0 e Z'. We have shown Z' = Z(B) and the result follows. □ Corollary 4.2. There exists an F-algebra isomorphism Z (B) ^ F[Ai, A2] that sends Cs ^ Ai, Ct ^ A2. 5 Uniform posets Recall the algebras Aq (s, t, f ) from Definition 2.1. In this section we discuss how these algebras are related to the uniform posets [10]. Throughout this section we assume that F is the complex number field C. Let P denote a finite ranked poset with fibers {Pi}N=0 [10, p. 194]. Let CP denote the vector space over C with basis P. Let End(CP) denote the C-algebra consisting of all C-linear maps from CP to CP. We now define three elements in End(CP) called the lowering, raising, and q-rank operators. For x e P, the lowering operator sends x to the sum of the elements in P that are covered by x. The raising operator sends x to the sum of the elements in P that cover x. The q-rank operator sends x to qN-2ix where x e Pi. In [10] we introduced a class of finite ranked posets said to be uniform. We refer the reader to that article for a detailed description of these posets. See also [2, p. 306] and [9], [11]. In [10, Section 3] we gave eleven examples of uniform posets. We are going to show that six of these examples give an Aq(s, t, f )-module. These six examples are listed in the first six rows of the table below. The remaining row of the table contains an example Polarbop(N, e) which is defined as follows. Start with the poset Polarb(N, e) which we denote by P. Using P we define an undirected graph r as follows. The vertex set of r consists of the top fiber PN of P. Vertices y, z e PN are adjacent in r whenever they are distinct and cover a common element of P. The graph r is often called a dual polar graph [4, p. 274], [12, Section 16]. Fix a vertex x e PN. Using x we define a partial order < on PN as follows. For y, z e PN let y < z whenever d(x, y) + d(y, z) = d(x, z), where d denotes path-length distance in r. We have turned PN into a poset. We call this poset Polarbop(N, e). Using [12, Proposition 26.4] one checks that Polarb°p(N, e) is uniform. Theorem 5.1. In each row of the table below we give an example of a uniform poset P. For each example we display integers s < t and a Laurent polynomial f e F[A, A-1]S t. In each case the vector space CP becomes an Aq(s, t, y)-module such that the generator e (resp. f ) (resp. k) acts on CP as the lowering (resp. raising) (resp. q-rank) operator for P. For convenience, for each example we display the element $ G F[A, A-1]s,t such that y = $s,t. example s t y $ Polarb(N, e) 0 1 -(q + q-1)(q2N+1+2eA2 + qN-3A-1) q2N+2«A2+qN-1A-1 (q-q-1)2 A b(N,M ) -1 0 -(q + q-1)qN+2M +1A qN + 2M-1 (q-q-1)2 A Altb(N ) -2 -1 -(q + q-1)q2N +1 q2N-2 (q-q-1)2 Herq (N ) -2 -1 -(q + q-1)q2N+2 q2N-1 (q-q-1)2 Quadb(N ) -2 -1 -(q + q-1)q2N+3 q2N (q-q-1)2 Hemb(N ) -2 -1 -(q + q-1)q2N +1 q2N-2 (q-q-1)2 Polarb°p(N, e) -2 -1 -(q + q-1)q2N+3+2e q2N+2e (q-q-1)2 In the above table b = q2. Proof. For each example except the last, our assertions follow routinely from [10, Theorem 3.2]. For the last example Polarjb°p(N, e) our assertions follow from [12, Theorem 1.10]. Note that the parameter denoted e in [12, Theorem 1.10] is one more than the parameter denoted e in [10, p. 201]. □ 6 Acknowledgement The main inspiration for the ADU algebra concept comes from the second author's thesis [12] concerning the uniform poset PolarJb°p(N, e). To be more precise, it was his discovery of two central elements that he called C1, C2 [12, Section 28] that suggested to us how to define an ADU algebra. References [1] V. Bavula and D. A. Jordan, Isomorphism problems and groups of automorphisms for generalized Weyl algebras, Trans. Amer. Math. Soc. 353 (2001), 769-794. [2] G. Benkart and T. Roby, Down-up algebras, J. Algebra 209 (1998), 305-344. [3] G. Bergman, The diamond lemma for ring theory, Adv. Math. 29 (1978), 178-218. [4] A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer-Verlag, Berlin, 1989. [5] T. Cassidy and B. Shelton, Basic properties of generalized down-up algebras, J. Algebra 279 (2004), 402-421. [6] J. C. Jantzen, Lectures on quantum groups, Graduate Studies in Mathematics 6, Amer. Math. Soc., Providence RI, 1996. [7] D. A. Jordan, Down-up algebras and ambiskew polynomial rings, J. Algebra 228 (2000), 311— 346. [8] D. A. Jordan and I. Wells, Simple ambiskew polynomial rings, J. Algebra 382 (2013), 46-70. [9] S. Miklavic and P. Terwilliger, Bipartite Q-polynomial distance-regular graphs and uniform posets, J. Algebraic Combin. 38 (2013), 225-242. [10] P. Terwilliger, The incidence algebra of a uniform poset, in: D. Ray-Chaudhuri (ed.), Coding theory and design theory, Part I, IMA Vol. Math. Appl., 20, Springer, New York, 1990, 193212. [11] P. Terwilliger, Quantum matroids, in: E. Bannai and A. Munemasa (eds.), Progress in algebraic combinatorics (Fukuoka, 1993), Adv. Stud. Pure Math., 24, Math. Soc. Japan, Tokyo, 1996, 323-441. [12] C. Worawannotai, Dual polar graphs, the quantum algebra Uq(sl2), and Leonard systems of dual q-Krawtchouk type, Linear Algebra Appl. 438 (2013), 443-497. d MFA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 419-433 Simplicial arrangements revisited Branko Grünbaum Department of Mathematics, University of Washington Seattle, WA 98195-4350, USA Received 13 March 2013, accepted 12 November 2013, published online 2 December 2013 Abstract In connection with the publication of the catalogue [7] of known simplicial arrangements of lines in the real projective plane, and the note [8] about small simplicial arrangements of pseudolines, several developments of these topics deserve to be mentioned. The present paper puts these results in perspective, and provides appropriate illustrations. Keywords: Simplicial arrangement. Math. Subj. Class.: 51M16 1 Simplicial arrangements of pseudolines Very significant new results on simplicial arrangements of pseudolines are contained in the publications [1] by L. W. Berman and [3] by M. Cuntz. We recall that an arrangement of pseudolines is a family of simple curves in the real projective plane such that each differs from a straight line in a finite part only, and every two have a single point in common at which they cross transversally. Throughout, we model or interpret the real projective plane as the extended Euclidean plane, with added points "at infinity" and the line "at infinity" (indicated by to if included in a diagram) consisting of all the points at infinity. Developing an idea of Eppstein [4], Berman described a method of construction of simplicial arrangements of pseudolines that has a very general applicability; moreover, it is very easily adapted for investigation of linear simplicial arrangements (that is, consisting of straight lines). To explain this approach, we start with the case of linear arrangements. (It needs to be noted that our explanation differs somewhat from Berman's; we shall return to this later on.) Starting with the lines of mirror symmetry of a regular k-gon (k > 2) centered at the origin, we select one of the 2k wedges (angular regions) determined by a pair of adjacent rays formed by these k lines. Considering these rays as mirrors, we shine a (laser) ray (or several such rays) into the wedge, and let it (them) reflect on the two mirrors according to the laws of reflection; this generates a beam (or several beams). As is easily seen be elementary considerations, the laser ray will reflect only a finite number of times, and the final fate of each beam will be one of the following: E-mail address: grunbaum@math.washington.edu (Branko Griinbaum) (i) The final segment will be perpendicular to one of the reflecting rays; this includes what can be considered a limiting case, where the starting laser ray is aimed at the origin; in particular, it includes the case where the mirrors are part of the arrangement. (ii) The last part of the beam will be a ray shooting out of the wedge. In this case there are two distinct portions of the beam — the incoming part and the outgoing part. Each of these parts is simple (has no selfintersections) but the two parts may have intersections. Such beams are called two-ended. In case of pseudoline arrangements, the same conditions are assumed, except that: • The reflections on the mirrors do not follow rules of optics but are simply endpoints of pairs of segments or rays; • Each segment or ray may be a pseudosegment or pseudoray (the purple line in Figure 1 is an example); • The orthogonality in (i) is waived, and each of the two parts in (ii) is assumed to be simple. See examples in Figures 1, 2, 3 and 4. In any case, if the beam(s) satisfy some additional conditions, as detailed in [1], repeated reflection in the 2k rays yields a linear or pseudoline simplicial arrangement. We call these kaleido arrangements, to distinguish them from more general simplicial arrangements. Examples of the latter kind (non-kaleido) are A( 14,3), A( 16,7), and others, in the notation of [7], as well as the linear arrangement in Figure 7. In Berman's paper [1], only beams satisfying (i) or its modification for pseudolines are accepted. Detailed discussion of the conditions that lead to linear simplicial arrangements (and of their pseudoline analogs) is presented in [1] for up to three beams other than the mirrors. It may be assumed that analogous investigations may determine conditions under which beams as defined here lead to simplicial arrangements, but I have not determined these conditions. The main reason for introducing condition (ii) in the definition of kaleido arrangements is that it leads to the following result: Theorem 1.1. Each simplicial arrangement, with k-fold dihedral symmetry such that all mirrors are lines of the arrangement, is a kaleido arrangement. The theorem is valid equally for linear arrangements and for pseudoline arrangements. Proof. Let all the beams be marked as far as possible, starting with the incoming rays; the claim is that there are no unmarked segments (of straight or pseudolines) or rays. If any such segment were present, its continuation by reflection in the mirrors would have to close on itself, which is impossible. □ In [3], Cuntz first enumerates simplicial arrangements of at most 27 pseudolines, and then investigates their stretchability, that is, the isomorphism to linear arrangements. The bound 27 is due to limitations of the computing power available, but even with this bound several notable results are obtained and several conjectures of the present writer are resolved. ^^ oc \ / \ Figure 1: The simplicial pseudoline arrangement B i (15) (adapted from [7]) is a kaleido arrangement with k = 2 and seven beams, one of which (red) is two-ended. The blue beam and the black ones are aimed at the origin, the purple one is a pseudoray, and the green and yellow ones are rays ending at mirrors. The mirrors are heavily drawn black lines. The enumeration of simplicial arrangements of pseudolines in [3] shows that all simplicial arrangements with at most 14 pseudolines are stretchable, thus confirming a conjecture made in [8]. The computer-assisted enumeration in [3] uses "wiring diagrams" introduced Goodman in [5], and elaborated in Goodman and Pollack [6] and other publications, together with innovative arguments to reduce the computational effort. The results, in particular, disprove another conjecture in [8]: Namely, that there is a single unstretchable simplicial arrangement of 15 pseudolines and four of 16 pseudolines. In the paper [3] Cuntz establishes that there are precisely two such arrangements with 15, and precisely seven with 16 pseudolines. The second 15-pseudoline arrangement is shown in Figures 7 and 8 in two forms. Figure 7 shows a "wiring diagram" of this pseudoline arrangement, modified from Figure 2 of [3]. The presentation in Figure 8 exhibits the 3-fold rotational symmetry of this arrangement in the extended Euclidean model of the real projective plane. The colors of the lines, and the labels, establish the isomorphism between the two diagrams in Figures 7 and 8. As no pseudolines in this example are mapped onto themselves by reflection, this is not a kaleido arrangement. 2 Simplicial arrangements of straight lines Another result of [3] is the discovery of four new simplicial arrangements of (straight lines. A short review of the historical background seems appropriate to explain the significance of Cuntz's results. The first introduction of the concept of simplicial arrangements of lines occurred in a 1/7 00 \ / / CD / / * / + * / 4 X / + s /+ s / ♦ X s /+ s ' + s \ * X 4 / / v ♦ \ \ y \ \ \ Figure 6: The simplicial linear kaleido arrangement A( 15, 2), with k = 4 and three beams, one of which is a mirror; one mirror is a virtual mirror. meet the mirror perpendicularly. On the other hand, our definition of kaleido arrangements could be extended to arrangements that are not simplicial. There seems to be no interesting information available about such more general arrangements, but the concept may well be worth investigating. Finally, another result of Cuntz and collaborators should be mentioned. They investigated a particular class of linear simplicial arrangements called "crystallographic arrangements"; their definition is too involved to be repeated here and readers are referred to [2] and the references given there. In contrast to the uncertainties discussed above, this class has the notable property that its members have been completely determined and classified. Acknowledgements The author is grateful for the helpful comments and suggestions by Professors L. W. Berman, M. Cuntz and J. Malkevitch, and a referee. He also acknowledges the stay at the Helen Riaboff Whiteley Center at the Friday Harbor Laboratories of the University of Washington, which provided the atmosphere and conditions that contributed to the work reported here. The author is indebted to Marko Boben for help in getting the paper to publication. Figure 7: A wiring diagram of the new simplicial arrangement B2(15) of 15 pseudolines found by Cuntz. Adapted from Figure 2 of [3]. Figure 8: A presentation of the simplicial arrangement B2(15) of 15 pseudolines in the extended Euclidean model of the real projective plane. The colors and labels of the pseudolines correspond to those in Figure 7. Figure 9: A version of the linear simplicial arrangement denoted A(25, 8) by Cuntz [3]. Any number of the three heavily drawn lines can be deleted, resulting in the simplicial arrangements labeled A(22,5), A(23,2), and A(24,4) in [3]. Figure 10: Cuntz's A(25,8) simplicial arrangement of lines is a kaleido arrangement with k = 3; it has two two-ended beams (red and green), and five other beams. Figure 11: The linear simplicial kaleido arrangement A(15,1) with k = 2 has five beams, three of which are two-ended. Figure 12: Isomorphic realizations with different symmetries. 00 / V\ Figure 13: A correct diagram of the simplicial arrangement A( 16,7); the diagram shown in [7] and labeled A(16,7) is not correct. References [1] L. W. Berman, Symmetric simplicial pseudoline arrangements, Electronic J. Combinatorics 15 (2008), #R13. [2] M. Cuntz, Crystallographic arrangements: Weyl grupoids and simplicial arrangements, Bull. London Math. Soc 43 (2011), 734-744. [3] M. Cuntz, Simplicial arrangements with up to 27 lines, Discrete Comput. Geom. 48 (2012), 682-701. [4] D. Eppstein, Simplicial pseudoline arrangements, http://11011110.livejournal. com/15749.html (Retrieved November 20, 2012). [5] J. E. Goodman, Proof of a conjecture of Burr, Griinbaum, and Sloane, Discrete Math. 32 (1980), 27-35. [6] J. E. Goodman and R. Pollack, Semispaces of configurations, cell complexes of arrangements. J. Comb. Theory A 37(1984), 257 293. [7] B. Grunbaum, A catalogue of simplicial arrangements in the real projective plane, Ars Math. Contemp. 2 (2009), 1-25. [8] B. Grunbaum, Small unstretchable simplicial arrangements of pseudolines, Geombinatorics 18 (2009), 153-160. [9] B. Grunbaum, Arrangements of hyperplanes, in: R. C. Mullin et al. (eds.), Proc. Second Louisiana Conf. on Combinatorics, Graph Theory and Computing, Louisiana State University, Baton Rouge 1971, Congressus Numerantium 3 (1971), 41-106. [10] B. Grünbaum, Arrangements and Spreads, CBMS Regional Conference Series in Mathematics, Number 10, Amer. Math. Soc., Providence, RI, 1972. [11] E. Melchior, Über Vielseite der projectiven Ebene, Deutsche Mathematik 5 (1941), 461-475. 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Marko Petkovsek Faculty of Mathematics and Physics University of Ljubljana lu 'it ARS MATHEMATICA CONTEMPORANEA Mathematical Chemistry Issue Dedicated to the Memory of Ante Graovac Split, 15.7.1945-Zagreb, 13.11.2012 In memory of our dear colleague, member of the advisory board of our journal, and a leading expert and promotor of mathematical chemistry Ante Graovac, we will publish a special issue of AMC dedicated to topics from mathematical chemistry with special emphasis on areas related to the work of Ante. Papers will be subject to our standard editorial procedure. In the pre-screening phase, one or two experts will be asked for a quick overall assessment of the contribution. If the opinion is reached that the paper does not fall within the scope of this special issue, the authors may be advised to submit it to a regular issue of the AMC or to some more appropriate journal. For papers that pass the initial screening, two referees will be assigned. 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