Volume 22, Number 4, Fall/Winter 2022, Pages 527–686 Covered by: Mathematical Reviews zbMATH (formerly Zentralblatt MATH) COBISS SCOPUS Science Citation Index-Expanded (SCIE) Web of Science ISI Alerting Service Current Contents/Physical, Chemical & Earth Sciences (CC/PC & ES) dblp computer science bibliography The University of Primorska The Society of Mathematicians, Physicists and Astronomers of Slovenia The Institute of Mathematics, Physics and Mechanics The Slovenian Discrete and Applied Mathematics Society The publication is partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications. Contents A practical algorithm for the computation of the genus Gunnar Brinkmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 Multivariate polynomials for generalized permutohedra Eric Katz, McCabe Olsen . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 A note on the k-tuple domination number of graphs Abel Cabrera Martínez . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 Growable realizations: a powerful approach to the Buratti-Horak-Rosa Conjecture M. A. Ollis, Anita Pasotti, Marco A. Pellegrini, John R. Schmitt . . . . . . 567 Ordering signed graphs with large index Maurizio Brunetti, Zoran Stanić . . . . . . . . . . . . . . . . . . . . . . . 595 The diameter of products of finite simple groups Daniele Dona . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609 S2 coverings by isosceles and scalene triangles – adjacency case II Catarina P. Avelino, Altino F. Santos . . . . . . . . . . . . . . . . . . . . . 617 Configured polytopes and extremal configurations Tibor Bisztriczky, Gyivan Lopez-Campos, Deborah Oliveros . . . . . . . . 637 Top-heavy phenomena for transformations Yaokun Wu, Yinfeng Zhu . . . . . . . . . . . . . . . . . . . . . . . . . . . 649 The number of rooted forests in circulant graphs Lilya A. Grunwald, Ilya Mednykh . . . . . . . . . . . . . . . . . . . . . . 675 Volume 22, Number 4, Fall/Winter 2022, Pages 527–686 xxvii ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P4.01 / 527–540 https://doi.org/10.26493/1855-3974.2320.c2d (Also available at http://amc-journal.eu) A practical algorithm for the computation of the genus Gunnar Brinkmann Ghent University, TWIST, Krijgslaan 281 S9, B9000 Ghent, Belgium Received 24 April 2020, accepted 28 June 2022, published online 29 July 2022 Abstract We describe a practical algorithm to compute the (orientable) genus of a graph, give re- sults of the program implementing this algorithm, and compare the performance to existing algorithms. The aim of this algorithm is to be fast enough for many applications instead of focusing on the theoretical asymptotic complexity. Apart from the specific problem and the results, the article can also be seen as an example how some design principles used to carefully develop and implement standard backtracking algorithms can still result in very competitive programs. Keywords: Genus, NP-complete, backtracking. Math. Subj. Class. (2020): 05C10, 05-04, 05C85 1 Introduction Algorithms are studied in two different ways. First, as underlying methods of computer programs. These algorithms only come to their right when implemented as a program and used as a tool. Second, as objects of research themselves. In the latter case the emphasis lies on determining the asymptotic complexity of a problem – that is: of optimal algorithms solving the problem – often without the intention or realistic possibility to transform the described algorithms to a useful computer program. Even if such algorithms could be implemented and used, many would be extremely inefficient for real problem sizes and outperform standard algorithms only for problem sizes far beyond the limit where either of them can be used. Of course there are also some nice cases – e.g. the linear time pla- narity algorithm described in [3] – where both aspects meet and algorithms with the best asymptotic behaviour also perform well in practice. E-mail address: gunnar.brinkmann@ugent.be (Gunnar Brinkmann) cb This work is licensed under https://creativecommons.org/licenses/by/4.0/ 528 Ars Math. Contemp. 22 (2022) #P4.01 / 527–540 The general difference between these two approaches can very well be illustrated at the example of the problem of determining the genus of a graph, which is defined as the genus of the smallest orientable 2-manifold so that the graph can be embedded on its surface without crossing edges. The problem is NP-complete [16], but for any fixed g there is a linear time algorithm that can compute the genus g′ ≤ g or decide that it has genus larger than g [14]. On the other hand, there is no algorithm implemented and available that is at least guaranteed to compute the genus of a single sparse graph with, say, 80 vertices, or determine that it has genus larger than, say, 20, in one year of CPU time. Nevertheless, the interests in genus computations remain alive. Even lately papers have been published that theoretically determine the genus of specific relatively small graphs or describe algorithms fine tuned for these graphs [7, 8, 12]. In addition to that, researchers have also started to develop general purpose genus computation programs. In [1] such an algorithm based on an integer linear programming approach was published. Later, an improved approach described in [6] was – although also being a general purpose algorithm – able to achieve many of the results formerly obtained by individual theoretical research, automatically in a reasonable amount of time. Unfortunately, some of these programs are neither publicly available nor easy to use. In this article we will describe an algorithm that – in spite of also being exponential al- ready for small genus – outperforms these approaches for moderate genus is widely usable. The program multi genus based on this algorithm has the options to compute the genus of a graph, one or all minimum genus embeddings, one or all embeddings on an orientable sur- face of given genus, or to filter large lists of graphs for those with genus at most or at least a given bound. When choosing for all embeddings, for graphs with a nontrivial symmetry group isomorphic embeddings can be generated, but no two embeddings that are (labelled) mirror images are generated. We use a carefully designed backtracking algorithm. 2 The algorithm and its implementation multi genus We assume all input graphs to be simple connected graphs. The embeddings are con- structed by interpreting each undirected edge as a pair of oppositely directed edges. We build a rotation system, that is a cyclic ordering of all directed edges starting in a vertex and interpret this ordering as clockwise. Faces of an embedded graph are defined by the usual face tracing algorithm (see e.g. [10]) starting from a directed edge (v, w) and con- structing the face containing (v, w) by going to the inverse edge (w, v), and then to the next edge in the orientation around w. This process is repeated until being back at (v, w). A face f is thus a set of oriented edges. We will use the following notation for a given embedded graph: with fv we denote the set of all vertices contained in one of the directed edges of a face f and with f(e) we denote the number of directed edges in the face containing the directed edge e. The set of all faces is denoted by F , the set of all vertices by V , and finally the set of all edges by E. Following the Euler formula, the genus g of the embedded graph is g = 1 + (|E| − |V | − |F |)/2. Preprocessing Vertices of degree 1 are irrelevant when computing the genus – they can simply be removed without any impact on the genus. Similarly vertices of degree 2 can be replaced by an edge connecting their two neighbours. If this operation produces a double edge, the new edge can be removed too without changing the genus of the graph. Except when all embed- G. Brinkmann: A practical algorithm for the computation of the genus 529 dings of a graph on a surface of a certain genus must be computed and there are at least three vertices, these operations are recursively applied before the real computation of the genus begins. This means that e.g. cycles, trees or complete bipartite graphs K2,n are all reduced to a single vertex. After having computed an embedding, reduced vertices are re- stored. For graphs with minimum degree at least 3 – which is almost always the case when mathematical research about the genus is done – this preprocessing step has of course no impact. When computing the genus, the algorithm works by first searching for plane embed- dings, then embeddings with genus 1, etc. until an embedding is found. The upper bound for the genus of the embedding that is to be constructed is used in the recursive routine embedding edges. When trying to embed the graph in genus g > 0, it has already been determined that there are no embeddings of genus at most g− 1 and the first embedding of genus g determines the genus of the graph. Sometimes – this also depends on chance – such an embedding can be found relatively fast and the real bottleneck is the complete search for embeddings of genus g − 1. By computing a lower bound on the genus, sometimes ex- pensive complete searches can be avoided, but the lower bound must be fast to compute in order to have an advantage over the complete search. We will now first describe a method to compute a (cheap) lower bound: Computing a lower bound for the genus When embedding a graph G = (V,E), the values of |V | and |E| are fixed, so a minimum genus embedding is in fact an embedding with a maximum number of faces and if f ′ is an upper bound on the number of faces in any embedding then g′ = ⌈1 + |E|−|V |−f ′ 2 ⌉ is a lower bound on the genus. A trivial upper bound on the number of faces is 2|E|3 as all faces have at least three edges. This lower bound comes for free and is always computed and used. Instead of the constant value 3, except for trees one could also use the girth of the graph, but that would also have to be computed. The following methods give a better bound if there are few cycles of minimum length. For a given embedding, let s[] denote the vector of size 2|E| indexed from 1 to 2|E| containing all values f(e) of directed edges e in non-decreasing order. Then |F | = F (s) =∑2|E| i=1 (1/s[i]). A vector s ′[] of size 2|E| with s′[i] ≤ s[i] for 1 ≤ i ≤ 2|E|, is said to be dominated by s[]. For a vector s′[] dominated by s[] we have F (s′) = ∑2|E| i=1 (1/s ′[i]) ≥ f . We call a cyclic sequence e0, . . . , ek−1 of k pairwise distinct directed edges a facial- like walk if and only if for 0 ≤ i < k the starting vertex of ei+1 (mod k) is the end vertex of ei and ei+1 (mod k) is the inverse (ei)−1 of ei if and only if the degree of the end vertex of ei is one. A first approximation s0[] of s[] is obtained by taking for each directed edge e the length fw(e) of the shortest facial-like walk containing e. The value of fw(e) can be easily computed by a Breadth First Search. As each facial walk in an embedded graph is also a facial like walk, we see immedi- ately that the non-decreasing sequence s0[] is dominated by s[], as fw(e) ≤ f(e) for each directed edge e. Especially s0[2|E|] ≤ s[2|E|] and as in s[] at least s[2|E|] edges – all di- rected edges in a longest facial walk – have value s[2|E|], we can replace the last s0[2|E|] values of s0[] with s0[2|E|] and get another sequence s1[] dominated by s[]. We use F (s1) as a first nontrivial upper bound on the number of faces. In fact the length of the shortest facial-like walk is the same for a directed edge and its 530 Ars Math. Contemp. 22 (2022) #P4.01 / 527–540 reverse, but unless the graph is a cycle, one facial-like walk that does not also contain the reverse edge, can only form a face for at most one of them. This observation might lead to a better approximation, but in order to keep the computation of the approximation easy and fast, the length of the shortest facial-like walk is used for a directed edge and its inverse. An angle α of a face is a pair of directed edges, following each other in the facial walk. The central vertex of the angle is the endpoint of the first edge – so except when this vertex has degree 1 it is the only common vertex of the two edges. In what follows we use that for an edge e and its inverse e−1 we have fw(e) = fw(e−1). Instead of summing over all edges, we can sum over all angles. With f(α) the size of the face that contains α and A(v) the set of all angles with central vertex v, we have |F | =∑ v∈V ( ∑ α∈A(v) 1/f(α)). If for a vertex v the sequence s ′ v[1], . . . , s ′ v[deg(v)] is the non- decreasing sequence of all f(α) with α ∈ A(v), then |F | = ∑ v∈V ( ∑ 1≤i≤deg(v) 1/s ′ v[i]). Taking for each vertex v a vector dominated by s′v[] we again get an upper bound on |F |. If we take for a vertex v and each angle α ∈ A(v) instead of f(α) the value max{fw(e), fw(e′)}, with e, e′ the edges in the angle, we get a non-decreasing sequence s′0,v[] dominated by s ′ v[]. If s ′ 1,v[] is the non-decreasing sequence of values of fw(e) with e starting at v, then we define s′2,v[] = s ′ 1,v[2], s ′ 1,v[3], . . . , s ′ 1,v[deg(v)], s ′ 1,v[deg(v)]. So we remove the smallest value of s′1,v[] and add a copy of the largest value. Remark 2.1. Let G = (V,E) be an embedded graph. Then for each vertex v ∈ V the sequence s′v[] dominates s ′ 2,v[]. Proof. We know that s′v[] dominates s ′ 0,v[]. We will show that s ′ 0,v[] dominates s ′ 2,v[]. As the maximum values of s′0,v[], s ′ 1,v[] and s ′ 2,v[] are the same, it is sufficient to prove s′0,v[i] ≥ s′2,v[i] = s′1,v[i+1] for i < deg(v). Let α1, . . . , αi be the angles (that is: pairs of edges) determining the values s′0,v[1], . . . , s ′ 0,v[i] and Si,v be the set of all directed edges starting at v, so that e or e−1 is in at least one of these angles. Then the value of s′0,v[i] is max{fw(e)|e ∈ Si,v} (here we use that fw(e) = fw(e−1)) and as |Si,v| ≥ i+ 1, we have that s′0,v[i] ≥ s′1,v[i+ 1]. We use ∑ v∈V ( ∑ 1≤i≤deg(v) 1/s ′ 2,v[i]) as a second nontrivial upper bound on the num- ber of faces. These upper bounds on the number of faces and the corresponding lower bounds on the genus are relatively fast to compute. Nevertheless they do not always speed up the program. Especially for small graphs or small genus they can even slow down the program, as the embedding algorithm can exclude low genus embeddings very fast. While for few small graphs this is no problem, for very large lists of small graphs it can have some impact. Of course the bounds can only increase the running time of the program by a small factor and never by a large factor, but they can sometimes speed the program up a lot: In this article, all running times for the C-program multi genus implementing the al- gorithm described here are on an Intel Core i7-9700 CPU @ 3.00GHz (running on one core at 4.4-4.7 Ghz). The prefix multi stands for the graph coding that is accepted as input: multi code. Examples for the impact of the computation of a lower bound when computing the genera of graphs are: All bipartite graphs on 14 vertices with degrees between 5 and 6 (73 graphs, genus 3 to 5): without lower bound 60.9 seconds, with lower bound 0.035 seconds. G. Brinkmann: A practical algorithm for the computation of the genus 531 All cubic graphs on 22 vertices (7, 319, 447 graphs, genus 0 to 3): without lower bound 300 seconds, with lower bound 364 seconds. Checking 1, 000, 000 random cubic graphs on 50 vertices, generated by genrang (which is part of the nauty-package [13]) for being planar: without lower bound 18.5 seconds, with lower bound 56.2 seconds. Checking the same 1, 000, 000 random cubic graphs on 50 vertices for having genus at most 1: without lower bound 144.8 seconds, with lower bound 85.5 seconds. The default is that the nontrivial bounds are used, but the use can be switched off by an option to multi genus. Constructing an embedding We begin by relabeling the graph in a BFS way. The time necessary to compute the genus can differ a lot for isomorphic graphs depending on the labelling. In some cases a BFS la- belling results in a better performance, in others it slows down the program. We have cho- sen for the BFS labeling as the results for different, but isomorphic, input graphs often differ less when always using such a labeling. An example showing the large differences that can still occur can be seen when computing a genus 7 embedding (that is a minimum genus embedding [11, 4]) of C3□C3□C3. Taking the first graph of the file ucay27_05_k=06 provided by Gordon Royle in his list of Cayley graphs (and doing BFS), it takes 0.19 sec- onds to find an embedding, taking the same graph from a program constructing cartesian products and not doing BFS, it takes 6.2 seconds. Taking the graph from the second source and doing BFS, it takes 281 seconds. So even when relabeling the graph in a BFS manner the time still depends on the labeling of the input graph. The algorithm works by first greedily embedding a subgraph so that for each embedding or its mirror image, the induced embedding of this subgraph is the one constructed. It has genus 0. If the maximum degree is smaller than 3, the graph is a path or a cycle, both of which can be uniquely embedded and only in the plane. Otherwise we construct the initial subgraph by taking a vertex with minimum degree among all vertices with degree at least 3, embedding this vertex and three of its edges in an arbitrary way (thereby fixing the orientation) and greedily extending the three edges – one after the other – to paths until they cannot be made longer. The result of this construction forms the root of the recursion tree of the Branch and Bound algorithm in which the remaining edges are added recursively . In branch and bound algorithms, the performance is often improved if one manages to reduce the branching at every node of the recursion tree. This is not a mathematical theorem, but more a rule of thumb, as in some cases more branching might be beneficial if it allows earlier bounding. Nevertheless in our case we have chosen to take small branching as the base (but not only) criterion for the order in which the edges are inserted. As an expensive choice of the next edge to insert is sometimes more costly than more branching, we work in three parts: (i) Before the recursion starts, the edges that are still to be embedded are sorted as {x1, y1}, {x2, y2}, . . . , {xk, yk}, so that with Si the initial subgraph with edges {x1, y1}, . . . , {xi, yi} added, for 1 ≤ i < k, we have for {xi, yi} that at least one of xi, yi is in Si and degSi−1(xi) · degSi−1(yi) ≤ degSi−1(xj) · degSi−1(yj) for all i < j ≤ k for which at least one of xj , yj is in Si. If one of degSi−1(xi),degSi−1(yi) is 0, then also degSi−1(xi) + degSi−1(yi) ≤ degSi−1(xj) + degSi−1(yj) for all j 532 Ars Math. Contemp. 22 (2022) #P4.01 / 527–540 for which one of degSi−1(xj),degSi−1(yj) is 0 and the other is in Si. Informally speaking: first all edges leading from the already embedded part to not yet included vertices are added and then the edges with both endpoints in the embedded subgraph are added. In each case we choose an edge that has the smallest number of possibili- ties how it can be added if there are no other restrictions. During the recursion we always have an upper bound max_genus on the genus. As long as this bound is not reached, edges can be inserted in all possible angles – sometimes increasing the genus and sometimes not. In order to reach this bound as fast as possible and therefore be able to reduce the branching, we check at each node of the recursion tree whether there is an edge that cannot be embedded in any existing face of the partial embedding – we call such an edge a critical edge – and therefore always increases the genus. If there is such an edge and the partial embedding has already genus max_genus, we can backtrack, otherwise the first such edge in the list gets priority above other edges that do not have this property and is inserted first. We distinguish two cases: (ii) If the recursion is still close to the root node and the genus of the partial embedding is still smaller than max_genus, we have relatively few nodes and the impact of a smaller branching is large. In this situation, also more expensive tests can pay and we do not only check for the existence of a critical edge, but do in fact look for an edge which has the smallest number of faces into which it can be embedded and take such an edge as the next one to be embedded. Among all edges with the same number of faces where they can be embedded, the first one in the sorted list is taken. Note that it is possible that an edge can be embedded into a face in more than one way, but this is not taken into account when counting the number of faces. (iii) Close to the leaves of the recursion tree we have many nodes and the impact of a smaller branching is small. In this situation, or when the genus of the partial embed- ding is already max_genus, we only check for the existence of a critical edge. The decision when we consider a node of the recursion to be close to the root or close to the leaves has an impact on the performance, but a simple rule for the optimal moment to switch is hard to determine. Tests on different kinds of graph showed that considering nodes where at most half of the edges (edges of the initial tree not counted) are embedded as close to the root and considering the others as close to a leave is often a good compromise. The method to find critical edges fast is crucial for the performance. An edge can be embedded into a face if both endpoints of the edge are in the same face. In the C- implementation we use bit vectors – that is C data types: integers of type unsigned long int (64 bit) or unsigned int128 to represent sets of vertices. Especially for graphs with up to 64 vertices this allows to determine whether an edge can be embedded in a face in few CPU cycles – provided the fact that the set of end vertices of the edge and the set of vertices of the face are represented as bit vectors. Unless otherwise mentioned, up to 64 vertices the version using unsigned long int is used for the timings and the version using unsigned int128 for larger graphs on up to 128 vertices. Lemma 2.2. Assume that an algorithm to embed a graph G = (V,E) starts with embed- ding a spanning tree and then inserts the remaining edges of a graph step by step, but in each step inserts non-critical edges only if there are no critical edges. Let G′ = (V ′, E′) G. Brinkmann: A practical algorithm for the computation of the genus 533 be a subgraph with at least one cycle that was embedded by this algorithm and let e be the last edge that was inserted in one face f and split it into two faces with vertex sets fv1 , f v 2 . If there is a critical edge ec for G′, then |ec ∩ (fv1 \ fv2 )| = |ec ∩ (fv2 \ fv1 )| = 1|. Proof. Note first that G′ need not have faces with vertex sets fv1 , f v 2 . The faces f1, f2 can not have been subdivided, as they are the result of the last subdivision, but after that subdivision they might have been united with other faces (or with each other) when an edge with endpoint in two different faces was inserted. If there is no critical edge, the statement is trivially true, so assume that there is a critical edge ec. Let G0 be the embedded subgraph into which e was embedded to form G1. As e was inserted into a face, there was no critical edge for G0, so ec could be embedded into a face f0 of G0 (so ec ⊂ fv0 ). After e only critical edges were inserted, so the vertex sets of all faces of G′ are unions of those of G1. If f ̸= f0, fv0 ⊆ f ′v0 for some face f ′0 of G′, so ec could be embedded into f ′0 and would not be critical. So we have f = f0. As (fv1 \ fv2 ) ∩ (fv2 \ fv1 ) = ∅ it is sufficient to show that ec ∩ (fv1 \ fv2 ) and ec ∩ (fv2 \ fv1 ) are both not empty. Assume that (w.l.o.g.) ec ∩ (fv1 \ fv2 ) = ∅. As fv1 ∪ fv2 = fv , we have ec ⊂ fv2 , but as fv2 is a subset of the vertex set of a face of G′, ec would not be critical. So ec ∩ (fv1 \ fv2 ) ̸= ∅. Finding critical edges is a nontrivial task. The straightforward way is a loop over all edges that still need to be inserted and inside this loop a loop over all faces of the embedded graph. The previous lemma gives a very cheap criterion to decide for many edges that they are not critical – without using the inner loop. In fact one could even make a list of all candidates for critical edges whenever a face is subdivided, but in the implementation this is not done. Performance There are some programs available, where the exact algorithm is not published – e.g. sim- ple connected genus backtracker in the computer algebra package sage. As a backtracking program it seems to be related to the algorithm described here. The manual says that it is an extremely slow but relatively optimized algorithm. This is “only” exponential for graphs of bounded degree, and feels pretty snappy for 3-regular graphs. It also says that K7 may take a few days, while multi genus takes less than 0.001 seconds for K7. So we tested it only for cubic graphs, but already for relatively small vertex numbers, it is very slow, e.g. more than 24 minutes for the unique cubic graph with girth 8 on 34 vertices (instead of less than 0.001 seconds of multi genus), so tests on a larger scale were not possible. The program used in the graph database House of Graphs – short HoG [5] at the mo- ment (it will be replaced by multi genus) is much faster. It is a Java program called Min- GenusEmbedder written by Jasper Souffriau as a student project and it is also a backtrack- ing algorithm using branch and bound. That program was also used for independent tests. For the generation of random graphs we use the program genrang [13] which allows to re- strict the generation to regular graphs of a given degree or to graphs with a given number of edges. As genrang also generates graphs that are not connected, we filtered them for con- nected graphs. If we say that we tested n random graphs generated by genrang, this means that we generated random graphs by genrang and took the first n connected ones. In order to have the results completely reproducible, we always fixed the seed used by genrang to 0. For 2000 random cubic graphs on 30 vertices, MinGenusEmbedder needed 16.3 sec- onds (compared to 0.6 seconds of multi genus – a factor of 27) and for 2000 random cubic 534 Ars Math. Contemp. 22 (2022) #P4.01 / 527–540 graphs on 40 vertices, MinGenusEmbedder needed 435 seconds (compared to 12.5 sec- onds, a factor of 34). For larger degrees the ratio grows. For 30 quartic graphs on 30 vertices MinGenusEmbedder already needs 792 seconds (compared to 6.6 seconds, a fac- tor of 120) and for 30 5-regular graphs on 22 vertices 2, 226 seconds (compared to 13.7 seconds, a factor of 163). For 6-regular graphs, testing 30 graphs on 19 vertices already took quite some time: 10.56 hours for MinGenusEmbedder and 2 minutes for multi genus (a factor of 315). Of course such small samples are not sufficient for reliable results and we should see these numbers just as a hint what the relation of the running times might be. Unfortunately the running times do not allow tests on large sets of data. The fastest published general purpose program is the integer linear programming ap- proach described in [6] and implemented in the program ILPDReal . The program ILP D Real is not publicly available, so we compare the running times for a data set they used in [6]: the Rome graphs, which can be downloaded from http://graphdrawing.org/d ata.html. This set of graphs contains 11, 534 graphs with (at least indicated by the file names) up to 100 vertices which are used e.g. for graph drawing and are said to come from practical applications. Among these graphs, 3 are disconnected and 3, 279 planar. In [6] only nonplanar graphs were tested. Note that the set of Rome graphs not only con- tains isomorphic graphs, but even identical copies. Some files also seem obscure: e.g. in grafo6975.39.graphml due to the otherwise used convention, there should be a graph with 39 vertices. Nevertheless it has 105 vertices. We filtered out the disconnected graphs, the planar graphs, and the obscure ones and – like [6] – received a list of 8, 249 nonplanar graphs for which the genus had to be computed. It should be mentioned that the sizes of the Rome graphs are a bit misleading when it comes to estimating the complexity of computing the genus: many of the graphs have vertices of degree 1 and 2, which do not increase the complexity of the computation of the genus. In [6] a Xeon Gold 6134 CPU was used to compute the genus of these graphs. For each graph a time limit of 10 minutes and a memory limit of 8 GB was given. With these restrictions ILPDReal was able to decide 82% of the instances. For multi genus the memory consumption is negligible. On the Core i7 it could determine the genus of 98.57% of the graphs within the same time limit of 10 minutes and even with a time limit of only 0.25 seconds for each graph, it can decide 83.7% of the cases. In Figure 1 the development of the running times for random cubic and quartic graphs are given. For all sizes the version for more than 64 vertices using unsigned int128 was used in order not to have some misleading behaviour around 64 vertices. As expected, for given fixed degree of the vertices, the measured times depend exponentially on the number of vertices. If we fix the number of vertices, but vary the number of edges, we get – as expected – again an exponential growth, as shown in Figure 2. Together with the number of vertices and edges, also the average genus increases, so it is also interesting to know how the running times develop, when it is only tested whether the graphs can be embedded in a surface of given genus – similar to planarity testing. In Figure 3 the running times for testing whether 1000 random cubic resp. quartic graphs can be embedded in a surface of genus at most 3 are given. In fact for practically all of the larger graphs tested, the answer was no. Nevertheless it is astonishing that from a certain point on the time necessary to test a graph decreases again. If one wanted to apply the algorithm to perform very well when testing only planarity, one could apply the reasoning of Demoucron, Malgrange and Pertuiset [9] as soon as a G. Brinkmann: A practical algorithm for the computation of the genus 535 0.001 0.01 0.1 1 10 100 1000 10000 10 20 30 40 50 60 3-regular graphs 4-regular graphs Ti m e fo r 1 00 0 ra nd om g ra ph s ( se c) Number of vertices Figure 1: The running times for computing the genus of 1000 random cubic, resp. quartic graphs. Note that the time scale is logarithmic. 0.01 0.1 1 10 100 1000 10000 45 50 55 60 65 70 75 Ti m e fo r 1 00 0 ra nd om g ra ph s ( se c) Number of edges Figure 2: The running times for computing the genus of 1000 random graphs with 32 vertices and a given number of edges. Note that the time scale is logarithmic. spanning tree and the first cycle is formed. Of course that would not be useful, as special- ized and very efficient algorithms for planarity testing exist. Although no special adaptation for the planar case is implemented and this is not the task the algorithm was developed for, a comparison with specialized programs for planarity testing might be interesting. In [3] a practical linear time algorithm for planarity testing is presented. In the program planarg [13] an implementation of this algorithm by Paulette Lieby is available. Testing 100, 000 random cubic graphs on 50 vertices with planarg and multi genus with 0 as an upper bound for the genus, the times are 2.29 (planarg) resp. 5.5 (multi genus) sec- onds. Testing 100, 000 random cubic graphs on 100 vertices, planarg is more than 4 times faster than multi genus (5.0 seconds to 22.1 seconds). All these graphs were non-planar. Filtering the 11, 529 rome graphs for non-planar ones took 0.36 seconds with planarg and 0.29 seconds with multi genus. In this case about 28% of the graphs were planar. In order 536 Ars Math. Contemp. 22 (2022) #P4.01 / 527–540 0 5 10 15 20 25 30 35 40 10 20 30 40 50 60 70 80 90 100 3-regular graphs 4-regular graphs Ti m e fo r 1 00 0 ra nd om g ra ph s ( se c) Number of vertices Figure 3: The running times for testing 1000 random cubic, resp. quartic graphs for having genus at most 3. to test how fast embeddings are found if they exist, once sparse planar graphs – that is cu- bic graphs – were tested and once dense graphs – that is triangulations. For these tests, for multi genus no lower bound for the genus was computed. We used the 285, 914 fullerenes on 100 vertices and their duals (triangulations on 52 vertices) randomly relabeled in order to avoid an impact of the embedding produced by the generation program. For the fullerenes, planarg needed 19.5 seconds (compared to 19.1 seconds for multi genus) and for the trian- gulations planarg needed 11.85 seconds (compared to 11.1 seconds for multi genus). Testing In order to test the implementation, independent programs were used and the results were compared to the results of multi genus. For all cases tested the results agreed. In order to test the option to compute the genus, the genus of each (connected) graph in the following set was computed by multi genus and MinGenusEmbedder and the result was compared. The sets of graphs are: all graphs on up to 11 vertices, all 3-regular graphs on up to 24 vertices, all 4-regular graphs on up to 16 vertices, all 5-regular graphs on up to 14 vertices, all 3-regular graphs on up to 26 vertices with girth at least 5, all 3-regular graphs on up to 28 vertices with girth at least 6, all 3-regular graphs on up to 34 vertices with girth at least 7, all 3-regular graphs on up to 44 vertices with girth 8, all (3, 9)-cages – that is all 3-regular graphs on 58 vertices with girth 9, all 4-regular graphs on up to 24 vertices with girth 5, and finally all graphs with valence vector (2, 2, 3, 3, 3). In order to test the option that makes multi genus generate all embeddings of a certain genus, a simple independent program was implemented that generated all combinations of all vertex orders around the vertices. Filtering the embeddings generated this way for those with a given genus we had a very slow but independent test. Then for each graph in one of the following sets, the range of possible genera was computed by the Euler formula and for each graph and each possible genus the embeddings of this genus were independently generated and the number of embeddings as well as the number of non-isomorphic embed- dings (computed by an isomorphism checking program using lists) were compared. Due G. Brinkmann: A practical algorithm for the computation of the genus 537 to the enormous number of embeddings already for small graphs, not too many graphs and no large graphs could be tested. The sets of graphs are: all 3-regular graphs on up to 18 vertices, all graphs on 7 vertices with 6 to 17 edges, all graphs on 8 vertices with 14 and with 15 edges, all graphs with valence vector (1, 1, 1, 5), and finally all graphs with valence vector (0, 2, 3, 2, 3) and girth at least 4. Results obtained or confirmed by multi genus Times given for multi genus in this section are again on an Intel Core i7-9700 CPU @ 3.00GHz running on one core at 4.4-4.7 Ghz. In [15] Plummer and Zha prove a theorem describing the cases when Kc+1 is the unique c-connected graph with smallest genus – except for the two cases c = 9 and c = 13 which are not decided and posed as an open question. This question is answered in [2] showing that in these cases the complete graphs are not unique, but that in these cases the graphs Mc+2 on c + 2 vertices, obtained by deleting a maximum matching from Kc+2, have the same genus as Kc+1. The embeddings given in that article were computed by multi genus. Computing the genus g(M11) = 4 takes 0.005 seconds and computing the genus g(M15) = 10 takes 7 hours and 6 minutes. In [7] Conder and Grande determine all circulant graphs of genus 1 and 2. A large part of the proof discusses 12 specific circulant graphs and in order to prove that 11 of these graphs have genus larger than 2, next to several pages of theoretical argumentation also more than 80 CPU hours were needed. The program described in [1] confirms these results in 180 hours of CPU time (without additional theoretical arguments), and ILPDReal computes the genera in a matter of seconds (the exact value isn’t given). Multi genus confirms the results of the paper in less than 0.03 seconds. Computing the exact genera (once genus 2, 7 times genus 3, 3 times genus 4, and once genus 5) and minimum genus embeddings takes 6.4 seconds. In [12] the genus of the Gray graph is theoretically determined by a nontrivial con- struction. ILPDReal confirms this result within 42 hours. Multi genus confirms this in 28.3 seconds. In order to determine all 258, 696 (labeled) minimum genus embeddings (219 non-isomorphic), multi genus needed a bit less than 10 minutes. Isomorphism rejection is done by an independent program simply storing canonical embeddings in lists. In [8] the genus (and also non-orientable genus) of several graphs was determined. They describe four specialized approaches they apply to some special graphs that have in general a large symmetry group. One of them – they call it the subgroup orbit method – is especially suited for as they write graphs on surfaces with a certain degree of symmetry and works well for graphs that allow an embedding with a face-transitive automorphism group. So the approach is not intended for general graphs and the program is also not available for everybody. Our general approach cannot reproduce their results for the Hoffman-Singleton graph, the Ljubljana graph or the Iofinova-Ivanov graph – at least not without an excessive amount of time and/or special adaptations. Some of the other examples they give can also be solved and sometimes extended by our general approach without any manual interfer- ence – just by piping the graph into multi genus. The instances for which the results could be confirmed and sometimes extended are: The graph C3□C3□C3: In [8] it says that with a natural vertex labeling the subgroup orbit method takes only a couple of minutes to find a genus 7 embedding. The method described here takes – depending on the labeling – from 0.19 seconds to 281 seconds 538 Ars Math. Contemp. 22 (2022) #P4.01 / 527–540 to find an embedding. Of course there may also be labelings that take even less or even more time. In total there are 188, 211, 024 minimum genus embeddings, 145, 468 of them pairwise non-isomorphic, but computing these took almost 3 weeks of CPU time (on another, much older, machine used for the large memory available for isomorphism rejection). Not only constructing a genus 7 embedding, but also proving its minimality by ex- cluding the existence of an embedding of smaller genus takes between 1.5 and 4 hours depending on the labeling (both with BFS numbering first). In [11] bounds for the genus of the cartesian product of four or five triangles are given. Determining the genus or useful bounds for the genus of these graphs or of C3□C3□C3□K2 is out of reach of the program described here. The Tutte graph (or (3, 8)-cage): In [8] no running times are given, but they construct a genus 4 embedding with cyclic automorphism group of order 3. The present ap- proach takes 0.005 seconds to determine the genus as 4 and 0.13 seconds to construct all 13, 440 embeddings. Among these embeddings there are 15 non-isomorphic em- beddings – 4 with a group of order 1, 10 with a group of order 2 (2 of them allowing a reflection) and one with a group of order 3. The Gray graph: The running times for the Gray graph were already given. In [8] it is reported that there are minimum genus embeddings with an automorphism group of order 6. Checking all possible embeddings, the result is that there are 186 non- isomorphic embeddings with trivial symmetry, 23 with a group of order 2, 4 with a group of order 3, 4 with a group of order 6, and 2 with a group of order 18. The Folkman graph: For the Folkman graph, in [8] minimum genus embeddings with a group of order 8 are constructed. The method described here takes less than 0.001 seconds to determine the genus as 3 and 0, 037 seconds to construct all 7, 680 minimum genus embeddings. Among these embeddings there are 7 pairwise non- isomorphic, 2 with a group of order 2, 3 with a group of order 4, and 2 with a group of order 8. All groups contain a reflection. The Doyle-Holt graph: For the Doyle-Holt graph [8] describes a genus 5 embedding with an automorohism group of order 2. The present approach needs 0.23 seconds to determine the genus of the graph and 7.3 seconds to determine all 1, 107 minimum genus embeddings. There are 24 pairwise non-isomorphic embeddings – 17 with a trivial group and 7 with a group of order 2. The dual Menger graph of the Gray configuration: For this 6-regular graph on 27 vertices, the present approach needs 20 seconds to determine the genus as 6. In a bit more than 26 minutes it constructed all 216 minimum genus embeddings – which turned out to be isomorphic. So the minimum genus embedding of the dual Menger graph of the Gray configuration is unique. It has an automorphism group of order 6. Conclusion The program described in this article can be a useful tool and has – among other applica- tions – e.g. be used to solve an old question of Plummer and Zha [15] about the uniqueness of certain complete graphs Kc+1 as the only c-connected graphs embeddable in a surface of G. Brinkmann: A practical algorithm for the computation of the genus 539 minimal genus into which Kc+1 can be embedded [2], determine the genus of the Georges- graph, etc. . . . Nevertheless computing the genus of a graph is a very difficult problem and as can be deduced from the figures, the time grows fast with the genus and the size of the graph. Also in the future, theoretical approaches – maybe combined with specialized computer programs – will be necessary to determine the genus of some graphs of special interest. ORCID iDs Gunnar Brinkmann https://orcid.org/0000-0003-4168-0877 References [1] S. Beyer, M. Chimani, I. Hedtke and M. Kotrbčı́k, A practical method for the minimum genus of a graph: Models and experiments, in: A. V. Goldberg and A. S. Kulikov (eds.), Experimental Algorithms, Springer International Publishing, Cham, 2016 pp. 75–88, doi:10.1007/978-3-31 9-38851-9 6. [2] D. Bokal, G. Brinkmann and C. T. Zamfirescu, The connectivity of the dual, J. Graph Theory (2022), doi:10.1002/jgt.22819. [3] J. Boyer and W. Myrvold, Stop minding your p’s and q’s: A simplified O(n) planar embedding algorithm, in: Tenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 140– 146, 1999. [4] M. Brin and C. Squier, On the genus of Z3 × Z3 × Z3, Eur. J. Comb. 9 (1988), 431–443. [5] G. Brinkmann, K. Coolsaet, J. Goedgebeur and H. Mélot, House of Graphs: a database of interesting graphs, Discrete Appl. Math. 161 (2013), 311–314, doi:10.1016/j.dam.2012.07.018. [6] M. Chimani and T. Wiedera, Stronger ILPs for the graph genus problem, in: 27th annual Euro- pean symposium on algorithms, Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Wadern, p. 15, 2019, doi:10.4230/lipics.esa.2019.30, id/No 30. [7] M. Conder and R. Grande, On embeddings of circulant graphs, Electron. J. Comb. 22 (2015), research paper p2.28, doi:10.37236/4013. [8] M. Conder and K. Stokes, New methods for finding minimum genus embeddings of graphs on orientable and non-orientable surfaces, Ars Math. Contemp. 17 (2019), 1–35, doi:10.26493/1 855-3974.1800.40c. [9] G. Demoucron, Y. Malgrange and R. Pertuiset, Graphes Planaires: Reconnaissance et Con- struction de Representations Planaires Topologiques, Rev. Franç. Rech. Opér. 8 (1964), 33–47. [10] J. L. Gross and T. W. Tucker, Topological Graph Theory, John Wiley and Sons, 1987. [11] M. Kotrbčı́k and T. Pisanski, Genus of the Cartesian product of triangles, Electron. J. Comb. 22 (2015), research paper p4.2, doi:10.37236/2951. [12] D. Marušič, T. Pisanski and S. Wilson, The genus of the Gray graph is 7, Eur. J. Comb. 26 (2005), 377–385, doi:10.1016/j.ejc.2004.01.015. [13] B. McKay, Nauty and Traces, {https://users.cecs.anu.edu.au/˜bdm/nauty /}. [14] B. Mohar, A linear time algorithm for embedding graphs in an arbitrary surface, SIAM J. Dis- crete Math. 12 (1999), 6–26, doi:10.1137/s089548019529248x. [15] M. D. Plummer and X. Zha, On the connectivity of graphs embedded in surfaces, J. Comb. Theory, Ser. B 72 (1998), 208–228, doi:10.1006/jctb.1997.1809. 540 Ars Math. Contemp. 22 (2022) #P4.01 / 527–540 [16] C. Thomassen, The graph genus problem is NP-complete, J. Algorithms 10 (1989), 568–576, doi:10.1016/0196-6774(89)90006-0. ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P4.02 / 541–559 https://doi.org/10.26493/1855-3974.2044.fd1 (Also available at http://amc-journal.eu) Multivariate polynomials for generalized permutohedra* Eric Katz † Department of Mathematics, The Ohio State University, Columbus, Ohio, United States McCabe Olsen ‡ Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute, Indiana, United States Received 9 July 2019, accepted 8 September 2021, published online 3 August 2022 Abstract Using the notion of a Mahonian statistic on acyclic posets, we introduce a q-analogue of the h-polynomial of a simple generalized permutohedron. We focus primarily on the case of nestohedra and on explicit computations for many interesting examples, such as Sn-invariant nestohedra, graph associahedra, and Stanley-Pitman polytopes. For the usual (Stasheff) associahedron, our generalization yields an alternative q-analogue to the well- studied Narayana numbers. Keywords: Generalized permutohedron, h-polynomial, q-analogues. Math. Subj. Class. (2020): 52B12, 05A15, 06A07, 05C31 1 Introduction Given any combinatorially defined polynomial, a common theme in enumerative combina- torics is to consider multivariate analogues which further stratify and enrich the encoded data by an additional combinatorial statistic. A notable example of is the Euler–Mahonian polynomial An(t, q) = ∑ π∈Sn tdes(π)qmaj(π) *The authors thank Vic Reiner for helpful comments at the beginning of this project. The authors also thank the anonymous referee for helpful comments and suggestions. †Partially supported by NSF DMS 1748837. ‡Corresponding author. E-mail addresses: katz.60@osu.edu (Eric Katz), olsen@rose-hulman.edu (McCabe Olsen) cb This work is licensed under https://creativecommons.org/licenses/by/4.0/ 542 Ars Math. Contemp. 22 (2022) #P4.02 / 541–559 which is a bivariate generalization of the more foundational Eulerian polynomial An(t) = ∑ π∈Sn tdes(π), both of which are specializations of the n− 1 variable polynomial An(t1, t2, . . . , tn−1) = ∑ π∈Sn ∏ i∈Des(π) ti. In this case, we further stratify the descent statistic on permutations by the additional data of the major index. Such a generalization is commonly referred to as a q-analogue in reference to usual choice of added variable. Given a convex polytope P ⊂ Rn, the h-polynomial is an encoding of the face numbers of P obtained as a linear change of variables of the generating function for the face num- bers. If P is simple or simplicial, then the Dehn–Sommerville equations for P are reflected in the palindromicity of the h-polynomial. For simple rational polytopes, the h-polynomial is the Poincaré polynomial of the cohomology groups of the toric variety attached to the polytope. Moreover, for simplicial polytopes, the h-polynomial is the generating function for facets of P according to the size of their restriction sets [25, Section 8.3]. Generalized permutohedra are a broad class of convex polytopes which exhibit many nice properties. First introduced by Postnikov [20], these polytopes have been the subject of much study and are of wide interest in many areas of algebraic and enumerative com- binatorics, including the combinatorics of Coxeter groups, cluster algebras, combinatorial Hopf algebras and monoids, and polyhedral geometry (see, e.g., [2, 4, 15, 16]). Of particular interest for our purposes, Postikov, Reiner, and Williams [21] give a com- binatorial description of the h-polynomial for any simple generalized permutohedron using an Eulerian descent statistic on posets. Moreover, they provide a formula for well-behaved, special cases of generalized permutohedra. We give a bivariate generalization of their de- scription for any simple generalized permutohedron: for P a simple generalized permuta- hedron and Qσ the cone poset for a full dimensional cone σ in the normal fan N (P ) (See Definition 2.1), we define hP (t, q) := ∑ σ∈N (P ) tdes(Qσ)qmaj(Qσ) where des and maj are statistics defined below. Furthermore, we are able to be more explicit when restricting to particular classes of generalized permutahedra, specifically Sn- invariant nestohedra, graph associahedra, and Stanley–Pitman polytopes. Our definition of the bivariate h-polynomial, which specializes to the usual h-polynomial is justified by analogy with the Euler–Mahonian polynomial. Other possible definitions ex- ist. An inequivalent definition is the principal specialization of the Frobenius characteristic of the permutohedral toric variety. This definition does not extend to generalized permuto- hedra and is not discussed in the body of the paper. However, it does make use of the major index. The structure of this note is as follows. In Section 2, we provide a review of necessary background and terminology on permutations, posets, polyhedral geometry, and general- ized permutohedra. Section 3 defines and discusses the the q-analogue for the h-polynomial of any simple generalized permutohedron. In Section 4, we focus on general results for a E. Katz and M. Olsen: Multivariate polynomials for generalized permutohedra 543 large class of simple generalized permutohedra called nestohedra, including a palidromic- ity result for special cases. Section 5 is devoted to several explict examples, including Sn-invariant nestohedra, graph associahedra, the classical associahedron, the stellohedron, and the Stanley–Pitman polytope. These examples produce some alternative q-analogues of some well-known combinatorial sequences, including the Narayana numbers. 2 Background In this section, we provide a brief review of basic properties of permutations statistics, posets, polytopes and normal fans, and generalized permutohedra. 2.1 Permutation statistics Let A = {a1 < a2 < . . . < an} be a set of n elements. The symmetric group on A, denoted SA, is the set of all permutations of the elements of A. In the case of A = [n], we will simply write Sn. Given π = π1π2 · · ·πn ∈ SA, the descent set of π is Des(π) = {i ∈ [n− 1] : πi > πi+1}, the descent number of π is des(π) = |Des(π)|, and the major index of π is maj(π) = ∑ i∈Des(π) i. The descent statistic is commonly referred to as an Eulerian statistic, due to the con- nection to polynomial first studied by Euler [14]. The Eulerian polynomial An(t) is the unique polynomial which satisfies∑ k≥0 (k + 1)ntk = An(t) (1− t)n+1 However, this polynomial can be interpreted entirely combinatorially as An(t) = ∑ π∈Sn tdes(π). The major index, on the other hand, is commonly known as a Mahonian statistic, as it was introduced by MacMahon [18]. The descent statistic and major index statistic are naturally linked as they both encode information regarding the descent set of a permutation. Thus, it is fruitful to consider the joint distribution of these statistics, which motivates the Euler– Mahonian polynomial An(t, q) = ∑ π∈Sn tdes(π)qmaj(π), which specializes to the Eulerian polynomial under the substitution q = 1. This polynomial and various generalizations are widely of interest (see, e.g., [1, 5, 7, 9]). 2.2 Posets Let Q be a partially ordered set (poset) on [n] with relation Z j} and thus the descent number of Q is des(Q) := |Des(Q)|. If Q is a graded poset on [n] with minimal rank function ρ, we further have a notion of major index of Q maj(Q) := ∑ (i,j)∈Des(Q) ρ(j). We note that if Q is a totally ordered set with labels π1 N j} and des(T ) = |Des(T )|. Given x ∈ T , we say that the depth of x, denoted dp(x), is the length of the unique path from x to the root. The depth of T is depth(T ) := maxx∈T dp(x). The major index of T is maj(T ) := ∑ (i,j)∈Des(T ) (depth(T )− dp(j)) Remark 4.5. Note that for any x ∈ T , the quantity depth(T ) − dp(x) is precisely ρ(x) where ρ is the minimal rank function on the poset representation of T . Proposition 4.6 ([21, Corollary 8.4]). For any connected building set B on [n], the h- polynomial of the generalized permutohedron PB is hB(t) = ∑ T tdes(T ) where the sum is over B-trees T . Given connected building sets B1, . . . ,Br on pairwise disjoint sets S1, . . . , Sr, we can form the combined connected building set B on S = ⋃r i=1 Si by B = ( ⊔r i=1 Bi) ⊔ {S}. We will now give a formula for the h-polynomial of such a building set. Proposition 4.7. Let B1, . . . ,Br be connected building sets on the pairwise disjoint sets S1, . . . , Sr, and let B be the combined connected building set on S = ⋃r i=1 Si. Then hB(t) = (1 + t+ · · ·+ tr−1) r∏ i=1 hBi(t). Proof. Without loss of generality, let S = [n] and let the sets S1, . . . , Sr partition [n] such that if x ∈ Si and y ∈ Sj , x < y if and only if i < j for every 1 ≤ i, j ≤ r. Let T be a B-tree with vertex i as the root. Suppose that i ∈ Sj for some j. By Proposition 4.4, T is formed by connecting the root i to the roots of trees on the connected components of B|[n]\{i}. Note that the connected components are precisesly Bk where k ̸= j and the connected components of Bj |Sj\{i}. Therefore, T is formed by Bk-trees T1, T2, . . . , Tr such that for all k ̸= j, the root of Tk is connected to the root of Tj for some j = 1, 2, . . . , r. Additionally, given any collection of Bk-trees, we can form a B- tree by simply choosing one of the trees Tj to contain the root. Therefore, we will consider T as being partitioned into Bk-trees T1, T2, . . . , Tr with root in Tj in this way. Now, it is a 550 Ars Math. Contemp. 22 (2022) #P4.02 / 541–559 straightforward computations to note that des(T ) = r−j+ ∑r k=1 des(Tk) as the construc- tion preserves all existing descents in each tree Tk and introduces exactly one new descent between Tj and Tk where k > j. Since we the choices of trees for each k are independent, the contribution of all trees where Tj has the root to the h-polynomial is tr−j ∏r k=1 hBk(t). Thus, summing over all choices of j gives us the desired expression. Now we give a different characterization of the q-h-polynomial of the generalized per- mutohedron. This description comes from specializing Definition 3.1 to the case of nesto- hedra, making use of alternative descriptions of the descent set and major index. Proposition 4.8. For any connected building set B on [n], the q-h-polynomial of the gen- eralized permutohedron PB is hB(t, q) = ∑ T tdes(T )qmaj(T ) where the sum is over B-trees T . Define the statistic µ(T ) := ∑ (i,j)∈T (depth(T )− dp(j)). Note that this statistic depends only on the isomorphism type of the rooted tree T not on the labeling. With this, we introduce a trivariate analogue of the h-polynomial of a nestohedron on connected building set hB(t, q, u) := ∑ T tdes(T )qmaj(T )uµ(T ) By the Dehn-Sommerville relations, we have that the h-polynomial is palindromic. In certain cases, we can provide a multivariate analogue of palindromicity. Theorem 4.9. Let B be a connected building set on [n] which is invariant under the invo- lution ω : [n] → [n] such that ω(i) = n− i+1. Then the h-polynomial for the nestohedron PB is hB(t, q, u) = t n−1hB(t −1, q−1, qu) Proof. Let B be a building set such that ω(B) = B. Suppose that T is a B-tree. By Proposition 4.4, there exists a B-tree T̃ such that T and T̃ such that T̃ = ω(T ). That is, the trees are isomorphic as unlabeled rooted trees, and one can obtain the appropriate labels of one tree by applying the involution. It is clear that Des(T̃ ) = {(i, j) : (i, j) ̸∈ Des(T )}. Hence des(T̃ ) = n− 1− des(T ) and maj(T̃ ) = µ(T )−maj(T ). This gives the equality above. 5 Examples We conclude with a section computing explicit examples of q-h-polynomials for nestohe- dra of interest. Included in the list are Sn-invariant nestohedra, graph associahedra, the associahedron, the stellahedron, and the Stanley–Pitman polytope. 5.1 Sn-invariant nestohedra We will now specialize to the case of building sets which are invariant under the action of Sn on the ground set [n]. Note that a connected building set B on [n] is Sn-invariant if and only if B = { {1}, . . . , {n}, ( [n] j ) , j = k, . . . , n } E. Katz and M. Olsen: Multivariate polynomials for generalized permutohedra 551 for some 2 ≤ k ≤ n. Therefore, for a fixed n and fixed 2 ≤ k ≤ n, we will denote this building set Bkn. Proposition 5.1. Let Bkn be the Sn-invariant connected building set of [n] with minimal nonsingleton set of cardinality k. Suppose that T1 and T2 are any two B-trees. Then T1 and T2 are isomorphic as unlabeled rooted trees. Moreover, for any B-tree T , T ∼= Ak−1⊕Cn−k+1 as a poset, where Ai is an antichain on i elements, Cj is a totally ordered chain on j elements, and ⊕ is ordinal sum. Proof. This follows from Proposition 4.4 with the observation that Bkn|[n]\{i} ∼= Bk−1n−1 which is a connected building set. Continuing in this fashion, repeated restictions will result in connected building sets until we arrive at Bkn|[n]\W where W ⊂ [n] with |W | = n−k+1, which consists only of singleton elements. Theorem 5.2. Let Bkn be the Sn-invariant connected building set on [n] with minimal nonsingleton set of cardinality k. The q-h-polynomial for the nestohedron PBkn is hBkn(t, q) = ∑ A∈( [n]n−k+1) ∑ π∈SA tdes(π)+|{j∈[n]\A : j>π1}|qmaj(π)+des(π)+|{j∈[n]\A : j>π1}| Moreover, this polynomial satisfies hBkn(t, q) = t n−1q k2−2kn−k+n2+3n−2 2 hBkn(t −1, q−1). Prior to giving the proof of this formula, it is instructive to give concrete example of enumerating the descents in Bkn-trees. Example 5.3. Consider the B58-tree T given in Figure 3. The descents which occur along the chain are precisely the descents of the permutation π = 5481 ∈ S{1,4,5,8} which has Des(5418) = {1, 3} and des(5418) = 2. Moreover, there are descents which oc- cur between the antichain and the chain itself. The number of such descents is precisely the number of elements of [8] \ {1, 4, 5, 8} which are larger than 5. There are precisely 2, and hence yielding des(T ) = des(5481) + |{j ∈ [8] \ {1, 4, 5, 8} : j > 5}| = 4. When computing the major index, we note that the contributions of π = 5418 is∑ i∈Des(5418)(i + 1) = maj(5418) + des(5418) = 4 + 2 = 6, to account for the cor- rect rank. Moreover, every descent between the antichain and the chain has rank 1, so this contributes a total of 2. Thus, maj(T ) = maj(5418)+des(5418)+|{j ∈ [8]\{1, 4, 5, 8} : j > 5}| = 8. Proof. By Proposition 5.1, we know that any T has the poset structure of Ak−1⊕Cn−k+1. So any labeled tree is described by an n− k+1-element subset A of [n] and a permutation π ∈ SA. The permutation labels Cn−k+1, and the remaining elements of [n] \ A label the antichain Ak−1. There are two types of descents in the labeling: descents in Cn−k+1 which are enumerated by des(π), and descents where a label on the antichain Ak−1 is greater than π1 which is enumerated by |{j ∈ [n] \ A : j > π1}|. To compute maj(T ), note that if i ∈ Des(π) this corresponds to (j, ℓ) ∈ Des(T ) such that ρ(ℓ) = i + 1. So the contribution from descents of this form is qmaj(π)+des(π). The other descents are of the form (i, π1) ∈ Des(T ) and since ρ(π1) = 1, this contributes q|{j∈[n]\A : j>π1}|. To see the palidromicity statement, note that since Bkn is Sn-invariant, then it is invariant under the involution ω(i) = n − i + 1. It is clear that µ(T ) = k − 2 + ∑n−k+1 i=1 i = 552 Ars Math. Contemp. 22 (2022) #P4.02 / 541–559 k2−2kn−k+n2+3n−2 2 for any B k n-tree T . Subsequently, applying the result of Theorem 4.9 and setting u = 1 yields the desired statement. 1 8 4 5 2 3 6 7 Figure 3: An example of a B58-tree T as appears in Example 5.3. By directly applying the definitions of descent and major index statistics, we can see that des(T ) = 4 and maj(T ) = 8. 5.2 Graph associahedra We now consider a large family of examples of nestohedra arising from graphs. Given a graph G = ([n], E), a tube of G is a proper, nonempty subset I ⊂ [n] such that the induced subgraph G|I is connected. A k-tubing of G, χ, is a a collection of k distinct tubes subject to: 1. For all incomparable A1, A2 ∈ χ, A1 ∪A2 ̸∈ χ (non-adjacency); 2. For all incomparable A1, A2 ∈ χ, A1 ∩A2 = ∅ (non-intersecting). We do, however, allow for A1 ⊂ A2, which is called a nesting. We say that a tubing χ is maximal if it cannot add any additional tubes to χ, or equivalently, if |χ| = n − 1. Given a graph G, the graph associahedron of G is the polytope PG whose face lattice is given by the set of all tubings of G where χ < χ′ if χ is obtained from χ′ by adding tubes. Subsequently, the vertices of PG correspond to maximal tubings. This notion of graph associahedra originates with Carr and Devadoss [12, 13] and has been a well-studied family of examples of simple generalized permutohedra (see, e.g., [3, 6, 10, 11, 19]). Remark 5.4. Given a simple graph G = ([n], E), the graph associahedron PG is an ex- ample of nestohedron on a connected building set, even when G is not a connected graph. The graphical building set of G, B(G) is the collection of nonempty J ⊆ [n] such that the induced subgraph G|J is connected. While the building set B(G) is connected if and only if G is connected (c.f. [21, Example 6.2]), the graph associahedra PG using the notions of Carr and Devadoss [12, 13] is the nestahedron with building set B̂(G) = B(G)∪ [n] which is always connected and B̂(G) = B(G) if G connected. In light of Remark 5.4, we can specialize Proposition 4.7 to determine the h-polynomial of a disconnected graph. E. Katz and M. Olsen: Multivariate polynomials for generalized permutohedra 553 Corollary 5.5. Let G be a simple graph on [n] with connected components G1, G2, . . . , Gk. Then hG(t) = (1 + t+ · · ·+ tk−1) k∏ i=1 hGi(t). Let G = ([n], E) be a simple graph and let χ be a maximal tubing of G. Given i ∈ [n], the nesting index of i, denoted νχ(i), is the number of tubes containing i. The nesting number of χ is nest(χ) := maxi∈[n] νχ(i). Given any maximal χ, observe that for any tube Aj ∈ χ, there exists a unique element αj ∈ Aj such that for any tube Ak ⊂ Aj , we have αj ̸∈ Ak. For convenience, we will write Ak ⋖ Aj if Ak ⊂ Aj and there is no tube Aℓ such that Ak ⊂ Aℓ ⊂ Aℓ. Let αn denote the unique element which is not contained in any tube of χ. The nesting descent set is NestDes(χ) :={(αk, αj) : αk > αj and Ak ⋖Aj} ∪{(αℓ, αn) : αℓ > αn and Aℓ ̸⊂ Ap for any Ap}. The nesting descent number is nestDes(χ) := |NestDes(χ)| and the nesting major index is nestMaj(χ) := ∑ (αk,αj)∈NestDes(χ) (nest(χ)− νχ(αj)) We now state a formula for the q-h-polynomial of graph associahedra in terms of graph tubings. Proposition 5.6. Let G be a simple graph. The q-h-polynomial is hG(t, q) = ∑ χ tnestDes(χ)qnestMaj(χ) where the sum is taken over all maximal tubings χ. Proof. This follows by unpacking the definitions of B-trees in terms of graph tubings and applying Proposition 4.8. Remark 5.7. As was the case with nestohedra in general, we should note that this poly- nomial is invariant only under labeled graph automorphisms. Under most circumstance, a different choice of labeling of the vertices G will produce a different bivariate polynomial. However, the specialization under q = 1 is invariant under permutation of the ground set. Remark 5.8. As with nestohedra, we can similarly define a trivariant polynomial for graph associahedra, namely hG(t, q, u) = ∑ χ tnestDes(χ)qnestMaj(χ)uµ(χ) where the sum ranges over all maximal and µ(χ) = ∑ (αk,αj) (nest(T ) − νT (αj)) where this sum is over all pairs (αk, αj) such that Ak ⋖Aj , which is a direct translation of the µ 554 Ars Math. Contemp. 22 (2022) #P4.02 / 541–559 statistic for nestohedra. If the involution ω : [n] → [n] such that ω(i) = n− i+1 produces a labeled graph automorphism, then Theorem 4.9 gives us that palindromicity statement hG(t, q, u) = t n−1hG(t −1, q−1, qu). There are only two Sn-invariant graphs, namely the complete graph Kn and the null graph Nn = Kn (i.e. the edgeless graph), which produce only the simplest examples of gen- eralized permutohedra. PKn is the usual permutohedron Πn, and hence hKn(t, q) is the usual Euler–Mahonian polynomial. PNn is simply an n− 1 dimensional simplex and thus hNn(t, q) = ∑n−1 i=0 (tq) i. 5.3 The associahedron and a new q-analogue of Narayana numbers The associahedron A(n), which first appeared in the work of Stasheff [24], as well as the notable work of Lee [17], is the graph associahedron for G = Path(n), where the vertices are labeled linearly. It is well-known that hPath(n)(t) = n∑ k=1 N(n, k)tk−1 where N(n, k) = 1n ( n k )( n k−1 ) is the Narayana number, which refine the Catalan numbers. That is, hPath(n)(1) = Cn. To verify this formula, one should note that B-trees, or graph tubings on Path(n), are in bijection with binary trees on n vertices (See [20, Section 8.2]). The bijection sends descents in a B-tree to right edges in an unlabeled binary tree and N(n, k) is known to enumerate the number of unlabeled binary trees on n vertices with k − 1 right edges. Subsequently, we will phrase all formulae in terms of binary trees. Let T be a binary tree. Given an edge e ∈ T , let dp(e) be the length of the path from the root vertex to the closest vertex incident with e. Let depth(T ) = maxe∈T dp(e). The right multiset of T is the multiset R(T ) := {dp(e) : e is a right edge of T} . The right number of T is r(T ) = |R(T )| and the right index of T is rindex(T ) := depth(T )r(T )− ∑ j∈R(T ) j. By translating the general results for nestohedra into the above language for binary trees, we have the following: Corollary 5.9. The q-h-polynomial for the associahedron is hPath(n)(t, q) = ∑ T tr(T )qrindex(T ) where the sum ranges over all rooted unlabelled binary tree T on n vertices. Remark 5.10. This theorem gives rise to a q-analogue of the Narayana numbers. We say the (alternative) q-Narayana number is N(n, k, q) = ∑ T r(T )=k−1 qrindex(T ). E. Katz and M. Olsen: Multivariate polynomials for generalized permutohedra 555 It is clear that the substitution q = 1 yields N(n, k) as desired. We call these the alter- native q-Narayana numbers because, while this is the natural q-analogue in the context of generalized permutohedra as it arises from the major index, this does not agree with the usual q-Narayana number in the literature (see, e.g., [8, 22]). 5.4 The stellahedron The star graph on n + 1 vertices is the complete bipartite graph K1,n. The stellohedron is the graph associahedron associated to K1,n. Let K1,n be labeled such that the center vertex is labeled n + 1. Recall that a partial permutation of [n] is a linear ordering of a k-subset L ⊆ [n] for some k = 1, 2, . . . n. The B-trees for K1,n are in bijection with partial permutations of [n]. In particular, the structure of a B-tree is given by the ordinal sum of an antichain with a totally ordered chain An−k−1 ⊕Ck+1 for some k = 0, . . . , n such that the minimal element of Ck+1 has label n+ 1. 2 35 6 1 4 7 2 3 4 5 6 1 7 Figure 4: A tubing of K1,6 and its corresponding B-tree. To see this, note that we can identify the B-trees with graph tubings. Any tubing of K1,n is either (i) the tubing where each vertex i = 1, 2, . . . , n is in a singleton tube and n + 1 is the root, or (ii) some vertex i is the root and we have a tube containing all other vertices. In the case of (ii), once i is chosen, then the tubing directly arises from a tubing of K1,n−1 on the labels [n+ 1] \ {i}. Thus, by induction, we will have B-trees of the proposed form. For example, consider the tubing and B-tree given in Figure 4, which corresponds to the partial permutation π = 61 on [6]. Subsequently, the elements of the Ck+1 above the n+1 are the partial permutation (see [21, Section 10.4]) With this in mind, we can state the q-analogue of the h-polynomial for the stellohedron. Proposition 5.11. The q-h-polynomial for the stellohedron is hK1,n(t, q) = 1 + ∑ w tdes(w)+1qmaj(w)+2des(w)+2 where the sum is over all nonempty partial permutations of [n]. 556 Ars Math. Contemp. 22 (2022) #P4.02 / 541–559 Proof. The labels on Ck+1 correspond to a partial permutation of w̃ of [n + 1] where w̃1 = n + 1. Thus, we consider w to be the partial permutation of [n] with this first element omitted. If w = ∅, the corresponding B-tree has no descents. If w ̸= ∅, then the corresponding B-tree T has precisely des(w) + 1 descents, due to the guaranteed descent between n + 1 and w1. When computing the major index, note that if i ∈ Des(w), this means that we have an element of rank i + 2 where a descent occurs in T . Hence, the contribution to the major index is ∑ i∈Des(w)(i + 2) = maj(w) + 2des(w). Additionally, the descent between n+ 1 and w1 contributes 2, as ρ(w1) = 2. Thus, we have the desired formula. 5.5 The Stanley-Pitman polytope Introduced by Stanley and Pitman in [23], the Stanley-Pittman polytope is a integral poly- tope defined by the equations PS(n) := { x ∈ Rn : xi ≥ 0 and j∑ i=1 xi ≤ j for each 1 ≤ j ≤ n } . This polytope is combinatorially equivalent to an n-cube, as illustrated in Figure 5. How- ever, this polytope is of particular interest as it appears naturally when studying empirical distributions in statistics and has connections to many combinatorial objects, such as park- ing functions and plane trees. Postnikov [20, Section 8.5] observed that this polytope can be realized as the nestohedron from the building set BPS = {[i, n], {i} : i ∈ [n]}, where [i, n] = {i, i + 1, . . . , n}. Notably, this is not a graph associahedron. Given that this polytope is combinatorially equivalent to an n-cube, we have hBPS(t) = (1 + t) n−1 [23, Theorem 20]. We now give the q-analogue. x1 x2 x1 x2 x3 Figure 5: PS(2) and PS(3). Proposition 5.12. The q-h-polynomial for the Stanley-Pitman polytope is hBPS(t, q) = n−2∑ ℓ=0 ( n− 2 ℓ ) tℓq ℓ2+3ℓ+2 2 ( t+ qℓ ) . E. Katz and M. Olsen: Multivariate polynomials for generalized permutohedra 557 3 4 6 75 21 3 4 7 65 21 Figure 6: Two BPS-trees for n = 7 from the increases sequences I1 = {3 < 4 < 6 < 7} and I2 = {3 < 4 < 7}. Alternatively, these are the two trees from the set {3, 4} ⊂ [5]. Proof. First note that hBPS(t, 1) = (t + 1) n−1, so this agrees with the known results. To compute this, we will need BPS-trees, which as determined by Postnikov, Reiner, and Williams [21, Section 10.5], are formed in the following way. Given any increasing se- quence of positive integers I = {i1 < i2 < · · · < ik = n} where we let i1 be the root and form the chain of edges (i1, i2), (i2, i3), . . . , (ik−1, ik) and for all j ∈ [n] \ I we have the edge (is, j) where is is the minimal element of I such that is > j. An example can be seen in Figure 6. It is clear that all descents will be occur along the chain of edges. So, we must consider two cases: (i) ik−1 = n− 1 and (ii) ik−1 ≤ n− 2. In case (i), for convenience let ℓ = k − 2. We form a tree T by choosing a subset J ∈( [n−2] ℓ ) and arranging it increasing order to form a chain of edges which ends in (iℓ, n−1), (n− 1, n). By definition, depth(T ) = ℓ+ 1, des(T ) = ℓ+ 1, and maj(T ) = (ℓ+ 1)2 −∑ℓ i=0 i = ℓ2+3ℓ+2 2 . So, the contribution of trees of this form to the q-h-polynomials is n−2∑ ℓ=0 ( n− 2 ℓ ) tℓ+1q ℓ2+3ℓ+2 2 . (5.1) In case (ii) where ik−1 ̸= n − 1, for ease of notation, let ℓ = k − 1. Similarly, we form such a tree T by choosing J ∈ ( [n−2] ℓ ) and arranging it increasing order to form a chain of edges which ends in (iℓ, n). Note that, when including the elements not in the chain, we gain edges from the vertex n going away from the root, in particular, the edge (n, n− 1). So, we again have depth(T ) = ℓ+1. However, we now have des(T ) = ℓ, and maj(T ) = (ℓ + 1)2 − ∑ℓ−1 i=0 i = ℓ2+5ℓ+2 2 . So the contribution of trees of this type to the q-h-polynomial is n−2∑ ℓ=0 ( n− 2 ℓ ) tℓq ℓ2+5ℓ+2 2 . (5.2) Summing (5.1) and (5.2) and simplifying gives the desired expression. 558 Ars Math. Contemp. 22 (2022) #P4.02 / 541–559 Remark 5.13. We conclude our discussion by noting that our computation produces an alternative q-analogue of ( n−1 ℓ ) , namely( n− 2 ℓ− 1 ) q ℓ2+ℓ 2 + ( n− 2 ℓ ) q ℓ2+5ℓ+2 2 . This, of course, reduces to ( n−1 ℓ ) when q = 1 and arises quite naturally from generalizing the major index statistic. However, this is not the usual q-analogue of a binomial coefficient which arises in many natural ways, such as bit string inversions and lattice path areas. References [1] R. M. Adin, F. Brenti and Y. Roichman, Descent numbers and major indices for the hyperocta- hedral group, Adv. Appl. Math. 27 (2001), 210–224, doi:10.1006/aama.2001.0731. [2] M. Aguiar and F. Ardila, Hopf monoids and generalized permutohedra, 2017, arXiv:1709.07504 [math.CO]. [3] F. Ardila, V. Reiner and L. Williams, Bergman complexes, Coxeter arrangements, and graph associahedra, Sém. Lothar. Comb. 54A (2006), Art. B54Aj, https://www.mat.univie. ac.at/˜slc/. [4] D. Armstrong, Generalized noncrossing partitions and combinatorics of Coxeter groups, Mem. Am. Math. Soc. 202 (2009), x+159, doi:10.1090/s0065-9266-09-00565-1. [5] E. Bagno and R. Biagioli, Colored-descent representations of complex reflection groups G(r, p, n), Isr. J. Math. 160 (2007), 317–347, doi:10.1007/s11856-007-0065-z. [6] E. Barnard and T. McConville, Lattices from graph associahedra and subalgebras of the Malvenuto-Reutenauer algebra, Algebra Univers. 82 (2021), Paper No. 2, 53, doi:10.1007/ s00012-020-00689-z. [7] M. Beck and B. Braun, Euler-Mahonian statistics via polyhedral geometry, Adv. Math. 244 (2013), 925–954, doi:10.1016/j.aim.2013.06.002. [8] P. Brändén, q-Narayana numbers and the flag h-vector of J(2×n), Discrete Math. 281 (2004), 67–81, doi:10.1016/j.disc.2003.07.006. [9] B. Braun and M. Olsen, Euler-Mahonian statistics and descent bases for semigroup algebras, Eur. J. Comb. 69 (2018), 237–254, doi:10.1016/j.ejc.2017.11.005. [10] J. Cardinal, S. Langerman and P. Pérez-Lantero, On the diameter of tree associahedra, Electron. J. Comb. 25 (2018), Paper 4.18, doi:10.37236/7762. [11] M. Carr, S. L. Devadoss and S. Forcey, Pseudograph associahedra, J. Comb. Theory Ser. A 118 (2011), 2035–2055, doi:10.1016/j.jcta.2011.04.004. [12] M. P. Carr and S. L. Devadoss, Coxeter complexes and graph-associahedra, Topol. Appl. 153 (2006), 2155–2168, doi:10.1016/j.topol.2005.08.010. [13] S. L. Devadoss, A realization of graph associahedra, Discrete Math. 309 (2009), 271–276, doi:10.1016/j.disc.2007.12.092. [14] L. Euler, Remarques sur un beau rapport entre les series des puissances tant direct que re- ciproques, Mem. L’Acad. Sci. Berlin 17 (1768), 83–106, https://scholarlycommons. pacific.edu/euler-works/352/. [15] E. M. Feichtner and B. Sturmfels, Matroid polytopes, nested sets and Bergman fans, Port. Math. (N.S.) 62 (2005), 437–468, https://www.emis.de/journals/PM/62f4/3.html. E. Katz and M. Olsen: Multivariate polynomials for generalized permutohedra 559 [16] S. Fomin and N. Reading, Generalized cluster complexes and Coxeter combinatorics, Int. Math. Res. Not. (2005), 2709–2757, doi:10.1155/imrn.2005.2709. [17] C. W. Lee, The associahedron and triangulations of the n-gon, Eur. J. Comb. 10 (1989), 551– 560, doi:10.1016/s0195-6698(89)80072-1. [18] P. A. MacMahon, Combinatory Analysis, Two volumes (bound as one), Chelsea Publishing Co., New York, 1960, http://name.umdl.umich.edu/ABU9009.0001.001. [19] T. Manneville and V. Pilaud, Graph properties of graph associahedra, Sém. Lothar. Comb. 73 (2015), Art. B73d, https://www.mat.univie.ac.at/˜slc/wpapers/ s73mannpil. [20] A. Postnikov, Permutohedra, associahedra, and beyond, Int. Math. Res. Not. IMRN (2009), 1026–1106, doi:10.1093/imrn/rnn153. [21] A. Postnikov, V. Reiner and L. Williams, Faces of generalized permutohedra, Doc. Math. 13 (2008), 207–273, https://elibm.org/article/10000114. [22] V. Reiner and E. Sommers, Weyl group q-Kreweras numbers and cyclic sieving, Ann. Comb. 22 (2018), 819–874, doi:10.1007/s00026-018-0408-y. [23] R. P. Stanley and J. Pitman, A polytope related to empirical distributions, plane trees, parking functions, and the associahedron, Discrete Comput. Geom. 27 (2002), 603–634, doi:10.1007/ s00454-002-2776-6. [24] J. D. Stasheff, Homotopy associativity of H-spaces. I, II, Trans. Am. Math. Soc. 108 (1963), 293–312, doi:10.1090/s0002-9947-1963-0158400-5. [25] G. M. Ziegler, Lectures on Polytopes, volume 152 of Graduate Texts in Mathematics, Springer- Verlag, New York, 1995, doi:10.1007/978-1-4613-8431-1. ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 5 (2022) #P4.03 / 561–565 https://doi.org/10.26493/1855-3974.2600.dcc (Also available at http://amc-journal.eu) A note on the k-tuple domination number of graphs* Abel Cabrera Martı́nez Universidad de Córdoba, Departamento de Matemáticas, Campus de Rabanales, 14071, Córdoba, Spain Received 9 April 2021, accepted 19 December 2021, published online 3 August 2022 Abstract In a graph G, a vertex dominates itself and its neighbours. A set D ⊆ V (G) is said to be a k-tuple dominating set of G if D dominates every vertex of G at least k times. The minimum cardinality among all k-tuple dominating sets is the k-tuple domination number of G. In this note, we provide new bounds on this parameter. Some of these bounds generalize other ones that have been given for the case k = 2. Keywords: k-domination, k-tuple domination. Math. Subj. Class. (2020): 05C69 1 Introduction Throughout this note we consider simple graphs G with vertex set V (G). Given a vertex v ∈ V (G), N(v) denotes the open neighbourhood of v in G. In addition, for any set D ⊆ V (G), the degree of v in D, denoted by degD(v), is the number of vertices in D adjacent to v, i.e., degD(v) = |N(v) ∩D|. The minimum and maximum degrees of G will be denoted by δ(G) and ∆(G), respectively. Other definitions not given here can be found in standard graph theory books such as [12]. Domination theory in graphs have been extensively studied in the literature. For in- stance, see the books [9, 10, 11]. A set D ⊆ V (G) is said to be a dominating set of G if degD(v) ≥ 1 for every v ∈ V (G) \ D. The domination number of G is the minimum cardinality among all dominating sets of G and it is denoted by γ(G). We define a γ(G)-set as a dominating set of cardinality γ(G). The same agreement will be assumed for optimal parameters associated to other characteristic sets defined in the paper. *We are grateful to the anonymous reviewers for their useful comments on this note that improved its presen- tation. E-mail address: acmartinez@uco.es (Abel Cabrera Martı́nez) cb This work is licensed under https://creativecommons.org/licenses/by/4.0/ 562 Ars Math. Contemp. 5 (2022) #P4.03 / 561–565 In 1985, Fink and Jacobson [4, 5] extended the idea of domination in graphs to the more general notion of k-domination. A set D ⊆ V (G) is said to be a k-dominating set of G if degD(v) ≥ k for every v ∈ V (G) \D. The k-domination number of G, denoted by γk(G), is the minimum cardinality among all k-dominating sets of G. Subsequently, and as expected, several variants for k-domination were introduced and studied by the scientific community. In two different papers published in 1996 and 2000, Harary and Haynes [7, 8] introduced the concept of double domination and, more generally, the concept of k-tuple domination. Given a graph G and a positive integer k ≤ δ(G) + 1, a k-dominating set D is said to be a k-tuple dominating set of G if degD(v) ≥ k − 1 for every v ∈ D. The k-tuple domination number of G, denoted by γ×k(G), is the minimum cardinality among all k-tuple dominating sets of G. The case k = 2 corresponds to double domination, in such a case, γ×2(G) denotes the double domination number of graph G. In this note, we provide new bounds on the k-tuple domination number. Some of these bounds generalize other ones that have been given for the double domination number. 2 New bounds on the k-tuple domination number Recently, Hansberg and Volkmann [6] put into context all relevant research results on mul- tiple domination that have been found up to 2020. In that chapter, they posed the following open problem. Problem 2.1 ([6, Problem 5.8, p. 194]). Give an upper bound for γ×k(G) in terms of γk(G) for any graph G of minimum degree δ(G) ≥ k − 1. A fairly simple solution for the problem above is given by the straightforward rela- tionship γ×k(G) ≤ kγk(G), which can be derived directly by constructing a set of ver- tices D′ ⊆ V (G) of minimum cardinality from a γk(G)-set D such that D ⊆ D′ and degD′(x) ≥ k − 1 for every vertex x ∈ D. From this construction above, it is easy to check that D′ is a k-tuple dominating set of G and so, γ×k(G) ≤ |D′| = |D|+ |D′ \D| ≤ |D|+ (k − 1)|D| = kγk(G). This previous inequality was surely considered by Hansberg and Volkmann and, in that sense, they have established the previous problem assuming that γ×k(G) < kγk(G) for every graph G with δ(G) ≥ k − 1. We next confirm their suspicions and provide a solution to Problem 2.1. Theorem 2.2. Let k ≥ 2 be an integer. For any graph G with δ(G) ≥ k − 1, γ×k(G) ≤ kγk(G)− (k − 1)2. Proof. Let D be a γk(G)-set. As γ×k(G) ≤ |V (G)| we assume, without loss of generality, that k|D| − (k− 1)2 ≤ |V (G)|. Now, let U = {u1, . . . , uk−1} ⊆ V (G) \D, D′ = D ∪U and D0 = {v ∈ D : degD′(v) < k − 1}. The following inequalities arise from counting arguments on the number of edges joining U with D0 and U with D \D0, respectively. ∑ v∈D0 degD′(v) ≥ k−1∑ i=1 degD0(ui) and |D \D0|(k − 1) ≥ k−1∑ i=1 degD\D0(ui). A. Cabrera Martı́nez: A note on the k-tuple domination number of graphs 563 By the previous inequalities and the fact that D is a k-dominating set of G, we deduce that ∑ v∈D0 degD′(v) + |D \D0|(k − 1) ≥ k−1∑ i=1 degD0(ui) + k−1∑ i=1 degD\D0(ui) = k−1∑ i=1 degD(ui) ≥ k(k − 1). Now, we define D′′ ⊆ V (G) as a set of minimum cardinality among all supersets W of D′ such that degW (x) ≥ k − 1 for every vertex x ∈ D. Since degD′(x) ≥ k − 1 for every x ∈ D \ D0, the condition on W is equivalent to that every vertex v ∈ D0 has at least k−1−degD′(v) neighbours in W \D. Hence, by the minimality of D′′ and the inequality chain above, we deduce that |D′′ \D′| ≤ |D0|(k − 1)− ∑ v∈D0 degD′(v) = |D|(k − 1)− (∑ v∈D0 degD′(v) + |D \D0|(k − 1) ) ≤ |D|(k − 1)− k(k − 1). Moreover, it is easy to check that D′′ is a k-tuple dominating set of G because each vertex in V (G) \D is dominated k times by vertices of D ⊆ D′′ (recall that D is a k-dominating set of G) and the construction of D′′ ensures that each vertex in D is dominated k times by vertices of D′′. Hence, γ×k(G) ≤ |D′′| = |D′|+ |D′′ \D′| ≤ |D|+ k − 1 + |D|(k − 1)− k(k − 1) = kγk(G)− (k − 1)2, which completes the proof. The bound above is tight. For instance, it is achieved by any complete bipartite graph Kk,k′ with k′ ≥ k, as γ×k(Kk,k′) = 2k−1 and γk(Kk,k′) = k. When k = 2, Theorem 2.2 leads to the relationship γ×2(G) ≤ 2γ2(G)− 1 given in 2018 by Bonomo et al. [1]. A set D ⊆ V (G) is a 2-packing of a graph G if N [u] ∩ N [v] = ∅ for every pair of different vertices u, v ∈ D. The 2-packing number of G, denoted by ρ(G), is the maximum cardinality among all 2-packings of G. The next theorem relates the k-tuple domination number with the 2-packing number of a graph. Note that the bounds given in this result are generalizations of the bounds γ×2(G) ≥ 2ρ(G) due to Chellali et al. [3], and γ×2(G) ≤ |V (G)| − ρ(G) due to Chellali and Haynes [2]. Theorem 2.3. Let k ≥ 2 be an integer. For any graph G of order n and δ(G) ≥ k, kρ(G) ≤ γ×k(G) ≤ n− ρ(G). 564 Ars Math. Contemp. 5 (2022) #P4.03 / 561–565 Proof. Let D be a ρ(G)-set and S a γ×k(G)-set. Since degS(v) ≥ k for every v ∈ D \ S, and degS(v) ≥ k − 1 for every v ∈ D ∩ S, we deduce that γ×k(G) = |S| ≥ ∑ v∈D\S degS(v) + ∑ v∈D∩S (degS(v) + 1) ≥ k|D| = kρ(G), and the lower bound follows. Next, let us proceed to prove that V (G) \ D is a k-tuple dominating set of G. Since δ(G) ≥ k, N(D) ∩ D = ∅ and degD(x) ≤ 1 for every x ∈ V (G) \ D, we deduce that degV (G)\D(v) ≥ k for every v ∈ D and degV (G)\D(v) ≥ k − 1 for every v ∈ V (G) \D. Hence, V (G) \D is a k-tuple dominating set of G, as desired. Therefore, γ×k(G) ≤ |V (G) \D| = n− ρ(G), which completes the proof. Let H be the family of graphs Hk,r defined as follows. For any pair of integers k, r ∈ Z, with k ≥ 2 and r ≥ 1, the graph Hk,r is obtained from a complete graph Kkr and an empty graph rK1 such that V (Hk,r) = V (Kkr) ∪ V (rK1), V (Kkr) = {v1, . . . , vkr} and V (rK1) = {u1, . . . , ur} and E(Hk,r) = E(Kkr) ∪ ( ⋃r−1 i=0 {ui+1vki+1, . . . , ui+1vki+k}). Figure 1 shows a graph of this family. Observe that |V (Hk,r)| = r(k+1), γ×k(Hk,r) = kr and ρ(Hk,r) = r for every Hk,r ∈ H. Therefore, for these graphs the bounds given in Theorem 2.3 are tight, i.e., γ×k(Hk,r) = kρ(Hk,r) = |V (Hk,r)| − ρ(Hk,r). Figure 1: The graph H4,2 ∈ H. In [8], Harary and Haynes showed that γ×k(G) ≥ 2kn−2mk+1 for any graph G of order n and size m with δ(G) ≥ k − 1. The next result is a partial refinement of the bound above because it only considers graphs with minimum degree at least k. Proposition 2.4. Let k ≥ 2 be an integer. For any graph G of order n and size m with δ(G) ≥ k, γ×k(G) ≥ (δ(G) + k)n− 2m δ(G) + 1 . Proof. Let S be a γ×k(G)-set and S = V (G) \ S. Hence, 2m = ∑ v∈S degS(v) + 2 ∑ v∈S degS(v) + ∑ v∈S degS(v) = ∑ v∈S degS(v) + ∑ v∈S degS(v) + ∑ v∈S degV (G)(v) ≥ (k − 1)|S|+ k(n− |S|) + δ(G)(n− |S|) = (k − 1)|S|+ (δ(G) + k)(n− |S|) = (δ(G) + k)n− (δ(G) + 1)|S|, A. Cabrera Martı́nez: A note on the k-tuple domination number of graphs 565 which implies that |S| ≥ (δ(G)+k)n−2mδ(G)+1 . Therefore, the proof is complete. The bound above is tight. For instance, it is achieved for the join graph G = Kk + Ck obtained from the complete graph Kk and the cycle graph Ck, with k ≥ 3. For this case, we have that γ×k(G) = k, |V (G)| = 2k, δ(G) = k + 2 and 2|E(G)| = 3k2 + k. Also, it is achieved for the complete graph Kn (n ≥ 3) and any k ∈ {2, . . . , n− 1}. ORCID iDs Abel Cabrera Martı́nez https://orcid.org/0000-0003-2806-4842 References [1] F. Bonomo, B. Brešar, L. N. Grippo, M. Milanič and M. D. Safe, Domination parameters with number 2: interrelations and algorithmic consequences, Discrete Appl. Math. 235 (2018), 23– 50, doi:10.1016/j.dam.2017.08.017. [2] M. Chellali and T. W. Haynes, On paired and double domination in graphs, Util. Math. 67 (2005), 161–171. [3] M. Chellali, A. Khelladi and F. Maffray, Exact double domination in graphs, Discuss. Math. Graph Theory 25 (2005), 291–302, doi:10.7151/dmgt.1282. [4] J. F. Fink and M. S. Jacobson, n-domination in graphs, in: Graph Theory with Applications to Algorithms and Computer Science (Kalamazoo, Mich., 1984), Wiley, New York, Wiley- Intersci. Publ., pp. 283–300, 1985. [5] J. F. Fink and M. S. Jacobson, On n-domination, n-dependence and forbidden subgraphs, in: Graph Theory with Applications to Algorithms and Computer Science (Kalamazoo, Mich., 1984), Wiley, New York, Wiley-Intersci. Publ., pp. 301–311, 1985. [6] A. Hansberg and L. Volkmann, Multiple domination, in: Topics in Domination in Graphs, Springer, Cham, volume 64 of Dev. Math., pp. 151–203, 2020, doi:10.1007/ 978-3-030-51117-3\ 6. [7] F. Harary and T. W. Haynes, Nordhaus-Gaddum inequalities for domination in graphs, volume 155, pp. 99–105, 1996, doi:10.1016/0012-365x(94)00373-q. [8] F. Harary and T. W. Haynes, Double domination in graphs, Ars Comb. 55 (2000), 201–213. [9] T. W. Haynes, S. T. Hedetniemi and M. A. Henning (eds.), Topics in domination in graphs, volume 64 of Developments in Mathematics, Springer, Cham, 2020, doi:10.1007/ 978-3-030-51117-3. [10] T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Domination in Graphs, Advanced Topics, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 1998, doi:10.1201/9781315141428. [11] T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of Domination in Graphs, vol- ume 208 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 1998, doi:10.1201/9781482246582. [12] D. B. West, Introduction to Graph Theory, Prentice Hall, Inc., Upper Saddle River, NJ, 1996, https://faculty.math.illinois.edu/˜west/igt/index.html. ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P4.04 / 567–594 https://doi.org/10.26493/1855-3974.2659.be1 (Also available at http://amc-journal.eu) Growable realizations: a powerful approach to the Buratti-Horak-Rosa Conjecture* M. A. Ollis † Marlboro Institute for Liberal Arts and Interdisciplinary Studies, Emerson College, Boston, MA 02116, USA Anita Pasotti DICATAM, Sez. Matematica, Università degli Studi di Brescia, Via Branze 43, I 25123 Brescia, Italy Marco A. Pellegrini Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via della Garzetta 48, I 25133 Brescia, Italy John R. Schmitt Mathematics Department, Middlebury College, Middlebury, VT 05753, USA Received 17 June 2021, accepted 16 December 2021, published online 3 August 2022 Abstract Label the vertices of the complete graph Kv with the integers {0, 1, . . . , v − 1} and define the length of the edge between the vertices x and y to be min(|x−y|, v−|x−y|). Let L be a multiset of size v− 1 with underlying set contained in {1, . . . , ⌊v/2⌋}. The Buratti- Horak-Rosa Conjecture is that there is a Hamiltonian path in Kv whose edge lengths are exactly L if and only if for any divisor d of v the number of multiples of d appearing in L is at most v − d. We introduce “growable realizations,” which enable us to prove many new instances of the conjecture and to reprove known results in a simpler way. As examples of the new method, we give a complete solution when the underlying set is contained in {1, 4, 5} or in {1, 2, 3, 4} and a partial result when the underlying set has the form {1, x, 2x}. We believe that for any set U of positive integers there is a finite set of growable realizations that implies the truth of the Buratti-Horak-Rosa Conjecture for all but finitely many multisets with underlying set U . Keywords: Hamiltonian path, complete graph, edge-length, growable realization. *The second and the third author were partially supported by INdAM-GNSAGA. †Corresponding author. cb This work is licensed under https://creativecommons.org/licenses/by/4.0/ 568 Ars Math. Contemp. 22 (2022) #P4.04 / 567–594 Math. Subj. Class. (2020): 05C38, 05C78 1 Introduction Let Kv be the complete graph on v vertices, labeled with the integers {0, 1, . . . , v − 1}. For two vertices x and y, define the length of the edge between them to be ℓ(x, y) = min(|x− y|, v − |x− y|), which is an integer in the range 1 ≤ ℓ(x, y) ≤ ⌊v/2⌋. A Hamiltonian path h = [h1, h2, . . . , hv] in Kv uses v − 1 edges and gives a multi- set L = {ℓ(hi, hi+1) : 1 ≤ i ≤ v − 1} of edge-lengths. Call h a realization of L or say that h realizes L. For example, with v = 7 the Hamiltonian path [0, 5, 1, 2, 6, 3, 4] has edge-length sequence [2, 3, 1, 3, 3, 1] and hence realizes the multiset {12, 2, 33} (where exponents indicate multiplicity). Given a multiset L, its underlying set is given by U = {x : x ∈ L}. The focus of our inquiry is the Buratti-Horak-Rosa Conjecture, or BHR Conjecture: Conjecture 1.1. Let L be a multiset of size v − 1 with underlying set U contained in {1, . . . , ⌊v/2⌋}. Then there is a realization of L in Kv if and only if for any divisor d of v the number of multiples of d in L is at most v − d. When v is prime, in which case the condition on divisors is always satisfied, we have the original Buratti Conjecture, see [1, 11]. Horak and Rosa [5] generalize this to composite v and show that the condition on divisors is necessary; Pasotti and Pellegrini [9] reformulate Horak and Rosa’s statement into the one in Conjecture 1.1. Call a multiset L of size v − 1 admissible if it has underlying set U ⊆ {1, . . . , ⌊v/2⌋} and it satisfies the divisor condition of the BHR conjecture. Denote the BHR Conjecture for L by BHR(L). Much work has been done on the BHR Conjecture. Theorem 1.2 captures the main progress that has been made to date. Theorem 1.2. Let L be a multiset of size v − 1 with underlying set U . In each of the following cases, if L is admissible, then it is realizable. (1) |U | ≤ 2 [3, 5], (2) U = {1, 2, 4}, {1, 2, 6}, {1, 2, 8} [10], (3) U ⊆ {1, 2, 3, 5} [2, 9], (4) L = {1a, 2b, 3c, 4d} with either a ≥ 3 and c, d ≥ 1 or a = 2 and b, c, d ≥ 1 [8], (5) L = {1a, 2b, xc} when x is even and a+ b ≥ x− 1 [10], (6) L = {1a, xb, (x + 1)c} when x is odd and either a ≥ min(3x − 3, b + 2x − 3) or a ≥ 2x− 2 and c ≥ 4b/3 [8], E-mail addresses: matt ollis@emerson.edu (M. A. Ollis), anita.pasotti@unibs.it (Anita Pasotti), marcoantonio.pellegrini@unicatt.it (Marco A. Pellegrini), jschmitt@middlebury.edu (John R. Schmitt) M. Ollis et al.: Growable realizations: a powerful approach to the Buratti-Horak-Rosa . . . 569 (7) L = {1a, xb, (x + 1)c} when x is even and either a ≥ min(3x − 1, c + 2x − 1) or a ≥ 2x− 1 and b ≥ c [8], (8) U ⊆ {1, 2, 4, . . . , 2x} and {12x−1, 2x} ⊆ L [8], (9) U ⊆ {1, 2, 4, . . . , 2x, 2x+ 1} and {16x−1, 2x+ 1} ⊆ L [8], (10) L = {1a1 , 2a2 , . . . , xax} with a1 ≥ a2 ≥ · · · ≥ ax [7, 8], (11) L = M∪{1a} for any multiset M and a > aM , where aM is a constant that depends on M [5], (12) v ≤ 19 or v = 23 [6]. After proving Theorem 1.2(11), Horak and Rosa observe that “to get an explicit bound... one only needs refer to lemmas used in the proof” [5]. It turns out that their methodology can be used to give a bound that is linear in the elements of the underlying set and inde- pendent of their multiplicities, neither of which is clear from the statement of the result. We believe that this is of interest and so give an explicit bound with these properties in Theorem 1.3. Theorem 1.3. Let M be a multiset with underlying set U = {x1, . . . , xk}, where 1 < x1 < · · · < xk. Then L = M ∪ {1s} is realizable for any s ≥ 3xk − 5 + ∑k i=1 xi. Proof outline. We give the steps required to establish the bound, referring to [5] for the specific details. In the notation of [5, Theorem 3.4], we partition M as L1 ∪ L2 ∪ L3 ∪ L4 in a certain way and then M ∪ {1s} is realizable for all s ≥ s1 + s2 + s3 + s4 − 1, where each si is dependent on Li for 1 ≤ i ≤ 4. Let Ui be the underlying set of Li for 1 ≤ i ≤ 4. By [5, Lemma 3.12], we may take s1 = 1 − 2|U1| + ∑ x∈U1 x; hence s1 ≤(∑k i=1 xi ) −1. By [5, Lemma 3.9], we may take s2 = max(U2)−1; hence s2 ≤ xk−1. By [5, Lemma 3.7], we may take s3 = max(U3)−1; hence s3 ≤ xk−1. By [5, Lemma 3.13], we may take s4 = max(U4)− |U4|; hence s4 ≤ xk − 1. Combining these bounds we find that L is realizable for all s ≥ 3xk−5+ ∑k i=1 xi. The BHR Conjecture has close connections to many other problems and conjectures concerning sequences with distinct partial sums or subgraphs of Kv other than paths; see [8] for more discussion of this. A recent paper also makes a connection between the BHR Conjecture and the Traveling Salesman Problem [4]. We are frequently concerned with the congruence classes of multiple elements with respect to multiple integers. The following notation is useful for these situations: if xi ≡ yi (mod zi) for 1 ≤ i ≤ k, then write (x1, . . . , xk) ≡ (y1, . . . , yk) (mod (z1, . . . , zk)). We also need the notion of a translation of a sequence h = [h1, . . . , hv] by an integer m: h+m = [h1 +m, . . . , hv +m]. The translation of a sequence produces the same multiset of absolute differences as the original sequence. 570 Ars Math. Contemp. 22 (2022) #P4.04 / 567–594 We require a lot of small examples of realizations for particular multisets. These were mostly found using a heuristic algorithm implemented in GAP. Given a target multiset L of size v − 1, the algorithm starts from a random Hamiltonian path in Kv and keeps trying to move to a Hamiltonian path that is “closer” to realizing L—in the sense of trying to increase |L ∩ L′|, where L′ is the multiset realized by the path under consideration—by removing an edge from the path and reconnecting the two resulting paths in a different way. If it gets stuck before finding a realization of L, then it tries again from a different starting path. For the fairly small values of v in which we are interested, this simple algorithm is sufficient to find the desired realizations quickly. The programs are available on the ArXiv page for this paper. The most far-reaching components of Theorem 1.2 were proved using “linear” realiza- tions. A Hamiltonian path h = [h1, h2, . . . , hv] of Kv defines a multiset of absolute differ- ences L = {|hi − hi+1| : 1 ≤ i ≤ v − 1} with underlying set contained in {1, . . . , v − 1}. In this situation, h is a linear realization of L. If h1 = 0, then the linear realization is standard; if h1 = 0 and hv = v− 1, then the linear realization is perfect. To emphasize the distinction between linear realizations and realizations, realizations as defined above are sometimes referred to as cyclic realizations. Linear realizations are closely related to cyclic realizations. For example, if each ele- ment in a multiset L of size v − 1 is at most ⌊v/2⌋, then a linear realization of L is also a cyclic realization of L. See [5] for further discussion. What makes linear realizations so useful in addressing the BHR Conjecture, standard and perfect ones especially, is their ability to be combined and hence used in inductive arguments. This is the approach taken by Horak and Rosa in [5] and the multisets L given in Theorem 1.3 in fact have linear realizations that are also cyclic realizations for the same L. The main contribution of this work is to introduce an alternative object: the “growable” realization, which we define in the next section. These are cyclic realizations that can be used in inductive arguments in somewhat similar ways to linear ones. In Section 3 we reprove, with a much shorter proof, the result from [2] that BHR(L) holds when L has underlying set U = {1, 2, 3} to illustrate that this new tool is, in some ways, more powerful than existing ones. We go on to prove instances of the BHR Conjec- ture that seem beyond the reach of current techniques. In particular, we are able to add the following items to Theorem 1.2: • U = {1, 4, 5} (Section 3), • U ⊆ {1, 2, 3, 4} (Section 4), • L = {1a, xb, (2x)c} when a ≥ x− 2, c is even and b ≥ 5x− 2 + c/2 (Section 5), • L = {1a, 3b, 6c} when c is odd and b ≥ 18 + (c− 1)/2 (Section 5). 2 Growable realizations Growable realizations will let us move from solving BHR(L) to BHR(L ∪ {xx}) under certain circumstances. When this can be done for multiple choices of x, this is a powerful tool. Take x with 0 < x ≤ v/2. For a given m, with 0 ≤ m < v, we shall embed Kv into M. Ollis et al.: Growable realizations: a powerful approach to the Buratti-Horak-Rosa . . . 571 Kv+x as follows: y 7→ { y when y ≤ m, y + x otherwise. This embedding preserves some edge lengths and increases others. Call it the m-embedding of Kv into Kv+x. Remark 2.1. Let y, z be two vertices of Kv and assume, without loss of generality, that y < z. Let y′ and z′ be the images of y and z, respectively, by the m-embedding of Kv into Kv+x. One can check that ℓ(y, z) = ℓ(y′, z′) if and only if one of the following holds: • y < z ≤ m and z − y ≤ v2 ; • m < y < z and z − y ≤ v2 ; • y ≤ m < z and z − y > v2 . Definition 2.2. Let h = [h1, . . . , hv] be a cyclic realization of a multiset L. Take x and m with 0 < x ≤ v/2 and 0 ≤ m < v. If each y with m−x < y ≤ m is incident with exactly one edge whose length is increased by the m-embedding of Kv into Kv+x and there is no other edge whose length is increased, then say that h is x-growable at m. A realization is said to be x-growable, if it is x-growable at some m. If a realization is x-growable for each x ∈ X for some set X , then say that it is X-growable. Example 2.3. It is easy to see that the sequence [6, 4, 3, 0, 7, 1, 5, 2, 8] is a cyclic realization of {1, 22, 34, 4} and that it is 3-growable at 2. In fact, we can represent this Hamiltonian path of K9 writing in bold the vertices not increased by the 2-embedding of K9 into K12, and using the symbol − for each edge whose length does not change and the symbol · · · for each edge whose length increases by 3: 6− 4− 3 · · ·0− 7− 1 · · · 5 · · ·2− 8. Note that every vertex in bold is incident with exactly one edge · · · . Also, note that the edges 0 − 7, 7 − 1 and 2 − 8 do not change length, since their absolute differences are greater than ⌊ 9 2 ⌋ . Theorem 2.4 and its immediate consequence Theorem 2.6 are the core results for using growable realizations. Theorem 2.4. Suppose a multiset L has an X-growable realization. Then for each x ∈ X , the multiset L ∪ {xx} has an X-growable realization. Proof. Let g = [g1, . . . , gv] be an X-growable realization of a multiset L. Take x ∈ X and m such that g is x-growable at m. Each element y with m− x < y ≤ m is adjacent to exactly one element z such that the edge between them is lengthened by the m-embedding of Kv into Kv+x. Applying the embedding we obtain a sequence h′ = [h1, . . . , hv] in Kv+x. Each adjacent pair y, z in g as above becomes a subsequence (y, z+x) or (z+x, y) in h′. Obtain a new sequence h in Kv+x by replacing each subsequence (y, z+x) with (y, y+x, z+x) and each subsequence (z + x, y) with (z + x, y + x, y). As there is one pair for each y in the range m− x < y ≤ m, this adds the elements m+ 1, . . . ,m+ x to the sequence and hence h is a Hamiltonian path in Kv+x. 572 Ars Math. Contemp. 22 (2022) #P4.04 / 567–594 Now, h has the desired lengths because each pair of adjacent elements in g whose length was fixed by the embedding are still adjacent in h and each adjacent pair y, z whose length was not fixed is replaced by a triple whose lengths are the original length and x. There are x such pairs. We now show that h is x-growable at m. Let (a, b) be an edge of g and let (a′, b′) be the corresponding edge of h, obtained by the m-embedding of Kv into Kv+x. Clearly, we may assume a < b. Now, let (a′′, b′′) be the edge obtained from (a′, b′) applying the m-embedding of Kv+x into Kv+2x. First, suppose ℓ(a′, b′) = ℓ(a, b): we show that ℓ(a′′, b′′) = ℓ(a′, b′). We have to distinguish three cases and apply Remark 2.1: 1. If a′, b′ ≤ m then a′′ = a′ = a and b′′ = b′ = b: hence, b′′−a′′ = b−a ≤ v2 ≤ v+2x 2 . 2. If a′, b′ > m then a′ = a + x, b′ = b + x, a′′ = a′ + x and b′′ = b′ + x: hence b′′ − a′′ = b− a ≤ v2 < v+2x 2 . 3. If a′ ≤ m < b′ then a ≤ m and b > m, so a′ = a, b′ = b+ x and b− a > v2 : hence, b′′ − a′′ = b+ x− a > v+2x2 . In each case, by Remark 2.1, we get that ℓ(a′′, b′′) = ℓ(a′, b′) = ℓ(a, b). Now, suppose ℓ(a′, b′) ̸= ℓ(a, b). Then m − x < a ≤ m and b > m. Take the edge (a + x, b + x) of h: the corresponding edge (a′′, b′′) by the m-embedding of Kv+x into Kv+2x is such that a′′ = a + 2x, b′′ = b + 2x, whence b′′ − a′′ ≤ v2 . This implies that ℓ(a′′, b′′) = ℓ(a′, b′). Finally, consider the edge (a, a + x). Note that (a + x) − a = x ≤ v2 < v+x 2 , so the corresponding edge in Kv+2x has length which is increased by the m-embedding, and this is the unique edge whose length changes. We conclude that h is x-growable at m. With similar reasoning, but with many more tedious calculations, one can prove that if g is x′-growable at m′, then h is x′-growable at m′ if m′ ≤ m and x′-growable at m′ + x if m′ > m. Example 2.5. Applying the 2-embedding of K9 into K12 to the 3-growable realization of Example 2.3 we obtain the sequence 9− 7− 6 · · · 0− 10− 1 · · · 8 · · · 2− 11. Now, following the proof of Theorem 2.4, we insert the vertices 3, 4, 5, replacing the edges 6 · · · 0, 1 · · · 8 and 8 · · · 2 with 6 − 3 · · · 0, 1 · · · 4 − 8 and 8 − 5 · · · 2, respectively. In this way, the sequence 9− 7− 6− 3 · · · 0− 10− 1 · · · 4− 8− 5 · · · 2− 11 is a cyclic realization of {1, 22, 37, 4}, which is still 3-growable at 2. Theorem 2.6. Suppose a multiset L has a realization that is {x1, . . . , xk}-growable. Then the multiset L∪{xx1ℓ11 , x x2ℓ2 2 , . . . , x xkℓk k } has a {x1, . . . , xk}-growable realization for any ℓ1, ℓ2, . . . , ℓk ≥ 0. Proof. Repeatedly apply Theorem 2.4. M. Ollis et al.: Growable realizations: a powerful approach to the Buratti-Horak-Rosa . . . 573 Example 2.7. The sequence [0, 3, 6, 2, 1, 13, 10, 11, 14, 12, 9, 8, 5, 4, 7] is a cyclic realization of L = {14, 2, 38, 4}. It is 1-growable at 8 and 9; it is 2-growable at 3; it is 3-growable at 11; and it is 4-growable at 5. If we apply Theorem 2.4 four times with x = 2 and then three times with x = 3 we get the sequence [0, 3, 5, 7, 9, 11, 14, 10, 8, 6, 4, 2, 1, 30, 27, 24, 21, 18, 19, 22, 25, 28, 31, 29, 26, 23, 20, 17, 16, 13, 12, 15], which is a {1, 2, 3, 4}-growable realization of {14, 2, 38, 4} ∪ {28, 39} = {14, 29, 317, 4}. Any standard linear realization (and hence any perfect realization) is 1-growable at 0. Suppose we are investigating multisets that have underlying set U = {x1, . . . , xk}. Us- ing Theorem 2.6, a U -growable realization for a multiset L = {xa11 , . . . , x ak k } is sufficient to cover all multisets M = {xb11 , . . . , x bk k } with bi ≥ ai for each i and (b1, . . . , bk) ≡ (a1, . . . , ak) (mod (x1, . . . , xk)). This means that the task frequently breaks naturally into considering ∏k i=1 xi cases accord- ing to congruence modulo (x1, . . . , xk). We conclude this section with two lemmas that allow the expansion of the range of values for which realizations are growable. Lemma 2.8. Suppose L has an X-growable realization with 1 ∈ X and K has a Y - growable perfect linear realization. Then L ∪K has a (X ∪ Y )-growable realization. Proof. Suppose |K| = k and let g = [g1, . . . , gk+1] be a Y -growable perfect linear real- ization of K. Apply Theorem 2.4 k times with x = 1 to the X-growable realization of L to obtain an X-growable realization of L∪ {1k} with subsequence m,m+ 1, . . . ,m+ k. Replace this subsequence with g+m to obtain the desired (X ∪Y )-growable realization of L∪K. It is possible to take Y to be the empty set in Lemma 2.8 to construct an X-growable realization for L ∪K. Lemma 2.9. Suppose L has an X-growable realization with 2 ∈ X . Let y and z be even (possibly with y = z). Then L ∪ {1y+z−4, yy+1, zz+1} has an (X ∪ {y, z})-growable realization. Proof. Apply Theorem 2.4 y+z−1 times with x = 2 to the X-growable realization of L to obtain an X-growable realization of L∪{22(y+z−1)} with the following two subsequences: [m,m+ 2, . . . ,m+ 2y + 2z − 2], [m− 1,m+ 1, . . . ,m+ 2y + 2z − 3]. The sequence g = [1, y + 1, y + 2, 2, 3, y + 3, . . . , y − 1, 2y − 1, 2y + z − 1, 2y + 2z − 1] 574 Ars Math. Contemp. 22 (2022) #P4.04 / 567–594 uses the elements {1, 2, . . . , y − 1, y + 1, y + 2, . . . , 2y − 1, 2y + z − 1, 2y + 2z − 1} and has edge-lengths {1y−2, yy−1, z2}. The sequence h = [0, y, 2y, 2y + z, 2y + z + 1, 2y + 1, 2y + 2, 2y + z + 2, . . . , 2y + 2z − 2] uses the elements {0, y, 2y, 2y + 1, . . . , 2y + z − 2, 2y + z, 2y + z + 1, . . . , 2y + 2z − 2}, and has edge-lengths {1z−2, y2, zz−1}. The elements used by g and h together are exactly those used in the two subsequences from the realization of L ∪ {22(y+z−1)}. Replace the two subsequences with g + m − 1 and h + m − 1 respectively to obtain a realization of L ∪ {1y+z−4, yy+1, zz+1}. It is y-growable at m + y − 1 because each t in the range m − 1 < t < m + y − 1 is adjacent to t + y > m + y − 1 and to t ± 1 ≤ m + y − 1, and m + y − 1 is adjacent to m − 1 and m + 2y − 1. It is z-growable at m + 2y + z − 2 because each t in the range m + 2y − 2 < t < m + 2y + z − 2 is adjacent to t + z > m + 2y + z − 2 and to t±1 ≤ m+2y+z−2, and m+2y+z−2 is adjacent to m+2y−2 and m+2y+2z−2. 3 Complete solutions for U = {1,2,3} and U = {1,4,5} Given any fixed set U , we may use growable realizations to try to prove BHR(L) for all but finitely many multisets L with underlying set U . To do this, divide the problem into ∏ x∈U x cases, corresponding to the possible congruence classes of the number of occurrences of each element x (mod x). For each case, a finite number—possibly one— of growable realizations can show that all but finitely many—possibly zero—admissible L matching these congruence classes has a realization. The finitely many exceptions can then be dealt with directly. In this section we illustrate this process for U = {1, 2, 3} and U = {1, 4, 5}. When U = {1, 2, 3}, the BHR Conjecture is already known to hold [2]. However, the self-contained proof given here in Theorem 3.1 is significantly shorter, which gives an indication of the power of the method of growable realizations compared to existing tools. When U = {1, 4, 5}, from previous work we know that {1a, 4b, 5c} is realizable when a ≥ 11 or when both a ≥ 7 and b ≥ c [8]. However, the proof of Theorem 3.3 does not rely on this result. Theorem 3.1. Let L = {1a, 2b, 3c} be an admissible multiset with a, b, c ≥ 1. Then BHR(L) holds. Proof. We start with the {1, 2, 3}-growable cyclic realizations of {1, 2b, 3c} described in the first part of Table 1, which allow to cover all the 6 possibilities of the congruence class combinations of (b, c) (mod (2, 3)). Using Theorem 2.6 this proves BHR(L) for all a, b ≥ 1 and c ≥ 4. To complete the case b + c ≥ 4, we use the {1, 2}-growable realizations for (b, c) ∈ {(2, 2), (3, 1), (3, 2), (3, 3), (4, 1)} from the second part of Table 1 and the 1-growable realization of {1, 2, 33}, described in Table 2. Now, the cases when b + c < 4 can be solved using the 1-growable realizations of {1a, 2b, 3c}, described in Table 2. M. Ollis et al.: Growable realizations: a powerful approach to the Buratti-Horak-Rosa . . . 575 Table 1: {1, 2, 3}-growable cyclic realizations for {1, 2b, 3c}: they are x-growable at mx. The congruence classes of (b, c) are taken modulo (2, 3). Classes Realizations (b, c) (m1,m2,m3) Missing cases (0, 0) [2, 4, 1, 5, 3, 0, 6] (2, 3) (5, 1, 3) (0, 1) [3, 6, 0, 5, 2, 1, 7, 4] (2, 4) (2, 3, 4) c = 1 (0, 2) [6, 5, 2, 8, 1, 4, 7, 0, 3] (2, 5) (7, 5, 2) c = 2 (1, 0) [8, 5, 2, 3, 6, 0, 7, 1, 4] (1, 6) (1, 6, 3) c = 3 (1, 1) [2, 5, 1, 3, 6, 0, 4] (1, 4) (1, 3, 6) c = 1 (1, 2) [6, 1, 4, 7, 5, 0, 3, 2] (1, 5) (4, 1, 2) c = 2 (0, 1) [0, 2, 4, 1, 6, 5, 3] (4, 1) (5, 2, 3) (0, 2) [3, 1, 4, 5, 2, 0] (2, 2) (4, 1,−) (1, 0) [7, 4, 2, 0, 3, 1, 6, 5] (3, 3) (4, 5, 2) (b, c) = (1, 3) (1, 1) [4, 2, 5, 3, 1, 0] (3, 1) (1, 3,−) (1, 2) [2, 4, 6, 5, 1, 3, 0] (3, 2) (4, 1, 2) Table 2: 1-growable cyclic realizations for {1a, 2b, 3c}: they are 1-growable at m1. (a, b, c) Realizations m1 (a, b, c) Realizations m1 (1, 1, 3) [2, 5, 4, 1, 3, 0] 4 (2, 1, 2) [0, 3, 5, 4, 1, 2] 3 (2, 2, 1) [3, 1, 0, 5, 2, 4] 1 (3, 1, 1) [0, 5, 4, 1, 3, 2] 4 We now move on to U = {1, 4, 5}. Lemma 3.2. Let L = {1a, 4b, 5c} be an admissible multiset with a ≥ 2. Then BHR(L) holds. Proof. In view of Theorem 1.2(1), we may assume b, c ≥ 1. We start with the {1, 4, 5}- growable cyclic realizations of {12, 4b, 5c} described in the first part of Table 4 (note that in this case b + c ≥ 7). These realizations allow to cover all the 20 possibilities of the congruence class combinations of (b, c) (mod (4, 5)). Using Theorem 2.6, this proves BHR(L) for all a ≥ 2, b ≥ 7 and c ≥ 1. The case 2 ≤ b ≤ 6 with b+ c ≥ 8 can be solved using the {1, 5}-growable cyclic realizations of {12, 4b, 5c} provided by Table 4, with the exception of (b, c) ≡ (2, 4) (mod (4, 5)). Furthermore, the same table gives 5-growable cyclic realizations of {12, 4, 5c} for c ≥ 7 with c ̸≡ 1 (mod 5). Note that the multisets {12, 4, 55k+6} are not admissible. To complete the case b + c ≥ 7 we consider the 5-growable cyclic realization of {12, 42, 59} and the 1-growable cyclic realizations of {12, 4b, 57−b}, 2 ≤ b ≤ 6, given in Table 4, as well as the {1, 5}-growable cyclic realization of {13, 42, 59} given in Table 3. To conclude our proof, we use the 1-growable cyclic realizations of {1a, 4b, 5c} with a+ b+ c = 9, described in Table 3. Theorem 3.3. Let L = {1a, 4b, 5c} be an admissible multiset with a, b, c ≥ 0. Then BHR(L) holds. 576 Ars Math. Contemp. 22 (2022) #P4.04 / 567–594 Table 3: {1, 5}-growable cyclic realizations for {1a, 4b, 5c}, a ≥ 3: they are x-growable at mx. (a, b, c) Realizations (m1,m5) (3, 1, 6) [6, 7, 2, 1, 5, 0, 10, 4, 9, 3, 8] (9, 4) (3, 2, 9) [9, 14, 0, 10, 5, 4, 8, 13, 3, 7, 12, 2, 1, 11, 6] (3, 9) (3, 1, 5) [8, 3, 2, 7, 1, 6, 5, 0, 9, 4] (2,−) (3, 2, 4) [7, 2, 6, 1, 5, 0, 9, 8, 3, 4] (1,−) (3, 3, 3) [3, 2, 8, 4, 9, 0, 5, 1, 6, 7] (8,−) (3, 4, 2) [6, 2, 8, 7, 3, 4, 9, 0, 5, 1] (7,−) (3, 5, 1) [5, 6, 2, 8, 9, 4, 0, 1, 7, 3] (7,−) (4, 1, 4) [2, 1, 6, 7, 3, 8, 9, 4, 5, 0] (1,−) (4, 2, 3) [7, 8, 4, 9, 3, 2, 1, 6, 5, 0] (4,−) (4, 3, 2) [9, 5, 0, 6, 1, 2, 3, 4, 8, 7] (4,−) (4, 4, 1) [0, 9, 4, 3, 7, 8, 2, 6, 5, 1] (8,−) (5, 1, 3) [8, 3, 2, 7, 6, 5, 1, 0, 9, 4] (6,−) (5, 2, 2) [5, 4, 9, 0, 1, 2, 8, 3, 7, 6] (8,−) (5, 3, 1) [4, 5, 9, 8, 3, 7, 6, 2, 1, 0] (2,−) (6, 1, 2) [3, 4, 8, 9, 0, 5, 6, 7, 2, 1] (8,−) (6, 2, 1) [8, 4, 3, 2, 1, 7, 6, 5, 0, 9] (4,−) (7, 1, 1) [3, 2, 1, 0, 4, 9, 8, 7, 6, 5] (8,−) Proof. By Lemma 3.2 we are left with the case L = {1, 4b, 5c} with b, c ≥ 1. The mul- tiset L is admissible only if b + c ≥ 8. Also, the following multisets are not admissible: {1, 4, 55k+7}, {1, 42, 55k+6} and {1, 44k+1, 5}. The {4, 5}-growable cyclic realizations of {1, 4b, 5c} described in the first part of Table 5 allow to cover all the 20 possibilities of the congruence class combinations of (b, c) (mod (4, 5)). Using Theorem 2.6, this proves BHR(L) for all b ≥ 2 and c ≥ 6. To complete the case b = 1 we use the 5- growable cyclic realization of {1, 4, 511} given in Table 5. Finally, the case c ≤ 5 can be solved using the 4-growable cyclic realizations of Table 6, as well as the cyclic realization [0, 5, 9, 4, 8, 3, 7, 2, 1, 6] of {1, 43, 55}. 4 A complete solution for U ⊆ {1,2,3,4} In this section we prove BHR ( {1a, 2b, 3c, 4d} ) . In view of Theorem 1.2(2) and 1.2(3), we may assume c, d ≥ 1. Also, by Theorem 1.2(4) we have as a starting point that BHR(L) holds for a ≥ 3 and also for a = 2 when b ≥ 1. We begin by closing the case a = 2. Lemma 4.1. Let L = {12, 3c, 4d} be an admissible multiset with c, d ≥ 1. Then BHR(L) holds. Proof. First, note that L is admissible only if c + d ≥ 5. The first part of Table 7 collects {3, 4}-growable cyclic realizations for L in each of the 12 possibilities of congruence class combinations of (c, d) (mod (3, 4)). Using Theorem 2.6, this proves BHR(L) except in the following cases: d = 1, 2; d = 3 and c ̸≡ 0 (mod 3); d = 4 and c ≡ 1 (mod 3). So, we prove the validity of BHR(L) for these exceptional cases using the 3-growable M. Ollis et al.: Growable realizations: a powerful approach to the Buratti-Horak-Rosa . . . 577 Table 4: {1, 4, 5}-growable cyclic realizations for {12, 4b, 5c}: they are x-growable at mx. The congruence classes of (b, c) are taken modulo (4, 5). Classes Realizations (b, c) (m1,m4,m5) Missing cases (0, 0) [5, 9, 1, 6, 7, 2, 10, 3, 8, 4, 11, 0] (4, 5) (9, 4, 5) (0, 1) [5, 9, 1, 6, 2, 10, 11, 3, 7, 8, 4, 0] (8, 1) (9, 3, 5) b = 4 (0, 2) [5, 6, 1, 10, 9, 0, 4, 8, 12, 3, 7, 2, 11] (8, 2) (8, 3, 5) b = 4 (0, 3) [1, 11, 12, 2, 7, 3, 13, 4, 8, 9, 5, 0, 10, 6] (8, 3) (10, 5, 6) b = 4 (0, 4) [1, 6, 7, 2, 9, 5, 0, 10, 3, 8, 4] (4, 4) (9, 3, 4) (1, 0) [7, 8, 3, 11, 2, 10, 6, 1, 5, 9, 4, 0, 12] (5, 5) (10, 6, 7) b = 1 (1, 1) [10, 1, 6, 2, 11, 12, 3, 7, 8, 4, 0, 9, 5] (9, 1) (9, 3, 5) b = 1, 5 (1, 2) [5, 9, 13, 3, 7, 8, 4, 0, 10, 1, 6, 2, 12, 11] (9, 2) (9, 3, 5) b = 1, 5 (1, 3) [5, 9, 2, 7, 6, 1, 0, 4, 8, 3, 10] (5, 3) (9, 3, 5) b = 1 (1, 4) [0, 1, 5, 9, 4, 11, 3, 8, 7, 2, 10, 6] (5, 4) (10, 4, 6) b = 1 (2, 0) [4, 9, 13, 0, 10, 5, 1, 11, 6, 2, 3, 8, 12, 7] (6, 5) (1, 4, 7) b = 2 (2, 1) [7, 11, 1, 5, 9, 10, 6, 2, 12, 13, 3, 8, 4, 0] (10, 1) (11, 3, 7) b = 2, 6 (2, 2) [1, 6, 2, 9, 5, 0, 10, 3, 7, 8, 4] (6, 2) (9, 3, 5) b = 2 (2, 3) [5, 9, 1, 6, 2, 10, 3, 7, 8, 4, 11, 0] (6, 3) (9, 4, 5) b = 2 (2, 4) [5, 6, 1, 10, 9, 0, 8, 4, 12, 3, 7, 2, 11] (6, 4) (8, 4, 5) b = 2 (3, 0) [1, 6, 2, 8, 9, 3, 7, 0, 5, 4, 10] (3, 5) (7, 10, 4) (3, 1) [10, 3, 7, 8, 4, 0, 1, 6, 2, 9, 5] (7, 1) (9, 3, 5) b = 3 (3, 2) [11, 3, 7, 2, 10, 6, 1, 0, 4, 8, 9, 5] (7, 2) (10, 3, 5) b = 3 (3, 3) [11, 12, 3, 8, 4, 0, 9, 5, 1, 10, 2, 7, 6] (7, 3) (9, 3, 6) b = 3 (3, 4) [11, 12, 2, 7, 6, 1, 10, 0, 4, 8, 3, 13, 9, 5] (7, 4) (9, 3, 5) b = 3 (0, 1) [12, 11, 3, 8, 4, 0, 5, 9, 1, 10, 2, 7, 6] (4, 6) (9, 5, 6) (0, 2) [3, 13, 4, 9, 10, 5, 0, 1, 6, 11, 7, 2, 12, 8] (4, 7) (12, 7, 8) (0, 3) [12, 2, 6, 1, 11, 7, 3, 13, 8, 9, 14, 10, 0, 5, 4] (4, 8) (7, 3, 4) (1, 1) [0, 5, 9, 8, 4, 13, 12, 3, 7, 2, 11, 1, 10, 6] (5, 6) (10, 5, 6) (1, 2) [4, 9, 5, 1, 12, 7, 2, 3, 8, 13, 14, 10, 0, 11, 6] (5, 7) (1, 5, 8) (2, 0) [10, 0, 5, 4, 14, 9, 8, 13, 3, 7, 12, 2, 6, 1, 11] (2, 10) (7, 10, 4) (2, 1) [2, 7, 0, 6, 1, 8, 3, 4, 9, 10, 5] (2, 6) (1, 4, 5) (2, 2) [5, 10, 11, 6, 1, 9, 4, 3, 8, 0, 7, 2] (2, 7) (1, 4, 5) (2, 3) [5, 10, 1, 6, 11, 12, 7, 2, 3, 8, 0, 9, 4] (2, 8) (1, 4, 5) (3, 1) [2, 7, 0, 8, 3, 4, 9, 1, 5, 10, 11, 6] (3, 6) (1, 4, 6) (3, 2) [10, 2, 7, 3, 11, 12, 4, 8, 0, 9, 1, 6, 5] (3, 7) (8, 4, 5) (3, 3) [4, 9, 0, 1, 10, 5, 6, 11, 2, 12, 7, 3, 13, 8] (3, 8) (3, 6, 8) (3, 4) [0, 5, 10, 6, 1, 11, 7, 2, 12, 13, 3, 14, 4, 9, 8] (3, 9) (11, 7, 8) (1, 0) [11, 2, 7, 12, 3, 8, 13, 4, 9, 10, 6, 1, 0, 5] (1, 10) (−, 9, 4) (1, 2) [8, 3, 2, 7, 1, 6, 0, 10, 4, 9, 5] (1, 7) (−, 4, 5) (1, 3) [6, 11, 4, 9, 8, 1, 2, 7, 3, 10, 5, 0] (1, 8) (−, 5, 6) (1, 4) [5, 10, 2, 7, 8, 3, 11, 6, 1, 0, 9, 4, 12] (1, 9) (−, 4, 5) (2, 4) [6, 11, 2, 7, 12, 3, 8, 4, 5, 10, 1, 0, 9, 13] (2, 9) (−, 5, 6) (0, 3) [5, 0, 6, 1, 7, 2, 3, 9, 8, 4] (4, 3) (4,−,−) (1, 2) [4, 8, 3, 9, 5, 0, 6, 7, 1, 2] (5, 2) (4,−,−) (2, 0) [9, 4, 0, 5, 6, 1, 7, 2, 3, 8] (2, 5) (7,−,−) (2, 1) [9, 3, 4, 0, 6, 5, 1, 7, 2, 8] (6, 1) (6,−,−) (3, 4) [9, 4, 5, 0, 1, 6, 2, 8, 3, 7] (3, 4) (8,−,−) 578 Ars Math. Contemp. 22 (2022) #P4.04 / 567–594 Table 5: {4, 5}-growable cyclic realizations for {1, 4b, 5c}: they are x-growable at mx. The congruence classes of (b, c) are taken modulo (4, 5). Classes Realizations (b, c) (m4,m5) Missing cases (0, 0) [4, 9, 5, 0, 1, 6, 10, 3, 7, 2, 8] (4, 5) (3, 4) (0, 1) [9, 2, 7, 3, 10, 5, 1, 0, 8, 4, 11, 6] (4, 6) (4, 6) c = 1 (0, 2) [4, 9, 1, 5, 0, 8, 12, 11, 6, 10, 2, 7, 3] (4, 7) (3, 4) c = 2 (0, 3) [6, 10, 5, 0, 9, 13, 4, 8, 3, 12, 7, 2, 1, 11] (4, 8) (5, 6) c = 3 (0, 4) [3, 14, 4, 9, 5, 0, 10, 6, 1, 11, 12, 7, 2, 13, 8] (4, 9) (7, 8) c = 4 (1, 0) [6, 11, 3, 8, 0, 12, 7, 2, 10, 5, 1, 9, 4] (1, 10) (4, 6) c = 5 (1, 1) [4, 9, 5, 0, 8, 3, 12, 7, 11, 10, 1, 6, 2] (5, 6) (3, 4) b = 1 or c = 1 (1, 2) [12, 3, 7, 11, 2, 1, 6, 10, 0, 5, 9, 4, 13, 8] (5, 7) (6, 8) b = 1 or c = 2 (1, 3) [4, 9, 10, 3, 8, 2, 7, 1, 6, 0, 5] (1, 8) (3, 4) c = 3 (1, 4) [10, 3, 8, 1, 2, 7, 0, 5, 9, 4, 11, 6] (1, 9) (5, 6) c = 4 (2, 0) [7, 2, 11, 6, 1, 10, 5, 0, 9, 13, 12, 3, 8, 4] (2, 10) (3, 4) c = 5 (2, 1) [7, 11, 2, 12, 3, 8, 4, 13, 9, 5, 0, 1, 10, 6] (6, 6) (6, 7) b = 2 or c = 1 (2, 2) [1, 6, 0, 5, 10, 3, 7, 2, 8, 9, 4] (2, 7) (3, 4) c = 2 (2, 3) [9, 1, 2, 7, 0, 5, 10, 3, 8, 4, 11, 6] (2, 8) (5, 6) c = 3 (2, 4) [9, 4, 12, 3, 8, 0, 1, 6, 11, 7, 2, 10, 5] (2, 9) (4, 5) c = 4 (3, 0) [4, 9, 14, 3, 8, 13, 2, 7, 12, 1, 6, 11, 10, 0, 5] (3, 10) (3, 4) c = 5 (3, 1) [4, 9, 3, 8, 2, 6, 10, 0, 5, 1, 7] (3, 6) (3, 4) c = 1 (3, 2) [1, 6, 10, 3, 8, 4, 11, 0, 7, 2, 9, 5] (3, 7) (4, 5) c = 2 (3, 3) [10, 5, 0, 4, 9, 1, 6, 11, 2, 3, 8, 12, 7] (3, 8) (6, 7) c = 3 (3, 4) [11, 6, 1, 2, 7, 12, 3, 8, 4, 13, 9, 0, 10, 5] (3, 9) (4, 5) c = 4 (1, 1) [1, 6, 10, 11, 2, 7, 12, 3, 8, 13, 4, 9, 0, 5] (1, 11) (9, 4) cyclic realizations for the cases (c, d) ∈ {(4, 1), (5, 1), (6, 1), (3, 2), (4, 2), (5, 2), (2, 3), (4, 3), (4, 4)} given in Table 7. The last case left open is L = {12, 3, 44}, for which we take the following cyclic realization: [0, 4, 5, 1, 2, 6, 3, 7]. Lemma 4.2. Let L = {1, 2b, 3c, 4d} be an admissible multiset, where b ≥ 0 is even and c, d ≥ 1. Then BHR(L) holds. Proof. Suppose first d ≥ 5. We start with the {2, 3, 4}-growable cyclic realizations of {1, 3c, 4d} described in the first part of Table 8 (note that in this case c + d ≥ 6). These realizations allow to cover all the 12 possibilities of the congruence class combinations of (c, d) (mod (3, 4)). Using Theorem 2.6 this proves BHR(L) except when c = 1 and d ̸≡ 2 (mod 4). So, suppose c = 1. Table 8 also gives {2, 3, 4}-growable cyclic realizations for d = 7, 8, proving the validity of BHR(L) for b ≥ 0 even, c = 1 and d ≡ 0, 3 (mod 4). Hence, we may assume d ≡ 1 (mod 4). Note that the multiset L = {1, 3, 44k+5} does not satisfy the necessary condition, as 4k + 5 > v − 4 = 4k + 4. On the other hand, a {2, 3, 4}-growable cyclic realization of {1, 22, 3, 45} is given in Table 9. Next, we consider the cases 1 ≤ d ≤ 4. Table 8 provides {2, 3}-growable cyclic realizations for {1, 3c, 4d}, in the following cases: d = 1 and 5 ≤ c ≤ 7; d = 2 and 4 ≤ c ≤ 6; d = 3, 4 and 3 ≤ c ≤ 5. It also provides a 2-growable cyclic realization for the multiset {1, 32, 44}. This completes the analysis for the admissible multisets with M. Ollis et al.: Growable realizations: a powerful approach to the Buratti-Horak-Rosa . . . 579 Table 6: 4-growable cyclic realizations for {1, 4b, 5c}: they are 4-growable at m4. (b, c) Realizations m4 (4, 4) [7, 3, 8, 4, 9, 0, 5, 1, 6, 2] 4 (5, 3) [9, 4, 8, 3, 7, 2, 6, 0, 1, 5] 4 (5, 4) [9, 10, 3, 7, 1, 5, 0, 6, 2, 8, 4] 3 (5, 5) [4, 8, 9, 1, 6, 11, 3, 7, 2, 10, 5, 0] 3 (6, 2) [7, 3, 8, 4, 0, 9, 5, 1, 6, 2] 5 (6, 3) [2, 6, 1, 7, 0, 4, 8, 3, 10, 9, 5] 3 (6, 4) [5, 6, 1, 9, 4, 0, 8, 3, 11, 7, 2, 10] 3 (6, 5) [5, 10, 6, 1, 9, 0, 4, 8, 12, 11, 3, 7, 2] 4 (7, 1) [7, 3, 9, 4, 8, 2, 6, 0, 1, 5] 4 (7, 2) [8, 4, 0, 10, 3, 7, 1, 6, 2, 9, 5] 4 (7, 3) [4, 8, 0, 1, 5, 9, 2, 6, 11, 7, 3, 10] 7 (7, 4) [8, 3, 12, 0, 4, 9, 5, 1, 10, 2, 6, 11, 7] 6 (7, 5) [5, 10, 0, 9, 13, 4, 8, 7, 3, 12, 2, 6, 1, 11] 4 (8, 1) [7, 3, 10, 0, 4, 8, 1, 6, 2, 9, 5] 3 (8, 2) [4, 8, 0, 5, 9, 1, 6, 2, 10, 11, 3, 7] 3 (8, 3) [6, 10, 11, 2, 7, 3, 12, 8, 4, 0, 5, 9, 1] 5 (9, 2) [4, 9, 8, 12, 3, 7, 11, 2, 6, 10, 1, 5, 0] 3 (10, 1) [6, 10, 1, 5, 9, 0, 4, 8, 12, 11, 2, 7, 3] 3 b = 0. Now, Table 9 gives 2-growable cyclic realizations for the multiset {1, 22, 3c, 4d} when (c, d) ∈ {(1, 3), (1, 4), (2, 2), (2, 3), (3, 1), (3, 2), (4, 1)}, that is with 4 ≤ c+ d ≤ 5. This completes the analysis for the admissible multisets with b = 2. Finally, Table 9 also gives 2-growable cyclic realizations for the multiset {1, 24, 3c, 4d} for each (c, d) in the set {(1, 1), (1, 2), (2, 1)}, concluding our proof. Lemma 4.3. Let L = {1, 2b, 3c, 4d} be an admissible multiset, where b ≥ 1 is odd and c, d ≥ 1. Then BHR(L) holds. Proof. The first part of Table 10 gives {2, 3, 4}-growable cyclic realizations for {1, 2, 3c, 4d} for each of the 12 possibilities of congruence class combinations of (c, d) (mod (3, 4)). Note that c + d ≥ 5. Using Theorem 2.6, this proves BHR(L) except for the following cases: d = 1, 2; d = 3 and c ̸≡ 0 (mod 3); d = 4 and c ≡ 1 (mod 3). So, we prove the validity of BHR(L) for these exceptional cases using {2, 3}-growable cyclic realizations for {1, 2, 3c, 4d} and a 2-growable cyclic realization for {1, 2, 3, 44}, which can be found in Table 10. To conclude the proof we have to consider the cases when c + d ≤ 4. Table 10 also provides 2-growable cyclic realizations for {1, 23, 3c, 4d} when (c, d) is in the set {(1, 2), (1, 3), (2, 1), (2, 2), (3, 1)}, and for {1, 25, 3, 4}. Lemma 4.4. Let L = {2b, 3c, 4d} be an admissible multiset, where b ≥ 1 is odd and c, d ≥ 1. Then BHR(L) holds. 580 Ars Math. Contemp. 22 (2022) #P4.04 / 567–594 Table 7: {3, 4}-growable cyclic realizations for {12, 3c, 4d}: they are x-growable at mx. The congruence classes of (c, d) are taken modulo (3, 4). Classes Realizations (c, d) (m3,m4) Missing cases (0, 0) [3, 7, 1, 4, 0, 9, 2, 6, 5, 8] (3, 4) (2, 3) (0, 1) [3, 7, 10, 2, 6, 5, 1, 9, 8, 4, 0] (3, 5) (2, 5) d = 1 (0, 2) [6, 9, 10, 2, 1, 5, 8, 0, 4, 7, 3, 11] (3, 6) (5, 6) d = 2 (0, 3) [1, 5, 6, 2, 8, 0, 3, 7, 4] (3, 3) (3, 4) (1, 0) [3, 7, 11, 10, 6, 2, 1, 5, 9, 0, 4, 8] (1, 8) (2, 7) d = 4 (1, 1) [3, 6, 2, 7, 8, 4, 0, 1, 5] (1, 5) (2, 4) d = 1 (1, 2) [5, 9, 8, 4, 1, 7, 3, 2, 6, 0] (1, 6) (4, 5) d = 2 (1, 3) [4, 8, 7, 3, 0, 1, 5, 9, 2, 6, 10] (1, 7) (3, 6) d = 3 (2, 0) [4, 8, 0, 3, 7, 6, 1, 5, 2] (2, 4) (3, 4) (2, 1) [5, 9, 3, 6, 2, 8, 7, 4, 0, 1] (2, 5) (4, 5) d = 1 (2, 2) [4, 0, 10, 6, 9, 2, 5, 1, 8, 7, 3] (2, 6) (2, 3) d = 2 (2, 3) [8, 0, 4, 3, 11, 7, 6, 9, 1, 5, 2, 10] (2, 7) (8, 3) d = 3 (0, 1) [5, 8, 9, 2, 6, 3, 0, 1, 4, 7] (6, 1) (4, 5) (0, 2) [6, 3, 2, 5, 1, 4, 0, 7] (3, 2) (3,−) (1, 0) [8, 5, 6, 2, 10, 9, 1, 4, 0, 7, 3] (4, 4) (2, 3) (c, d) = (1, 4) (1, 1) [2, 5, 6, 3, 7, 4, 1, 0] (4, 1) (4,−) (1, 2) [3, 7, 8, 5, 2, 1, 4, 0, 6] (4, 2) (2, 3) (1, 3) [2, 6, 3, 0, 9, 5, 1, 8, 7, 4] (4, 3) (3, 5) (2, 1) [0, 4, 1, 7, 8, 2, 5, 6, 3] (5, 1) (2, 3) (2, 2) [5, 8, 1, 4, 7, 3, 2, 6, 9, 0] (5, 2) (4, 5) (2, 3) [7, 6, 2, 3, 0, 4, 1, 5] (2, 3) (2,−) M. Ollis et al.: Growable realizations: a powerful approach to the Buratti-Horak-Rosa . . . 581 Table 8: {2, 3, 4}-growable cyclic realizations for {1, 3c, 4d}: they are x-growable at mx. The congruence classes of (c, d) are taken modulo (3, 4). Classes Realizations (c, d) (m2,m3,m4) Missing cases (0, 0) [3, 6, 1, 4, 0, 5, 2, 7, 8] (3, 4) (6, 2, 3) (0, 1) [3, 6, 2, 8, 4, 1, 7, 0, 9, 5] (3, 5) (2, 4, 5) d = 1 (0, 2) [4, 5, 1, 8, 0, 3, 7, 10, 6, 2, 9] (3, 6) (7, 3, 4) d = 2 (0, 3) [3, 7, 6, 2, 10, 1, 5, 9, 0, 4, 8, 11] (3, 7) (9, 2, 6) d = 3 (1, 0) [4, 7, 0, 6, 3, 9, 8, 2, 5, 1] (4, 4) (7, 3, 4) c = 1 (1, 1) [6, 9, 2, 10, 3, 7, 4, 0, 1, 8, 5] (4, 5) (4, 5, 6) c = 1 or d = 1 (1, 2) [2, 6, 1, 5, 0, 3, 7, 8, 4] (1, 6) (1, 3, 4) d = 2 (1, 3) [4, 7, 1, 5, 0, 8, 2, 6, 3] (4, 3) (2, 3, 4) c = 1 (2, 0) [7, 11, 3, 4, 0, 8, 5, 1, 9, 6, 2, 10] (2, 8) (6, 8, 3) d = 4 (2, 1) [3, 7, 2, 6, 0, 1, 5, 8, 4] (2, 5) (2, 3, 4) d = 1 (2, 2) [4, 5, 1, 8, 2, 6, 0, 7, 3, 9] (2, 6) (7, 3, 4) d = 2 (2, 3) [4, 8, 1, 0, 7, 10, 3, 6, 2, 9, 5] (2, 7) (3, 4, 5) d = 3 (1, 0) [4, 5, 1, 8, 0, 7, 3, 10, 6, 2, 9] (1, 8) (7, 3, 4) (1, 3) [3, 7, 1, 4, 8, 2, 6, 0, 9, 5] (1, 7) (2, 4, 5) (0, 1) [7, 1, 4, 5, 8, 2, 6, 0, 3] (6, 1) (6, 2,−) (0, 2) [5, 6, 3, 9, 2, 8, 1, 4, 7, 0] (6, 2) (4, 5,−) (0, 3) [0, 3, 7, 6, 2, 5, 1, 4] (3, 3) (1, 2,−) (1, 1) [5, 2, 9, 8, 1, 4, 7, 0, 6, 3] (7, 1) (7, 4, 9) (1, 2) [0, 5, 2, 6, 1, 4, 3, 7] (4, 2) (2, 3,−) (2, 0) [10, 9, 2, 6, 3, 0, 7, 4, 1, 8, 5] (5, 4) (8, 4, 5) (c, d) = (2, 4) (2, 1) [2, 5, 0, 1, 6, 3, 7, 4] (5, 1) (3, 4,−) (2, 2) [2, 5, 8, 7, 3, 0, 6, 1, 4] (5, 2) (1, 3,−) (2, 3) [3, 0, 6, 7, 1, 4, 8, 5, 2, 9] (5, 3) (5, 2, 7) (2, 0) [2, 6, 7, 3, 0, 4, 1, 5] (2, 4) (1,−,−) Table 9: {2, 3, 4}-growable cyclic realizations for {1, 2b, 3c, 4d}, with b ≥ 2 even: they are x-growable at mx. The congruence classes of (c, d) are taken modulo (3, 4). Classes Realizations (b, c, d) (m2,m3,m4) (1, 1) [7, 8, 2, 6, 0, 4, 1, 9, 5, 3] (2, 1, 5) (6, 2, 3) (0, 1) [6, 4, 1, 2, 5, 7, 3, 0] (2, 3, 1) (5, 2,−) (0, 2) [4, 7, 5, 1, 8, 0, 3, 6, 2] (2, 3, 2) (1, 3, 4) (1, 0) [1, 5, 8, 6, 2, 7, 0, 4, 3] (2, 1, 4) (6, 2, 3) (1, 1) [3, 6, 0, 4, 1, 8, 7, 5, 2] (2, 4, 1) (6, 2, 3) (1, 3) [1, 0, 4, 6, 2, 5, 3, 7] (2, 1, 3) (3, 4,−) (2, 2) [7, 6, 2, 4, 1, 5, 3, 0] (2, 2, 2) (1, 3,−) (2, 3) [3, 5, 7, 8, 2, 6, 1, 4, 0] (2, 2, 3) (6, 2, 3) (1, 1) [0, 3, 1, 7, 5, 4, 2, 6] (4, 1, 1) (1, 2,−) (1, 2) [3, 5, 7, 8, 6, 2, 0, 4, 1] (4, 1, 2) (6, 2, 3) (2, 1) [1, 3, 5, 8, 2, 4, 0, 7, 6] (4, 2, 1) (6, 2,−) 582 Ars Math. Contemp. 22 (2022) #P4.04 / 567–594 Table 10: {2, 3, 4}-growable cyclic realizations for {1, 2b, 3c, 4d}, with b ≥ 1 odd: they are x-growable at mx. The congruence classes of (c, d) are taken modulo (3, 4). Classes Realizations (b, c, d) (m2,m3,m4) Missing cases (0, 0) [9, 2, 6, 0, 4, 1, 7, 8, 5, 3] (1, 3, 4) (6, 2, 3) (0, 1) [5, 8, 1, 9, 10, 2, 6, 4, 0, 7, 3] (1, 3, 5) (8, 4, 5) d = 1 (0, 2) [3, 5, 9, 1, 4, 0, 8, 7, 10, 6, 2, 11] (1, 3, 6) (6, 2, 3) d = 2 (0, 3) [4, 7, 3, 0, 1, 5, 8, 6, 2] (1, 3, 3) (1, 3, 4) (1, 0) [10, 6, 2, 11, 3, 7, 9, 1, 5, 4, 0, 8] (1, 1, 8) (7, 8, 4) d = 4 (1, 1) [2, 6, 7, 3, 0, 5, 1, 8, 4] (1, 1, 5) (1, 3, 4) d = 1 (1, 2) [3, 4, 0, 6, 2, 8, 1, 5, 9, 7] (1, 1, 6) (6, 2, 3) d = 2 (1, 3) [9, 2, 6, 5, 1, 10, 3, 7, 0, 8, 4] (1, 1, 7) (8, 3, 5) d = 3 (2, 0) [2, 6, 1, 4, 0, 8, 5, 7, 3] (1, 2, 4) (1, 2, 3) (2, 1) [8, 1, 5, 4, 0, 6, 2, 9, 7, 3] (1, 2, 5) (7, 2, 4) d = 1 (2, 2) [3, 7, 0, 4, 6, 10, 2, 5, 1, 8, 9] (1, 2, 6) (7, 2, 4) d = 2 (2, 3) [7, 11, 3, 4, 0, 9, 1, 5, 8, 10, 6, 2] (1, 2, 7) (6, 8, 3) d = 3 (0, 1) [4, 7, 9, 2, 6, 3, 0, 1, 8, 5] (1, 6, 1) (3, 4, 5) (0, 2) [2, 5, 1, 0, 6, 3, 7, 4] (1, 3, 2) (3, 4,−) (1, 0) [3, 7, 10, 8, 0, 4, 5, 1, 9, 6, 2] (1, 4, 4) (7, 2, 4) (c, d) = (1, 4) (1, 1) [0, 6, 3, 7, 4, 1, 2, 5] (1, 4, 1) (2, 4,−) (1, 2) [3, 7, 6, 0, 4, 1, 8, 5, 2] (1, 4, 2) (1, 2, 3) (1, 3) [3, 6, 9, 7, 0, 1, 5, 2, 8, 4] (1, 4, 3) (2, 3, 4) (2, 1) [3, 6, 7, 1, 4, 0, 2, 5, 8] (1, 5, 1) (6, 2, 3) (2, 2) [7, 1, 4, 0, 2, 5, 8, 9, 6, 3] (1, 5, 2) (6, 2, 3) (2, 3) [4, 0, 3, 1, 5, 2, 6, 7] (1, 2, 3) (1, 2,−) (1, 0) [3, 7, 6, 2, 0, 4, 1, 5] (1, 1, 4) (1,−,−) (0, 1) [3, 6, 8, 7, 5, 2, 0, 4, 1] (3, 3, 1) (6, 2, 3) (1, 0) [4, 6, 0, 8, 7, 3, 9, 1, 5, 2] (3, 1, 4) (7, 3, 4) (1, 1) [4, 2, 0, 8, 6, 3, 1, 5, 7] (5, 1, 1) (6, 3,−) (1, 2) [4, 6, 2, 5, 3, 7, 1, 0] (3, 1, 2) (3, 4,−) (1, 3) [2, 6, 7, 0, 3, 5, 1, 8, 4] (3, 1, 3) (1, 3, 4) (2, 1) [5, 2, 0, 1, 7, 3, 6, 4] (3, 2, 1) (3, 4,−) (2, 2) [4, 6, 0, 2, 5, 1, 8, 7, 3] (3, 2, 2) (2, 3, 4) M. Ollis et al.: Growable realizations: a powerful approach to the Buratti-Horak-Rosa . . . 583 Proof. The first part of Table 11 gives {2, 3, 4}-growable cyclic realizations for {2, 3c, 4d} in each of the 12 possibilities of congruence class combinations of (c, d) (mod (3, 4)). Note that c + d ≥ 6. Using Theorem 2.6, this proves BHR(L) except for the following cases: d = 1, 2, 3; d = 4, 5 and c ≡ 2 (mod 3); c = 1 and d ≡ 0, 1 (mod 4). Next, we consider the case c ≥ 2 and 1 ≤ d ≤ 5, using {2, 3}-growable cyclic realizations for {2, 3c, 4d}. For the exceptional case {2, 32, 44} we use a 2-growable cyclic realization. Now we complete the case c ≥ 2: Table 11 also provides 2-growable cyclic realization for the multisets {23, 3c, 4d} when (c, d) ∈ {(2, 2), (2, 3), (3, 1), (3, 2), (4, 1)}, and for the multiset {25, 32, 4}. Finally, we assume c = 1. Note that the multiset {2, 3, 44k+5} does not satisfy the necessary condition. For the multisets {2, 3, 44k+8} we use the 4-growable realization [2, 6, 10, 3, 7, 4, 0, 9, 5, 1, 8] of {2, 3, 48}. Table 11 gives {2, 4}-growable cyclic realiza- tions for the multisets {23, 3, 44} and {23, 3, 45}; it also gives 2-growable cyclic realiza- tions for the multisets {23, 3, 43}, {25, 3, 4} and {25, 3, 42}. Theorem 4.5. Let L = {1a, 2b, 3c, 4d} be an admissible multiset with a, b, c, d ≥ 0. Then BHR(L) holds. Proof. By Theorem 1.2(2)–(4) we may assume 0 ≤ a ≤ 2 and c, d ≥ 1. If a = 2 the result follows from Theorem 1.2(4) and Lemma 4.1. Suppose now a = 1. If b is even we apply Lemma 4.2, otherwise we apply Lemma 4.3. Finally, assume a = 0. By [5] we may assume b ≥ 1. If b is odd we apply Lemma 4.4 and so, we may also assume b ≥ 2 is even. We start with the {2, 3, 4}-growable cyclic realizations of L = {22, 3c, 4d} for each of the 12 possibilities of congruence class combinations of (c, d) (mod (3, 4)) described in the first part of Table 12. Note that c + d ≥ 5. Using Theorem 2.6, this proves BHR(L) except for the following cases: d = 1, 2; d = 3 and c ̸≡ 0 (mod 3); d = 4 and c ≡ 1 (mod 3). For these exceptions, Table 12 also gives {2, 3}-growable cyclic real- izations when (c, d) ∈ {(4, 1), (5, 1), (6, 1), (3, 2), (4, 2), (5, 2), (2, 3), (4, 3), (4, 4)} and a 2-growable cyclic realization for {22, 3, 44}. We are left to the cases c + d ≤ 4. This table also provides a 2-growable cyclic realization for the multiset {24, 3c, 4d} when (c, d) ∈ {(1, 2), (1, 3), (2, 1), (2, 2), (3, 1)}, and for the multiset {26, 3, 4}. The success in proving these small cases leads us to make the following conjecture, which says that the method of the previous section and this one is always successful. Conjecture 4.6. For any fixed set U , there is a finite set of growable realizations with un- derlying set U that implies the existence of realizations for all but finitely many admissible multisets L that have underlying set U . If Conjecture 4.6 is true, then the BHR Conjecture for any given underlying set can be proved with a finite set of realizations. 5 A partial solution for U = {1, x,2x} In previous sections we have seen how it is possible to completely prove the BHR Con- jecture for a fixed U by the construction of one or more base case realizations for each of∏ x∈U x cases. In this section we develop ways to produce general results with fewer base cases. 584 Ars Math. Contemp. 22 (2022) #P4.04 / 567–594 Table 11: {2, 3, 4}-growable cyclic realizations for {2b, 3c, 4d}, with b ≥ 1 odd: they are x-growable at mx. The congruence classes of (c, d) are taken modulo (3, 4). Classes Realizations (b, c, d) (m2,m3,m4) Missing cases (0, 0) [2, 6, 8, 5, 0, 4, 1, 7, 3] (1, 3, 4) (1, 2, 3) (0, 1) [2, 5, 1, 8, 4, 0, 6, 9, 7, 3] (1, 3, 5) (1, 2, 4) d = 1 (0, 2) [3, 6, 10, 7, 0, 9, 2, 5, 1, 8, 4] (1, 3, 6) (2, 3, 4) d = 2 (0, 3) [6, 10, 2, 5, 9, 0, 8, 4, 1, 3, 11, 7] (1, 3, 7) (5, 6, 7) d = 3 (1, 0) [3, 6, 9, 5, 2, 8, 0, 4, 1, 7] (1, 4, 4) (6, 2, 3) c = 1 (1, 1) [2, 5, 9, 6, 3, 10, 8, 1, 4, 0, 7] (1, 4, 5) (6, 7, 3) c = 1 or d = 1 (1, 2) [3, 7, 2, 6, 0, 5, 1, 8, 4] (1, 1, 6) (2, 3, 4) d = 2 (1, 3) [2, 6, 0, 4, 8, 1, 5, 9, 7, 3] (1, 1, 7) (1, 2, 5) d = 3 (2, 0) [10, 2, 6, 3, 11, 7, 5, 1, 9, 0, 8, 4] (1, 2, 8) (8, 3, 5) d = 4 (2, 1) [7, 11, 8, 12, 3, 5, 9, 0, 4, 1, 10, 6, 2] (1, 2, 9) (6, 7, 3) d = 1, 5 (2, 2) [2, 6, 0, 3, 7, 9, 5, 1, 8, 4] (1, 2, 6) (1, 3, 5) d = 2 (2, 3) [9, 2, 5, 1, 8, 0, 7, 3, 10, 6, 4] (1, 2, 7) (7, 3, 4) d = 3 (0, 1) [3, 6, 0, 4, 1, 7, 5, 2, 8] (1, 6, 1) (6, 2, 3) (0, 2) [4, 7, 1, 5, 2, 9, 6, 3, 0, 8] (1, 6, 2) (7, 3, 4) (0, 3) [0, 4, 1, 7, 3, 6, 2, 5] (1, 3, 3) (2, 3,−) (1, 1) [3, 6, 9, 2, 5, 1, 8, 0, 7, 4] (1, 7, 1) (7, 3, 4) (1, 2) [7, 1, 4, 0, 5, 2, 6, 3] (1, 4, 2) (2, 3,−) (1, 3) [3, 6, 0, 4, 2, 7, 1, 5, 8] (1, 4, 3) (6, 2, 3) (2, 0) [6, 10, 2, 9, 1, 4, 8, 0, 3, 7, 5] (1, 5, 4) (4, 5, 6) (c, d) = (2, 4) (2, 1) [2, 6, 1, 3, 7, 4, 0, 5, 8] (1, 2, 5) (1, 4,−) [1, 4, 7, 3, 6, 0, 5, 2] (1, 5, 1) (2, 4,−) (2, 2) [3, 6, 0, 4, 1, 8, 5, 2, 7] (1, 5, 2) (6, 2, 3) (2, 3) [9, 6, 2, 8, 1, 5, 3, 0, 7, 4] (1, 5, 3) (7, 3, 4) (2, 0) [7, 3, 6, 2, 4, 0, 5, 1] (1, 2, 4) (3,−,−) (0, 1) [1, 6, 0, 2, 5, 3, 7, 4] (3, 3, 1) (3, 4,−) (0, 2) [7, 1, 4, 0, 2, 6, 8, 5, 3] (3, 3, 2) (6, 2, 3) (1, 1) [3, 6, 8, 2, 5, 7, 0, 4, 1] (3, 4, 1) (6, 2, 3) (2, 1) [5, 7, 3, 1, 8, 2, 0, 6, 4] (5, 2, 1) (4, 5,−) (2, 2) [2, 4, 1, 5, 7, 3, 0, 6] (3, 2, 2) (1, 2,−) (2, 3) [3, 5, 7, 0, 4, 1, 6, 2, 8] (3, 2, 3) (6, 2, 3) (1, 0) [3, 7, 5, 0, 4, 1, 8, 6, 2] (3, 1, 4) (1, 2, 3) (1, 1) [5, 7, 1, 3, 0, 6, 2, 4] (5, 1, 1) (1, 2,−) (1, 2) [4, 6, 8, 1, 5, 2, 0, 7, 3] (5, 1, 2) (2, 3, 4) (1, 3) [1, 7, 3, 5, 2, 6, 4, 0] (3, 1, 3) (3, 4,−) (1, 4) [5, 9, 1, 7, 3, 0, 8, 2, 6, 4] (3, 1, 5) (3, 4, 5) Our main goal is Theorem 5.10, which says that BHR(L) holds for L = {1a, xb, (2x)c} when a ≥ x−2, c is even, and b ≥ 5x−2+c/2. When x is even, this covers many instances not covered by Theorem 1.2(8). When x is odd, the instances covered are all new. Lemma 5.1. Suppose L has an X-growable realization and take x ∈ X . Take i with 1 ≤ i ≤ x. Then L ∪ {x3x−2i, (2x)2i} has an X-growable realization. M. Ollis et al.: Growable realizations: a powerful approach to the Buratti-Horak-Rosa . . . 585 Table 12: {2, 3, 4}-growable cyclic realizations for {2b, 3c, 4d}, with b ≥ 2 even: they are x-growable at mx. The congruence classes of (c, d) are taken modulo (3, 4). Classes Realizations (b, c, d) (m2,m3,m4) Missing cases (0, 0) [2, 5, 9, 6, 8, 0, 4, 1, 7, 3] (2, 3, 4) (1, 2, 3) (0, 1) [8, 0, 7, 3, 10, 1, 5, 2, 9, 6, 4] (2, 3, 5) (7, 3, 4) d = 1 (0, 2) [8, 0, 4, 2, 10, 1, 5, 7, 11, 3, 6, 9] (2, 3, 6) (7, 8, 3) d = 2 (0, 3) [3, 6, 1, 5, 8, 2, 4, 0, 7] (2, 3, 3) (6, 2, 3) (1, 0) [4, 8, 0, 2, 6, 10, 1, 9, 5, 3, 11, 7] (2, 1, 8) (3, 6, 7) d = 4 (1, 1) [4, 8, 5, 1, 6, 2, 0, 7, 3] (2, 1, 5) (2, 3, 4) d = 1 (1, 2) [4, 8, 0, 2, 6, 9, 5, 1, 7, 3] (2, 1, 6) (2, 3, 5) d = 2 (1, 3) [4, 8, 1, 5, 7, 0, 9, 2, 6, 3, 10] (2, 1, 7) (8, 3, 5) d = 3 (2, 0) [2, 6, 0, 3, 7, 5, 1, 8, 4] (2, 2, 4) (1, 3, 4) (2, 1) [4, 7, 1, 5, 3, 9, 6, 2, 8, 0] (2, 2, 5) (7, 3, 4) d = 1 (2, 2) [5, 8, 10, 3, 7, 0, 4, 6, 2, 9, 1] (2, 2, 6) (8, 4, 5) d = 2 (2, 3) [10, 2, 6, 8, 0, 9, 5, 1, 11, 3, 7, 4] (2, 2, 7) (9, 3, 6) d = 3 (0, 1) [4, 7, 9, 6, 3, 0, 8, 1, 5, 2] (2, 6, 1) (7, 3, 4) (0, 2) [2, 5, 7, 3, 6, 0, 4, 1] (2, 3, 2) (2, 3,−) (1, 0) [0, 3, 7, 10, 8, 1, 5, 2, 9, 6, 4] (2, 4, 4) (7, 3, 4) (c, d) = (1, 4) (1, 1) [0, 3, 1, 6, 2, 5, 7, 4] (2, 4, 1) (5, 2,−) (1, 2) [3, 6, 1, 4, 0, 7, 5, 2, 8] (2, 4, 2) (6, 2, 3) (1, 3) [8, 1, 5, 2, 0, 7, 3, 9, 6, 4] (2, 4, 3) (7, 3, 4) (2, 1) [7, 5, 2, 8, 1, 4, 0, 6, 3] (2, 5, 1) (6, 2, 3) (2, 2) [9, 1, 5, 2, 8, 6, 3, 0, 7, 4] (2, 5, 2) (7, 3, 4) (2, 3) [7, 3, 6, 4, 0, 2, 5, 1] (2, 2, 3) (3, 4,−) (1, 0) [0, 4, 2, 6, 1, 5, 3, 7] (2, 1, 4) (3,−,−) (0, 1) [2, 5, 7, 0, 4, 1, 8, 6, 3] (4, 3, 1) (6, 2, 3) (1, 1) [7, 5, 2, 0, 4, 6, 8, 1, 3] (6, 1, 1) (6, 2,−) (1, 2) [7, 1, 3, 5, 2, 6, 4, 0] (4, 1, 2) (3, 4,−) (1, 3) [3, 7, 0, 4, 6, 8, 1, 5, 2] (4, 1, 3) (1, 2, 4) (2, 1) [1, 3, 0, 6, 2, 4, 7, 5] (4, 2, 1) (5, 2,−) (2, 2) [8, 6, 2, 0, 4, 1, 7, 5, 3] (4, 2, 2) (6, 2, 3) Proof. Apply Theorem 2.4 three times to the x-growable realization of L to obtain an X- growable realization of L ∪ {x3x} with each of the subsequences [m,m+ x,m+ 2x,m+ 3x], [m− 1,m− 1 + x,m− 1 + 2x,m− 1 + 3x], . . . , [m− x+ 1,m+ 1,m+ 1 + x,m+ 1 + 2x] appearing, possibly reversed. Each subsequence has differences {x3}. Take i of the sub- sequences and in each switch the middle two elements (so, for example, the first would become [m,m + 2x,m + x,m + 3x]). Each time we perform this operation we obtain a subsequence with differences {x, (2x)2} instead of {x3}. After performing it i times the new differences are {x3x−2i, (2x)2i}. These operations do not interfere with growability: if the original realization is y- growable at m′, then the new realization is y-growable at m′ if m′ ≤ m and at m′ + 3x 586 Ars Math. Contemp. 22 (2022) #P4.04 / 567–594 otherwise. Let L = {1a, xb, (2x)c}. When x = 1 or 2 BHR(L) follows from Theorem 1.2(1) or 1.2(2) respectively, so x = 3 is the first open case. We treat the x = 3 case first both as an illustration of the general method and because some of the later constructions require x > 3. Lemma 5.2. Let L = {1a, 3b, 6c}. If c is even and b ≥ 13 + c/2, then BHR(L) holds; if c is odd and b ≥ 18 + (c− 1)/2, then BHR(L) holds. Proof. By Theorem 1.2(1) we may assume that a, b, c ≥ 1. The multiset L is not admissible when a = 1 and b+ c ≡ 1 (mod 3). Table 13 gives {1, 3}-growable realizations for (a, b, c) ∈ {(2, 4, 0), (1, 5, 0), (1, 6, 0), (1, 10, 1), (1, 11, 1), (2, 9, 1)}. First we use Lemma 5.1 along with the realizations of Table 13 to obtain a {1, 3}-growable realization of L′ = {1a′ , 3b′ , 6c′} with a′ ∈ {1, 2}, b′ ≡ b (mod 3) and c′ ≤ 5 such that c′ ≡ c (mod 6). We have to distinguish cases according to the congruence class of c modulo 6. Table 13: {1, 3}-growable realizations of {1a, 3b, 6c}. Where they are 1- or 3-growable is indicated by (m1,m3). name realization (a, b, c) (m1,m3) g1 [6, 5, 1, 4, 0, 3, 2] (2, 4, 0) (4, 2) g2 [3, 0, 6, 2, 5, 1, 4] (1, 5, 0) (5, 2) g3 [5, 2, 7, 0, 3, 6, 1, 4] (1, 6, 0) (6, 2) g4 [2, 12, 9, 6, 3, 0, 10, 7, 4, 5, 8, 1, 11] (1, 10, 1) (3, 5) g5 [5, 2, 13, 10, 7, 8, 11, 0, 3, 6, 9, 12, 4, 1] (1, 11, 1) (6, 8) g6 [9, 6, 3, 0, 10, 7, 8, 11, 1, 4, 5, 12, 2] (2, 9, 1) (6, 8) When c is even, we start with g1, g2 or g3. If c ≡ 0 (mod 6), then start by taking g1, g2 or g3 according to whether b is congruent to 1, 2 or 0 (mod 3) respectively. If c ≡ 2 (mod 6), then start by taking g1, g2 or g3 according to whether b is congruent to 2, 0 or 1 (mod 3) respectively and apply Lemma 5.1 with i = 1. If c ≡ 4 (mod 6), then start by taking g1, g2 or g3 according to whether b is congruent to 0, 1 or 2 (mod 3) respectively and apply Lemma 5.1 with i = 2. When c is odd, we start with g4, g5 or g6. If c ≡ 1 (mod 6), then start by taking g4, g5 or g6 according to whether b is congruent to 1, 2 or 0 (mod 3) respectively. If c ≡ 3 (mod 6), then start by taking g4, g5 or g6 according to whether b is congruent to 2, 0 or 1 (mod 3) respectively and apply Lemma 5.1 with i = 1. If c ≡ 5 (mod 6), then start by taking g4, g5 or g6 according to whether b is congruent to 0, 1 or 2 (mod 3) respectively and apply Lemma 5.1 with i = 2. In each case we obtain the required realization of L′. Next, apply Lemma 5.1 (c−c′)/6 times with x = i = 3 to obtain a {1, 3}-growable realization of {1a′ , 3b′+(c−c′)/2, 6c}. Finally, complete to the required realization using a− a′ applications of Theorem 2.4 with x = 1 and b−b ′ 3 − c−c′ 6 applications with x = 3. M. Ollis et al.: Growable realizations: a powerful approach to the Buratti-Horak-Rosa . . . 587 When c is even, the method requires up to six 3’s in the gi, up to seven 3’s to adjust the congruency class of the number of 6’s, and then c/2 3’s to obtain the correct number of 6’s. Hence it always works for b ≥ 6+7+c/2 = 13+c/2. When c is odd, the method requires up to eleven 3’s in the gi, up to seven 3’s to adjust the congruency class of the number of 6’s, and then up to (c−1)/2 3’s to obtain the correct number of 6’s. Hence it always works for b ≥ 11 + 7 + (c− 1)/2 = 18 + (c− 1)/2. Example 5.3. Let L = {13, 318, 610}. Since b ≡ 0 (mod 3) and c ≡ 4 (mod 6), we start applying Lemma 5.1 with i = 2 to g1. In this way we obtain the realization [15, 14, 1, 7, 4, 10, 13, 0, 6, 3, 9, 12, 11, 8, 5, 2] of the multiset {12, 39, 64}. Now we apply Lemma 5.1 once with i = x = 3 to this new multiset and we get a realization of {12, 312, 610}: [24, 23, 1, 7, 4, 10, 16, 13, 19, 22, 0, 6, 3, 9, 15, 12, 18, 21, 20, 17, 14, 11, 5, 8, 2]. We now apply Theorem 2.4 twice with x = 3 to get a realization of {12, 318, 610} and then once with x = 1 to get a realization of L: [31, 30, 1, 4, 7, 13, 10, 16, 22, 19, 25, 28, 29, 0, 3, 6, 12, 9, 15, 21, 18, 24, 27, 26, 23, 20, 17, 11, 14, 8, 5, 2]. To use the method of proof of Lemma 5.2 for x > 3 we require {1, x}-growable realiza- tions for {1a, xb} for a as small as possible and for a b in each congruence class modulo x. Lemmas 5.4, 5.6 and 5.8 provide these. Each of the constructions has at least one subse- quence consisting of multiple instances of pairs [t, t + x], triples [t, t + x, t + 2x] or their reverses; we indicate these pairs and triples with underbraces to help illuminate the overall structure. Lemma 5.4. Let x ≥ 4. The multisets {1x−1, xx+1}, {1x−2, xx+2} and {1x−2, x2x} have {1, x}-growable realizations. Proof. First, we cover {1x−1, xx+1}, in which case v = 2x + 1. When x is even, the sequence [1, x+ 1, 0, 2x, x, x− 1, 2x− 1︸ ︷︷ ︸, 2x− 2, x− 2︸ ︷︷ ︸, x− 3, 2x− 3︸ ︷︷ ︸, . . . , x+ 2, 2︸ ︷︷ ︸] has edge-lengths [x, x, 1, x, 1, x, . . . , 1, x] and so realizes {1x−1, xx+1}. It is 1-growable at 1 and x-growable at x. When x is odd, the sequence [x, x+1, 1, 0, 2x, x−1, 2x−1, x− 2, 2x− 2︸ ︷︷ ︸, 2x− 3, x− 3︸ ︷︷ ︸, x− 4, 2x− 4︸ ︷︷ ︸, . . . , x+ 2, 2︸ ︷︷ ︸] has edge-lengths [1, x, 1, 1, x, x, x, x, 1, x, . . . , 1, x] and so realizes {1x−1, xx+1}. It is 1- growable at 2x− 1 and x-growable at x. Next, consider {1x−2, xx+2} and so v = 2x+ 1. When x is even, the sequence [x, 2x, 0, x+ 1, 1, x+ 2, 2, 3, x+ 3︸ ︷︷ ︸, x+ 4, 4︸ ︷︷ ︸, 5, x+ 5︸ ︷︷ ︸, . . . , x− 1, 2x− 1︸ ︷︷ ︸] 588 Ars Math. Contemp. 22 (2022) #P4.04 / 567–594 has edge-lengths [x, 1, x, x, x, x, 1, x, 1, x, . . . , 1, x] and so realizes {1x−2, xx+2}. It is 1-growable at 1 and x-growable at x. When x is odd, the sequence [0, x, x− 1, 2x, 2x− 1, x− 2, 2x− 2, x− 3, 2x− 3︸ ︷︷ ︸, 2x− 4, x− 4︸ ︷︷ ︸, x− 5, 2x− 5︸ ︷︷ ︸, . . . , x+ 1, 1︸ ︷︷ ︸] has edge-lengths [x, 1, x, 1, x, x, x, x, 1, x, 1, x, . . . , 1, x] so this realizes {1x−2, xx+2}. It is 1-growable at 2x− 2 and x-growable at x− 1. Finally, consider {1x−2, x2x} and so v = 3x− 1. When x is even, the sequence [0, x, 2x︸ ︷︷ ︸, 2x+ 1, x+ 1, 1︸ ︷︷ ︸, 2, x+ 2, 2x+ 2︸ ︷︷ ︸, . . . , x− 4, 2x− 4, 3x− 4︸ ︷︷ ︸, x− 3, 2x− 3, 2x− 2, x− 2, 3x− 3, 3x− 2, x− 1, 2x− 1] has edge-lengths [x, x, 1, x, x, 1, . . . , 1, x, x, x, x, 1, x, x, 1, x, x] and so realizes {1x−2, x2x}. It is 1-growable at 3x− 4 and x-growable at x− 1. When x is odd the sequence [3x− 2, x− 1, 2x− 1, 2x, x, 0︸ ︷︷ ︸, 1, x+ 1, 2x+ 1︸ ︷︷ ︸, 2x+ 2, x+ 2, 2︸ ︷︷ ︸, . . . , x− 4, 2x− 4, 3x− 4︸ ︷︷ ︸, x− 3, 2x− 3, 2x− 2, x− 2, 3x− 3] has edge-lengths [x, x, 1, x, x, 1, x, x, . . . , 1, x, x, x, x, 1, x, x] and so realizes {1x−2, x2x}. It is 1-growable at 3x− 4 and x-growable at x. Example 5.5. Let x = 8. Lemma 5.4 gives the {1, 8}-growable realizations [1, 9, 0, 16, 8, 7, 15, 14, 6, 5, 13, 12, 4, 3, 11, 10, 2], [8, 16, 0, 9, 1, 10, 2, 3, 11, 12, 4, 5, 13, 14, 6, 7, 15], [0, 8, 16, 17, 9, 1, 2, 10, 18, 19, 11, 3, 4, 12, 20, 5, 13, 14, 6, 21, 22, 7, 15] of {17, 89}, {16, 810} and {16, 816} respectively. Let x = 9. Lemma 5.4 gives the {1, 9}-growable realizations [9, 10, 1, 0, 18, 8, 17, 7, 16, 15, 6, 5, 14, 13, 4, 3, 12, 11, 2], [0, 9, 8, 18, 17, 7, 16, 6, 15, 14, 5, 4, 13, 12, 3, 2, 11, 10, 1], [25, 8, 17, 18, 9, 0, 1, 10, 19, 20, 11, 2, 3, 12, 21, 22, 13, 4, 5, 14, 23, 6, 15, 16, 7, 24] of {18, 910}, {17, 911} and {17, 918} respectively. Lemma 5.6. Let x ≥ 4 be even. There is a {1, x}-growable realization for {1x−2, xb} for b in range x+ 3 ≤ b ≤ 2x− 1. Proof. We consider odd b and even b separately, starting with odd b. Take r in the range 0 ≤ r ≤ (x − 4)/2. Write x = 2r + 2s + 4 for some s ≥ 0. We construct a realization for L = {12r+2s+2, x4r+2s+7} = {1x−2, xx+2r+3}. We have v = (2r + 2s+ 2) + (4r + 2s+ 7) + 1 = 6r + 4s+ 10. M. Ollis et al.: Growable realizations: a powerful approach to the Buratti-Horak-Rosa . . . 589 We build the required realization by concatenating three sequences. First: [2r + 1, 4r + 2s+ 5︸ ︷︷ ︸, 4r + 2s+ 6, 2r + 2︸ ︷︷ ︸, 2r + 3, 4r + 2s+ 7︸ ︷︷ ︸, . . . , 4r + 4s+ 6, 2r + 2s+ 2︸ ︷︷ ︸], which has 2s+ 2 pairs and produces edge-lengths {12s+1, x2s+2}. Second: [6r + 4s+ 8, 4r + 2s+ 4, 2r︸ ︷︷ ︸, 2r − 1, 4r + 2s+ 3, 6r + 4s+ 7︸ ︷︷ ︸, 6r + 4s+ 6, 4r + 2s+ 2, 2r − 2︸ ︷︷ ︸, . . . , 4r + 4s+ 8, 2r + 2s+ 4, 0︸ ︷︷ ︸], which has 2r + 1 triples and produces {12r, x4r+2}. Third: [6r + 4s+ 9, 2r + 2s+ 3, 4r + 4s+ 7] which produces {x2}. Upon concatenation we have a difference of x where the first and second sequences join and a difference of 1 where the second and third join. Hence we have a realization of L = {12s+1, x2s+2} ∪ {12r, x4r+2} ∪ {x2} ∪ {1, x} = {1x−2, xx+2r+3}. It is 1-growable at v−2 = 6r+4s+8: when embedding with m = 6r+4s+8, the only lengthened edge is (2r+2s+2, 6r+4s+8). It is x-growable at x−1 = 2r+2s+3: when embedding with m = 2r + 2s+ 3 the lengthened edges are (i, i+ x) for 0 ≤ i ≤ x− 1. For the case with b even, first note that when x = 4 there are no values of b to be considered. Let x ≥ 6 be even and take r in the range 0 ≤ r ≤ (x − 6)/2. Write x = 2r + 2s+ 6 for some s ≥ 0. We construct a realization for L = {12r+2s+4, x4r+2s+10} = {1x−2, xx+2r+4}. We have v = (2r + 2s+ 4) + (4r + 2s+ 10) + 1 = 6r + 4s+ 15. We build the required realization by concatenating three sequences. First: [4r + 2s+ 11, 2r + 5︸ ︷︷ ︸, 2r + 6, 4r + 2s+ 12︸ ︷︷ ︸, 4r + 2s+ 13, 2r + 7︸ ︷︷ ︸, . . . , 4r + 4s+ 11, 2r + 2s+ 5︸ ︷︷ ︸], which has 2s+ 1 pairs and produces the edge-lengths {12s, x2s+1}. Second: [2r + 2s+ 6, 4r + 4s+ 12, 4r + 4s+ 13, 2r + 2s+ 7, 1, 4r + 2s+ 10, 2r + 4, 2r + 3, 4r + 2s+ 9, 0] which produces {12, x7}. Third: [6r + 4s+ 14, 4r + 2s+ 8, 2r + 2︸ ︷︷ ︸, 2r + 1, 4r + 2s+ 7, 6r + 4s+ 13︸ ︷︷ ︸, 6r + 4s+ 12, 4r + 2s+ 6, 2r︸ ︷︷ ︸, . . . , 4r + 4s+ 14, 2r + 2s+ 8, 2︸ ︷︷ ︸], 590 Ars Math. Contemp. 22 (2022) #P4.04 / 567–594 which has 2r + 1 triples and produces {12r, x4r+2}. Upon concatenation we have a difference of 1 at each of the joins. Hence we have a realization of L = {12s, x2s+1} ∪ {12, x7} ∪ {12r, x4r+2} ∪ {12} = {1x−2, xx+2r+4}. It is 1-growable at 1: when embedding with m = 1, the only lengthened edge is (1, 2r+ 2s+ 7). It is x-growable at x = 2r + 2s+ 6: when embedding with m = 2r + 2s+ 6 the lengthened edges are (i, i+ x) for 1 ≤ i ≤ x. Example 5.7. To construct a {1, 8}-growable realization of {16, 813} using the proof of Lemma 5.6 we take r = s = 1 to obtain [3, 11, 12, 4, 5, 13, 14, 6, 18, 10, 2, 1, 9, 17, 16, 8, 0, 19, 7, 15], which is 1-growable at 18 and 8-growable at 7. To construct a {1, 10}-growable realization of {18, 1016} we take r = s = 1 to obtain [17, 7, 8, 18, 19, 9, 10, 20, 21, 11, 1, 16, 6, 5, 15, 0, 24, 14, 4, 3, 13, 23, 22, 12, 2] which is 1-growable at 1 and 10-growable at 10. Lemma 5.8. Let x ≥ 5 be odd. There is a {1, x}-growable realization for {1x−2, xb} for b in range x+ 3 ≤ b ≤ 2x− 1. Proof. The constructions are similar to those of Lemma 5.6 in that they each are built from the concatenation of three sequences and we need to consider odd and even b separately. We start with odd b. Take r in the range 0 ≤ r ≤ (x − 5)/2. Write x = 2r + 2s + 5 for some s ≥ 0. We construct a realization for L = {12r+2s+3, (2r + 2s+ 5)4r+2s+9} = {1x−2, xx+2r+4}. We have v = (2r + 2s+ 3) + (4r + 2s+ 9) + 1 = 6r + 4s+ 13. The first sequence is [6r + 4s+ 10, 4r + 2s+ 5, 2r︸ ︷︷ ︸, 2r − 1, 4r + 2s+ 4, 6r + 4s+ 9︸ ︷︷ ︸, 6r + 4s+ 8, 4r + 2s+ 3, 2r − 2︸ ︷︷ ︸, . . . , 4r + 4s+ 10, 2r + 2r + 5, 0︸ ︷︷ ︸]. There are 2r+1 triples, so this sequence produces edge-lengths {12r, x4r+2}. The second sequence is [6r + 4s+ 12, 2r + 2s+ 4, 4r + 4s+ 9, 4r + 4s+ 8, 2r + 2s+ 3, 6r + 4s+ 11]. This has internal differences [2r+2s+5, 2r+2s+5, 1, 2r+2s+5, 2r+2s+5] and so produces {1, x4}. The third sequence is [4r + 2s+ 6, 2r + 1︸ ︷︷ ︸, 2r + 2, 4r + 2s+ 7︸ ︷︷ ︸, 4r + 2s+ 8, 2r + 3︸ ︷︷ ︸, . . . , 2r + 2s+ 2, 4r + 4s+ 7︸ ︷︷ ︸]. M. Ollis et al.: Growable realizations: a powerful approach to the Buratti-Horak-Rosa . . . 591 There are 2s+ 2 pairs, so this sequence produces {12s+1, x2s+2}. Upon concatenation, we have a difference of 1 generated where the first and second sequences join and a difference of 2r+2s+5 = x where the second and third join. Hence we have a realization of L = {12r, x4r+2} ∪ {1, x4} ∪ {12s+1, x2s+2} ∪ {1, x} = {1x−2, xx+2r+4}. The realization is 1-growable at v − 2 = 6r + 4s + 11: when embedding with m = 6r+ 4s+ 11, the only lengthened edge is (2r+ 2s+ 3, 6r+ 4s+ 11). It is x-growable at x − 1 = 2r + 2s + 4: when embedding with m = 2r + 2s + 4 the lengthened edges are (i, i+ x) for 0 ≤ i ≤ x− 1. Moving to even b, take r in the range 0 ≤ r ≤ (x − 5)/2 and write x = 2r + 2s + 5 for some s ≥ 0. We construct a realization for L = {12r+2s+3, (2r + 2s+ 5)4r+2s+8} = {1x−2, xx+2r+3}. We have v = (2r + 2s+ 3) + (4r + 2s+ 8) + 1 = 6r + 4s+ 12. The first sequence is [4r + 2s+ 6, 2r + 1︸ ︷︷ ︸, 2r + 2, 4r + 2s+ 7︸ ︷︷ ︸, 4r + 2s+ 8, 2r + 3︸ ︷︷ ︸, . . . , 4r + 4s+ 8, 2r + 2s+ 3︸ ︷︷ ︸]. There are 2s+3 pairs, so this sequence produces edge-lengths {12s+2, x2s+3}. The second sequence is the same as the first sequence of the previous construction: [6r + 4s+ 10, 4r + 2s+ 5, 2r︸ ︷︷ ︸, 2r − 1, 4r + 2s+ 4, 6r + 4s+ 9︸ ︷︷ ︸, 6r + 4s+ 8, 4r + 2s+ 3, 2r − 2︸ ︷︷ ︸, . . . , 4r + 4s+ 10, 2r + 2r + 5, 0︸ ︷︷ ︸]. As before, there are 2r + 1 triples, so this sequence produces {12r, x4r+2}. The third sequence is [6r + 4s+ 11, 2r + 2s+ 4, 4r + 4s+ 9] which has internal differences [2r + 2s+ 5, 2r + 2s+ 5] and so produces {x2}. Upon concatenation, we have a difference of 2r+2s+5 = x generated where the first and second sequences join and a difference of 1 where the second and third join. Hence we have a realization of L = {12s+2, x2s+3} ∪ {12r, x4r+2} ∪ {x2} ∪ {1, x} = {1x−2, xx+2r+3}. The realization is 1-growable at v − 2 = 6r + 4s + 10: when embedding with m = 6r+ 4s+ 10, the only lengthened edge is (2r+ 2s+ 3, 6r+ 4s+ 10). It is x-growable at x − 1 = 2r + 2s + 4: when embedding with m = 2r + 2s + 4 the lengthened edges are (i, i+ x) for 0 ≤ i ≤ x− 1. Example 5.9. To construct a {1, 13}-growable realization of {111, 1321} using the proof of Lemma 5.8 we take r = s = 2 to obtain [30, 17, 4, 3, 16, 29, 28, 15, 2, 1, 14, 27, 26, 13, 0, 32, 12, 25, 24, 11, 31, 18, 5, 6, 19, 20, 7, 8, 21, 22, 9, 10, 23], 592 Ars Math. Contemp. 22 (2022) #P4.04 / 567–594 which is 1-growable at 31 and 13-growable at 12. To construct a {1, 13}-growable realization of {17, 914} we take r = s = 1 to obtain [12, 3, 4, 13, 14, 5, 6, 15, 16, 7, 20, 11, 2, 1, 10, 19, 18, 9, 0, 21, 8, 17] which is 1-growable at 20 and 9-growable at 8. We can now prove the main result of the section. Theorem 5.10. Let L = {1a, xb, (2x)c}. If a ≥ x − 2, c is even and b ≥ 5x − 2 + c/2, then BHR(L) holds. Proof. If x ≤ 2, then BHR(L) holds without restriction and the case x = 3 is covered in Lemma 5.2, so assume x ≥ 4. We follow the method of proof of Lemma 5.2, with Lemmas 5.4, 5.6 and 5.8 providing the realizations to get started. Take i in the range 0 ≤ i < c/2 such that 2i ≡ c (mod 2x). To construct the required realization for L, start with the realization of {1a′ , xb′} that has b′ ≡ b + 2i (mod x) given by Lemma 5.4, 5.6 or 5.8. So a′ = x − 2, except when b+ 2i ≡ 1 (mod x) and admissibility forces us to use a′ = x− 1. If c ̸≡ 0 (mod 2x), then apply Lemma 5.1 using i to give a realization whose number of occurrences c′ of 2x differs from c by a multiple of 2x and whose number of occurrences of x differs from b by a multiple of x. (If c ≡ 0 (mod 2x), then this is already the case.) Apply Lemma 5.1 a further (c−c′)/2x times with i = x to obtain a {1, x}-growable re- alization of {1a′ , xb′′ , (2x)c} where b′′ ≡ b (mod x). Complete to the required realization using the appropriate number of applications of Theorem 2.4 with 1 and x. The method requires up to 2x occurrences of x in the initial realization, up to 3x − 2 occurrences of x to adjust the congruency class of the number of occurrences of 2x, and c/2 occurrences of x to obtain the correct number of occurrences of 2x. Hence it always works for b ≥ 2x+ 3x− 2 + c/2 = 5x− 2 + c/2. When c is odd, we are not aware of any reason why the same approach will not work. However, without new ideas, it will take more work to get weaker results than in the even case. This is because the starter realizations now need to be for {1x−2, xb, 2x}, which means that we must have v ≥ 4x, compared to the constructions here which all have v < 3x. As well as being larger, using the same approach as Lemmas 5.6 and 5.8 would probably take more cases to cover all required values of b. Some of these issues are already apparent in Lemma 5.2. Lemma 5.1 can be thought of as combining the notion of growability with that of a particular perfect realization. This can be generalized to other perfect realizations, which we now do. For a multiset L, define sL = {sy : y ∈ L}. When we apply Theorem 2.4 k times to an x-growable realization we produce a realization with the x subsequences [m+ 1− x,m+ 1,m+ 1 + x, . . . ,m+ 1 + (k − 1)x] + t for 0 ≤ t ≤ x − 1. If we have a perfect linear realization of length k of a multiset L, then we can multiply each element by x to get a sequence that realizes xL and then take a translate of it to replace a subsequence of the above form. Lemma 5.1 uses this process with the perfect linear realization [0, 2, 1, 3] of {1, 22}. In general the approach gives the following lemma. M. Ollis et al.: Growable realizations: a powerful approach to the Buratti-Horak-Rosa . . . 593 Lemma 5.11. Let L have an X-growable realization with x ∈ X . Let L1, . . . , Lx be multisets of size k − 1 that have perfect linear realizations. Then L ∪ xL1 ∪ · · · ∪ xLx has an X-growable realization. If we use the perfect linear realization [0, 1] of {1} in Lemma 5.11, then we end up back at Theorem 2.4. Example 5.12. Let x ≥ 3. In this section we have constructed {1, x}-growable realizations of the multisets {1x−1, xx+1} and {1x−2, xb} for x + 2 ≤ b ≤ 2x. Let c, d, e, f, g ≥ 0 with c + d + e + f + g ≡ 0 (mod x). Take c copies of the perfect linear realization [0, 1, 2, 3, 4, 5] of {15}, d copies of the perfect realization [0, 2, 1, 3, 4, 5] of {13, 22}, e copies of the perfect linear realization [0, 3, 1, 2, 4, 5] of {12, 22, 3}, f copies of the perfect linear realization [0, 3, 1, 4, 2, 5] of {22, 33}, and g copies of the perfect linear realization [0, 2, 4, 1, 3, 5] of {24, 3}. Lemma 5.11 proves BHR(L) for L = {1a, xb+5c+3d+2e, (2x)2d+2e+2f+4g, (3x)e+3f+g} for a ≥ x− 2 and b ≥ x+ 1. ORCID iDs Anita Pasotti https://orcid.org/0000-0002-3569-2954 Marco A. Pellegrini https://orcid.org/0000-0003-1742-1314 References [1] M. Buratti and F. Merola, Dihedral Hamiltonian cycle systems of the cocktail party graph, J. Comb. Des. 21 (2013), 1–23, doi:10.1002/jcd.21311. [2] S. Capparelli and A. Del Fra, Hamiltonian paths in the complete graph with edge-lengths 1, 2, 3, Electron. J. Comb. 17 (2010), research paper r44, doi:10.37236/316. [3] J. H. Dinitz and S. R. Janiszewski, On Hamiltonian paths with prescribed edge lengths in the complete graph, Bull. Inst. Comb. Appl. 57 (2009), 42–52. [4] S. C. Gutekunst and D. P. Williamson, The circlet inequalities: a new, circulant-based facet- defining inequality for the tsp, 2020, arXiv:2012.12363 [math.CO]. [5] P. Horak and A. Rosa, On a problem of Marco Buratti, Electron. J. Comb. 16 (2009), research paper r20, doi:10.37236/109. [6] M. Meszka, Private communication, reported in [5]. [7] F. Monopoli, Absolute differences along Hamiltonian paths, Electron. J. Comb. 22 (2015), research paper p3.20, doi:10.37236/5159. [8] M. A. Ollis, A. Pasotti, M. A. Pellegrini and J. R. Schmitt, New methods to attack the Buratti-Horak-Rosa conjecture, Discrete Math. 344 (2021), No 112486, doi:10.1016/j.disc. 2021.112486. [9] A. Pasotti and M. A. Pellegrini, A new result on the problem of Buratti, Horak and Rosa, Discrete Math. 319 (2014), 1–14, doi:10.1016/j.disc.2013.11.017. 594 Ars Math. Contemp. 22 (2022) #P4.04 / 567–594 [10] A. Pasotti and M. A. Pellegrini, On the Buratti-Horak-Rosa conjecture about Hamiltonian paths in complete graphs, Electron. J. Comb. 21 (2014), research paper p2.30, 20, www. combinatorics.org/ojs/index.php/eljc/article/view/v21i2p30. [11] D. B. West, Buratti’s Conjecture (2007), 2021, accessed January 22, 2021, {https:// faculty.math.illinois.edu/˜west/regs/buratti.html}. ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P4.05 / 595–608 https://doi.org/10.26493/1855-3974.2714.9b3 (Also available at http://amc-journal.eu) Ordering signed graphs with large index Maurizio Brunetti Dip. di Matematica e Applicazioni, Universitá di Napoli ‘Federico II’, P. le Tecchio 80, I-80125 Naples, Italy Zoran Stanić * Faculty of Mathematics, University of Belgrade, Studentski trg 16, Belgrade, Serbia Received 28 October 2021, accepted 5 January 2022, published online 3 August 2022 Abstract The index of a signed graph is the largest eigenvalue of its adjacency matrix. We estab- lish the first few signed graphs ordered decreasingly by the index in classes of connected signed graphs, connected unbalanced signed graphs and complete signed graphs with a fixed number of vertices. Keywords: Adjacency matrix, largest eigenvalue, edge relocation, unbalanced signed graph, com- plete signed graph. Math. Subj. Class. (2020): 05C50, 05C22 1 Introduction A signed graph Ġ is a pair (G, σ), where G = (V,E) is an unsigned graph, called the underlying graph, and σ : E −→ {1,−1} is the sign function or the signature. The number of vertices of Ġ is called the order and denoted by n. The edge set of Ġ is composed of the subset E+ of positive edges and the subset E− of negative edges. We interpret an unsigned graph as a signed graph with the all positive signature, that is the signature which assigns 1 to every edge. The adjacency matrix AĠ of Ġ is obtained from the standard adjacency matrix of its underlying graph by switching the sign of all 1’s which correspond to negative edges. The eigenvalues of Ġ are identified to be the eigenvalues of its adjacency matrix; they form *Corresponding author. Research of the second author is partially supported by the Serbian Ministry of Edu- cation, Science and Technological Development via the University of Belgrade. E-mail addresses: maurizio.brunetti@unina.it (Maurizio Brunetti), zstanic@matf.bg.ac.rs (Zoran Stanić) cb This work is licensed under https://creativecommons.org/licenses/by/4.0/ 596 Ars Math. Contemp. 22 (2022) #P4.05 / 595–608 the spectrum of Ġ. The largest eigenvalue of Ġ is called the index and denoted by λ1 (or λ1(Ġ)). If S is a set of vertices of Ġ, the switched signed graph ĠS is obtained from Ġ by reversing the signs of the edges in the cut [S, V (Ġ) \ S]. The signed graphs Ġ and ĠS are said to be switching equivalent. The switching equivalence is an equivalence relation that preserves the eigenvalues, and the switching class of Ġ is denoted by [Ġ]. A signed graph is said to be balanced if it switches to the signed graph with all positive signature. Otherwise, it is said to be unbalanced. Equivalently, Ġ is balanced if every cycle contained in Ġ is balanced [15]. Ordering of unsigned graphs by the largest eigenvalue of some associated matrix has received a great deal of attention in literature. Many results can be found in [12]. More re- cently, there has been a growing interest for extremal problems in the framework of signed graphs. For instance, in [9] Koledin and the second author studied connected signed graphs of fixed order, size and number of negative edges that maximize the index. In the wake of that paper, signed graph maximizing the index in suitable subsets of complete signed graphs have been studied in [2]. Let Un (resp. Bn) denote the class of unbalanced uni- cyclic (resp. bicyclic) signed graphs of order n. Akbari et al. [1] determined the signed graphs attaining the extremal indices in Un. Some of the same authors studied in [10] signed graphs achieving the maximum index among signed graphs in Un of fixed girth. The first five largest indices among signed graphs in Bn with n ⩾ 36 are detected by He et al. [8]. Signed graphs in Un and Bn with extremal spectral radius were identified in [4]. Finally, extremal graphs in Un and Bn with respect to the least Laplacian eigenvalue were studied in [5] and [3], respectively. In [6] we determined the unbalanced signed graph with largest index, for every order n. In this paper we continue this research by presenting a general method for ordering the signed graphs with a fixed number of vertices by their index. We demonstrate the method by determining the first few signed graphs ordered by the index in the class of connected signed graphs, or connected unbalanced signed graphs, or complete signed graphs with n vertices. The paper is organized as follows. Section 2 contains a preliminary setting related to the graphical representations of signed graphs in this paper along with terminology, nota- tion, a few known results and the proofs of two preliminary lemmas. The main result that provides the subsequent orderings is formulated in Theorem 3.2 of Section 3. Orderings in the mentioned classes are considered in Sections 3–5. Further computations, including orderings of signed graphs with a comparatively small number of vertices, are given in Section 6. 2 Preparatory We introduce a way of depicting signed graphs that will be used in the subsequent sections. For a signed graph of order n, we draw only the negative edges and the non-edges, along with the assumption that all non-depicted edges are positive. By convention, a negative edge is represented by a full line and a non-edge is represented by a dotted line. Accord- ingly, the complete signed graph with the all positive signature (i.e. the complete unsigned graph) is represented by an empty figure, the complete signed graph with a single negative edge is represented by a negative edge, and so on. The following lemmas are taken from [13, 14]. M. Brunetti and Z. Stanić: Ordering signed graphs with large index 597 Lemma 2.1 ([14]). For a connected signed graph Ġ = (G, σ), we have λ1(Ġ) ≤ λ1(G) with equality if and only if Ġ switches to G. Lemma 2.2 ([13]). For an eigenvalue λ of a signed graph Ġ, there is a switching equiv- alent signed graph for which the λ-eigenspace contains an eigenvector whose non-zero coordinates are of the same sign. For the sake of completeness we say that a signed graph of the previous lemma can be constructed by taking Ġ with AĠx = λx and considering D −1AĠD where D is the diagonal matrix of ±1s whose negative entries correspond to negative coordinates of x. We proceed with some notation. For a signed graph Ġ we denote by R(Ġ) the set of signed graphs obtained by taking a positive edge e of some signed graph of the switching class [Ġ], and then either removing e or reversing its sign. Let S = (Ġ1, Ġ2, . . . , Ġg) be a sequence which consists of the representatives of all switching equivalence classes of connected signed graphs with n vertices such that the representatives are ordered non-increasingly by the index and chosen in such a way that, for 1 ≤ i ≤ g, the λ1-eigenspace of Ġi contains an eigenvector whose non-zero coordinates are positive. (The existence of Ġi is provided by Lemma 2.2.) We now prove the following lemmas. They generalize known results for unsigned graphs that can be found in [12, Lemma 1.28]. Lemma 2.3 (Changing an edge). Let x = (x1, x2, . . . , xn)⊺ be an eigenvector associated with the index of a signed graph Ġ and let r, s be fixed vertices of Ġ. (i) If xrxs ≥ 0 and rs is a non-edge (resp. rs is a negative edge), then for a signed graph Ġ′ obtained by inserting a positive edge between r and s (resp. deleting rs or reversing its sign) we have λ1(Ġ′) ≥ λ1(Ġ). If at least one of xr, xs is non-zero, the previous inequality is strict. (ii) If xrxs < 0 and rs is a non-edge (resp. rs is a positive edge), then for a signed graph Ġ′ obtained by inserting a negative edge between r and s (resp. deleting rs or reversing its sign) we have λ1(Ġ′) > λ1(Ġ). Proof. We only demonstrate the proof of (i), as (ii) is proved analogously. If y is an eigenvector associated with λ1(Ġ′), using the Rayleigh principle we get λ1(Ġ ′)− λ1(Ġ) = y⊺AĠ′y − x ⊺AĠx ≥ x ⊺AĠ′x− x ⊺AĠx = x ⊺(AĠ′ −AĠ)x = { 2xrxs if ( rs /∈ E(Ġ) ∧ rs ∈ E+(Ġ′) ) ∨ ( rs ∈ E−(Ġ) ∧ rs /∈ E(Ġ′) ) , 4xrxs if rs ∈ E−(Ġ) ∧ rs ∈ E+(Ġ′). Hence, λ1(Ġ′) ≥ λ1(Ġ). Assume that xr ̸= 0 and, by way of contradiction, that λ1(Ġ′) = λ1(Ġ). In this case, the inequality in the previous chain reduces to equality, which means that x is an eigenvector afforded by λ1(Ġ′). Using the eigenvalue equations at vertex s in Ġ and Ġ′, we get 0 = ( λ1(Ġ ′)− λ1(Ġ) ) xs = ∑ i : is∈E(Ġ′) σĠ′(is)xi − ∑ i : is∈E(Ġ) σĠ(is)xi = αxr, (2.1) 598 Ars Math. Contemp. 22 (2022) #P4.05 / 595–608 where α depends on (r, s)-entries in AĠ and AĠ′ , but it is always non-zero; for example, if rs /∈ E(Ġ) ∧ rs ∈ E+(Ġ′) then α = σĠ′(rs) = 1, and similarly for the remaining possibilities listed in the statement formulation. Together with (2.1), this leads to xr = 0, which in turn contradicts the initial assumption and we are done. Let r, s, t, u be fixed vertices of a signed graph. A relocation Rot(r, s, t) (called a rotation) is realised in the adjacency matrix by replacing the entries ars, asr with the entries art, atr, and vice versa. In simple words, this relocation is realised by taking the object (which can be a positive edge, or a negative edge, or a non-edge) located between r and s and the object located between r and t and then inserting the first object between r and t and the second object between r and s. A relocation Shift(r, s, t, u) (called a shifting) is realised in the adjacency matrix by replacing the entries ars, asr with atu, aut, and vice versa. Lemma 2.4 (Rotation and shifting). Let x = (x1, x2, . . . , xn)⊺ be an eigenvector asso- ciated with the index of a signed graph Ġ and let r, s, t, u be fixed vertices of Ġ. (i) Let Ġ′ be obtained from Ġ by the relocation Rot(r, s, t). If ( xr ( xs−xt) > 0∨ (xs = xt ∧ xr ̸= 0) ) ∧ ( (rs is a non-edge ∧ rt is a positive edge) ∨ (rs is a negative edge and rt is a positive edge) ∨ (rs is a negative edge and rt is a non-edge) ) then λ1(Ġ ′) > λ1(Ġ). (ii) Let Ġ′ be obtained from Ġ by the relocation Shift(r, s, t, u). If ( xtxu > xrxs ∨ (xtxu = xrxs ∧ at least one of xr, xs, xt, xu is non-zero) ) ∧ ( (rs is a positive edge ∧ tu is a non-edge) ∨ (rs is a positive edge and tu is a negative edge) ∨ (rs is a non-edge and tu is a negative edge) ) then λ1(Ġ′) > λ1(Ġ). Proof. This proof is similar to the proof of the previous lemma. If rs and rt are as in (i), then we compute λ1(Ġ ′)− λ1(Ġ) ≥ x⊺(AĠ′ −AĠ)x = 2αxr(xs − xt), where α = 1 for the first and the third assumption on rs and rt, and α = 2 for the second assumption. Now, for xr(xs − xt) > 0 we get λ1(Ġ′) > λ1(Ġ). For xs = xt we have λ1(Ġ ′) ≥ λ1(Ġ). In case of equality, we have that x is afforded by λ1(Ġ′). Considering the eigenvalue equation at the vertex s in Ġ and Ġ′ , we get xr = 0, which completes (i). If rs and rt are as in (ii), then we compute λ1(Ġ ′)− λ1(Ġ) ≥ x⊺(AĠ′ −AĠ)x = 2α(xtxu − xrxs), with α ∈ {1, 2}, as before. We are done for xtxu > xrxs, while for xtxu = xrxs, using the previous reasoning we get that the equality between the indices necessarily leads to the conclusion that x takes zero at the corresponding four vertices. 3 Ordering signed graphs by the index We start our considerations with an example. Example 3.1. Clearly, there are just 3 connected signed graphs of order 3, up to switching: the positive triangle (with index 2), the 3-vertex path (with index √ 2) and the negative M. Brunetti and Z. Stanić: Ordering signed graphs with large index 599 triangle (with index 1). We know from [11] that there are exactly 12 connected signed graphs of order 4 (again, up to switching). Their indices are computed directly, and the corresponding ordering is given in Figure 1. 2. 2.56161. 3.0000 3. 2.2361 4. 2.1701 7. 1.7321 8. 1.6181 9. 1.5616 10. 1.4812 11. 1.4142 12. 1.00005-6. 2.0000 Figure 1: Connected signed graphs with 4 vertices ordered by the index. Here and in the subsequent graphical representations, signed graphs are depicted according to the conven- tion explained in Section 2. In what follows, we determine the first 5 connected signed graphs with n vertices or- dered by the index, for every n ≥ 5. In other words, we determine the signed graphs Ġ1 − Ġ5 of the sequence S defined in the previous section. First, Ġ1 is the complete signed graph with the all positive signature, which follows from the well-known Perron- Frobenius Theorem and Lemma 2.1. We now prove the following theorem, crucial for our considerations. Theorem 3.2. Let S ′ = (Ġk+1, Ġk+2, . . . , Ġk+ℓ) be a subsequence of S such that λ1(Ġk) > λ1(Ġk+1) = λ1(Ġk+2) = · · · = λ1(Ġk+ℓ) > λ1(Ġk+ℓ+1). (3.1) Then, for every Ġ ∈ S ′, we have Ġ ∈ R(Ḣ) where Ḣ ∈ {Ġ1, Ġ2, . . . , Ġk+ℓ} \ Ġ. In addition: (a) For at least one Ġ ∈ S ′ we have Ḣ ∈ {Ġ1, Ġ2, . . . , Ġk}; (b) If Ḣ /∈ {Ġ1, Ġ2, . . . , Ġk} then a non-negative λ1-eigenvector for Ġ has at least two zero coordinates and the same eigenvector is afforded by λ1(Ḣ). Proof. Assume by way of contradiction that for some Ġ ∈ S ′, Ġ /∈ R(Ḣ), for every Ḣ that belongs to the set given in the statement formulation. Let x = (x1, x2, . . . , xn)⊺ be an eigenvector with non-negative coordinates afforded by the index of Ġ. Assume that at least one of xr, xs is non-zero for some vertices r, s of Ġ. If rs is not a positive edge, then Lemma 2.3(i) produces a signed graph Ġ′ that differs from Ġ only in the positive edge rs, along with λ1(Ġ′) > λ1(Ġ). Since Ġ /∈ ⋃k i=1 R(Ġi), we have Ġ′ /∈ ⋃k i=1[Ġi], i.e. there are at least k+1 signed graphs whose index is larger than λ1(Ġ), which contradicts (3.1). Hence, rs is a positive edge. Further, if xr = xs = 0, then reversing the sign of rs, or removing rs, or adding rs do not affect the existence of the corresponding eigenvalue; indeed, it is afforded by the same eigenvector. Thus if for any such r, s there is no positive edge between them, as before we get Ġ′ with λ1(Ġ′) ≥ λ1(Ġ). Since Ġ /∈ ⋃k i=1 R(Ġi), we have λ1(Ġ′) = λ1(Ġ), but then Ġ ∈ ⋃ℓ i=1 R(Ġk+i) which together with the initial assumption leads to Ġ ∈ R(Ġ), i.e. Ġ is isomorphic to the signed graph obtained by inserting a positive edge rs. Replacing Ġ with this signed graph we get that rs is positive. Amalgamating the previous conclusions we get that Ġ is the complete signed graph with the all positive signature, which together with Lemma 2.1 contradicts (3.1) (since we assumed in (3.1) that Ġ is not Ġ1). 600 Ars Math. Contemp. 22 (2022) #P4.05 / 595–608 Consider now (a). Take an arbitrary Ġ ∈ S ′. If Ġ ∈ ⋃k i=1 R(Ġi), we are done. Assume that Ġ /∈ ⋃k i=1 R(Ġi). If x is the previously defined eigenvector, then there is a positive edge rs for every pair r, s such that at least one of xr, xs is non-zero, as otherwise by inserting a positive edge between such vertices we get Ġ ∈ ⋃k i=1 R(Ġi). If at most one coordinate of x is zero, then Ġ switches to a complete unsigned graph, which contradicts (3.1). Therefore there exist at least 2 vertices at which x takes zero. In addition, there is a negative edge between at least one such a pair, since otherwise Ġ has the all positive signature and then x has no zero coordinates by the Perron-Frobenius Theorem. If Ġ′ is obtained by switching the sign of such a negative edge, then λ1(Ġ′) = λ1(Ġ), as otherwise we get Ġ ∈ ⋃k i=1 R(Ġi). Moreover, λ1(Ġ′) is afforded by the same eigenvector, so we may repeat the previous consideration with Ġ′ in the role of Ġ. In this way we necessarily arrive at some Ġ ∈ S ′ ∩ (⋃k i=1 R(Ġi) ) since the number of negative edges strictly decreases in passing from Ġ to Ġ′. It remains to consider (b). Let Ġ ∈ R(Ḣ). If at most one coordinate of x is zero, then λ1(Ḣ) > λ1(Ġ) (by Lemma 2.3(i)), which implies Ḣ ∈ {Ġ1, Ġ2, . . . , Ġk}. Further, the assumption that Ġ ∈ R(Ḣ) together with λ1(Ḣ) = λ1(Ġ) leads to the conclusion that xr = xs = 0 for a non-positive (resp. positive) edge rs in Ġ (resp. Ḣ), and thus AḢx = AĠx = λ1(Ḣ)x. Remark 3.3. Theorem 3.2 gives a method for the ordering of signed graphs by the index. Let Ġ1, Ġ2, . . . , Ġk be the first k signed graphs ordered by the index such that all signed graphs (if any) sharing the index with Ġk are listed before it (so, as in the theorem). Then the sequence is extended by the signed graph(s) belonging to ⋃k i=1 R(Ġi). The candidates must be connected and the λ1-eigenspace for each of them must contain an eigenvector with non-negative coordinates. They are compared on the basis of an algebraic computation that relies on Lemmas 2.3 and 2.4. As long as we deal with signed graphs whose λ1- eigenspaces do not contain an eigenvector with at least two zero coordinates, there are no other candidates. In case of such eigenvectors, signed graphs with equal indices sharing the same eigenvector may appear. Remark 3.4. To determine the set R(Ġ) we need to consider the entire switching class of Ġ. For example, if Ġ is the complete signed graph with exactly one negative edge, say e, then the signed graphs obtained from Ġ by removing a positive edge or reversing its sign are the following 4: By making a switch at a vertex incident with e and either removing e or reversing its sign, we get 2 additional members of R(Ġ) that are not switching equivalent to the previous ones. But in both cases the λ1-eigenspace does not contain a non-negative eigenvector (the condition required in Remark 3.3). A method for computing the λ1-eigenvectors is demonstrated in the proof of the forthcoming Lemma 3.5. Now we proceed with the ordering. Lemma 3.5. Ġ2 is . M. Brunetti and Z. Stanić: Ordering signed graphs with large index 601 Proof. There are exactly two candidates for Ġ2: Ḟ obtained by removing an edge of Ġ1 and Ḣ obtained by reversing the sign of an edge of Ġ1. If the vertices joined by the unique negative edge of Ḣ are labelled by 1 and 2, using the eigenvalue equation for λ1(Ḣ) we get λ1a = − a+ (n− 2)b λ1b = 2a+ (n− 3)b which leads to the λ1(Ḣ)-eigenvector b(λ1+1λ1+3 , λ1+1 λ1+3 , 1, 1, . . . , 1)⊺, b ̸= 0. By virtue of Lemma 2.3(i) (applied to Ḣ), we have λ1(Ḟ ) > λ1(Ḣ). Hence Ġ2 ∼= Ḟ . Lemma 3.6. Ġ3 is . Proof. The candidates for Ġ3 are illustrated in Figure 2. They are obtained by considering R(Ġ1)∪R(Ġ2); we also include the transposes of the corresponding positive eigenvectors afforded by the index. b ( (λ1+1)(λ1−2) λ1(λ1+2)−4 , λ21 λ1(λ1+2)−4 , λ21+λ1−2 λ1(λ1+2)−4 , 1, 1, . . . , 1 ) b ( (λ21−1) λ1(λ1+2)−1 , λ1(λ1+1) λ1(λ1+2)−1 , λ1(λ1+1) λ1(λ1+2)−1 , 1, 1, . . . , 1 ) b ( λ1+1 λ1+2 , λ1+1 λ1+2 , λ1+1 λ1+2 , λ1+1 λ1+2 , 1, 1, . . . , 1 )1 2 3 4 Ḣ2 Ḣ4 Ḣ5 b ( λ1+1 λ1+3 , λ1+1 λ1+3 , 1, 1, . . . , 1 ) 1 2 Ḣ1 b ( λ1+1 λ1+3 , λ1+1 λ1+3 , λ1+1 λ1+2 , λ1+1 λ1+2 , 1, 1, . . . , 1 )1 2 3 4 Ḣ3 Figure 2: The candidates for Ġ3. From Lemma 2.3(i), we have λ1(Ḣ1) > max{λ1(Ḣ2), λ1(Ḣ3)}. We further apply Lemma 2.4(i) to Ḣ5 with (r, s, t) = (1, 2, 3) to conclude that λ1(Ḣ4) > λ1(Ḣ5). To show that Ġ3 ∼= Ḣ1 it remains prove that λ1(Ḣ1) > λ1(Ḣ4). The adjacency matrix of Ḣ1 is AḢ1 =  0 −1−1 0 J⊺ J AKn−2  , (where J is the all-1 matrix) which leads to the quotient matrix (i.e. the matrix of row sums in the corresponding blocks of AḢ1 ): QḢ1 = ( −1 n− 2 2 n− 3 ) . We know from [7] that every eigenvalue whose eigenspace does not contain an eigenvector orthogonal to the all-1 vector j belongs to the spectrum of the quotient matrix. In our case, this means that λ1(Ḣ1) is an eigenvalue of QḢ1 , i.e. λ1(Ḣ1) is the largest root of x2 + (4− n)x− 3n+ 7. (3.2) 602 Ars Math. Contemp. 22 (2022) #P4.05 / 595–608 In the same way, we get that λ1(Ḣ4) is the largest root of f(x) = x3 − (n− 3)x2 − (2n− 5)x+ n− 3. (3.3) Now, computing the largest root of (3.2) we get λ1(Ḣ1) = n+ √ (n−2)(n+6) 2 −2. Insert- ing it in (3.3), we get f (n+√(n−2)(n+6) 2 −2 ) = √ (n− 2)(n+ 6)−n > 0, which leads to the conclusion that either λ1(Ḣ1) > λ1(Ḣ4) or the two roots of f are larger than λ1(Ḣ1). The latter is not true because f(0) > 0, f(1) < 0 which means that f has a negative root and a root in (0, 1). To avoid repetitive proofs, in the remainder of this section and the next two sections we omit the parts in which we compute the λ1-eigenvectors of potential candidates since these are technical algebraic computations performed in exactly the same way as in the previous proof. Lemma 3.7. Ġ4 is . Proof. The 6 candidates for Ġ4 are (those of Figure 2 that have not passed for Ġ3 are included, of course): We note that there are two additional members of R(Ġ3) mentioned in Remark 3.3, but the non-existence of a required λ1-eigenvector eliminates them. The 3rd and the 5th candidate are eliminated since their indices are dominated by the index of the 1st one, while the 4th and the 6th are eliminated by the 2nd one in the same way – all this by virtue of Lemma 2.3(i). Finally, the fact that the index of the 1st signed graph is larger than that of the 2nd one is established in the proof of Lemma 3.6, and we are done. Lemma 3.8. Ġ5 is . Proof. The candidates for Ġ5 are the 5 signed graphs that are eliminated in the proof of the previous lemma (when we considered Ġ4) and the following 8 signed graphs: Lemma 2.3(i) eliminates all except the 2nd and the 3rd of the previous proof; in Figure 2 they are denoted by Ḣ5 and Ḣ2) and the 1st and the 3rd of the additional candidates (we denote them by Ḟ1 and Ḟ2). By Lemma 2.4, λ1(Ḟ2) dominates λ1(Ḟ1); we already had this in the proof of Lemma 3.6. Thus, it remains to prove that λ1(Ḣ5) > max{λ1(Ḣ2), λ1(Ḟ2)}. As in the proof of Lemma 3.6 we deduce that these indices are the largest roots of characteristic polynomials of the corresponding quotient matrices. These polynomials are: h5(x) = x 2 − (n− 3)x− 2(n+ 3) h2(x) = x 4 − (n− 4)x3 − (3n− 7)x2 + 2(n− 4)x+ 4(n− 3) f2(x) = x 3 − (n− 3)x2 − 2(n− 3)x+ 2(n− 4) M. Brunetti and Z. Stanić: Ordering signed graphs with large index 603 The largest root of h5 is 12 (n − 3 + √ (n− 3)(n+ 5)). Concerning h2 we get h2(−4) = 12(n + 11) > 0, h2(−2) = −4(n − 4) < 0, h2(0) = 4(n − 3) > 0 and h2(n − 2) = −(n − 1)(n − 4)2 < 0, which together with n − 2 < λ1(Ḣ5) leads to the conclusion that at least 3 roots of h2 are less than λ1(Ḣ5). Since h2(λ1(Ḣ5)) = (n − 4)(n − 3 +√ (n− 3)(n+ 5)) > 0, we conclude that the fourth root of h2 is also less than λ1(Ḣ5). Similarly, we have f2(−3) = −n − 26 < 0, f2(0) = 2(n − 4) > 0 and f2(1) = 2−n < 0, which means that that two roots of f2 are less than λ1(Ḣ5), while f2(λ1(Ḣ5)) = 2(n− 4) > 0 confirms the same for the third root, and we are done. Amalgamating the previous results we arrive at the following theorem. Theorem 3.9. The first 5 connected signed graphs with n ≥ 5 vertices ordered by their indices are: Ġ1 Ġ2 Ġ3 Ġ4 Ġ5 4 Unbalanced signed graphs Let now (U̇1, U̇2, . . . , U̇u) be the subsequence of S (defined in Section 2) containing only unbalanced signed graphs. In other words, the previous sequence ignores the balanced ones. In what follows, we determine U̇1 − U̇4 for n ≥ 6. We know from [6] that U̇1 is obtained by reversing the sign of a single edge in the complete graph of order n. Lemma 4.1. U̇2 is . Proof. The candidates for U̇2 are the last 4 signed graphs considered as the candidates in the proof of Lemma 3.7. (As before, it is not complicated to show that these are the only candidates with positive λ1-eigenvectors). The latter two candidates are eliminated by Lemma 2.3(i) – they are dominated by the 1st candidate. Observe that the λ1-eigenvector for the 2nd candidate is given in Figure 2. Using the same vertex labelling and applying the relocation Rot(3, 4, 1) we arrive at the result formulated in this statement. Lemma 4.2. U̇3 is . Proof. The candidates for U̇3 are the following 14 signed graphs: 2 1 3 4 1 2 3 J̇1 J̇2 J̇3 J̇4 604 Ars Math. Contemp. 22 (2022) #P4.05 / 595–608 All in the second row are easily eliminated on the basis of Lemma 2.3(i). The two in the third row are eliminated by Lemma 2.4(i). Namely, if we denote the vertices in the rep- resenting path by 1, 2, 3, 4 (in the natural order) and if x2 ≥ x3, then Rot(1, 2, 3) implies that the index of the signed graph under consideration is less than that of J̇3. Otherwise, we can apply Rot(4, 3, 2) with the same result. It remains to consider the indices of J̇1−J̇4. We first show that λ1(J̇2) > max{λ1(J̇3), λ1(J̇4)}. As in the proof of Lemma 3.5 we can show that J̇3 and J̇4 have a positive λ1- eigenvector. (Namely, we compute b ( (λ1+1)(λ1−3) λ1(λ1+2)−5 , (λ21+λ1−2) λ1(λ1+2)−5 , (λ21+λ1−2) λ1(λ1+2)−5 , λ21+1 λ1(λ1+2)−5 , 1, 1, . . . , 1 )⊺ for J̇3 which is positive for every b > 0, as λ1 > 3 when n ≥ 6. Similarly, we get b ( (λ1+1)2 λ1(λ1+4)+1 , λ1(λ1+1)λ1(λ1+4)+1 , λ1(λ1+1) λ1(λ1+4)+1 , 1, 1, . . . , 1 )⊺ for J̇4, which is positive for b > 0, as well.) Set J̇∗ ∈ {J̇3, J̇4}, and let x be a positive eigenvector afforded by λ1(J̇∗). Observe that J̇2 is obtained by inserting a positive edge 12 and a negative edge 13 in J̇∗. Therefore, we have λ1(J̇2)− λ1(J̇∗) ≥ x⊺(AJ̇2 −AJ̇∗)x = 2(x1x2 − x1x3) = 0, where the last equality follows since x2 = x3 (by the symmetry in J̇∗). Hence, λ1(J̇2) ≥ λ1(J̇∗). If λ1(J̇2) = λ1(J̇∗), then x is afforded by λ1(J̇2), but this is impossible since the eigenvalue equation at the vertex 2 cannot hold in J̇2 and J̇∗. Characteristic polynomials of quotient matrices of J̇1 and J̇2 are: j1(x) = x 3 − (n− 6)x2 − (5n− 17)x− 6n+ 20 j2(x) = x 3 − (n− 3)x2 − (2n− 3)x+ 7n− 23 We compute j(x) = j1(x)− j2(x) = 3x2 − (3n− 14)x− 13n+43, with roots: x1, x2 = 1 6 (3n − 14 ± √ 9n(8 + n)− 320). It follows that j1(x) < j2(x) for x ∈ (x1, x2). For the larger root x2 we have j1(x2) = j2(x2) = 127 ( (3n− 16) √ 9n(8 + n)− 320 + 9n2 − 96n+248 ) > 0 where the inequality follows since 9n2 − 96n+248 > 0 for n ≥ 7, while for n = 6 it is confirmed directly. Taking into account that x1 is negative (the easiest way to see this is to compute j(0)), we conclude that λ1(J̇1), λ1(J̇2) ∈ (x1, x2). Together with j1(x) < j2(x) on the same interval, this leads to λ1(J̇1) > λ1(J̇2). Lemma 4.3. U̇4 is . Proof. Besides the 13 signed graphs listed in the previous lemma, we have other 4 candi- dates for U̇4 (that arise from U̇3 but not from U̇1 or U̇2): The former two are eliminated by Lemma 2.3(i), the latter two by Lemma 2.4(i). Therefore, it remains to consider J̇2 − J̇4, but they have been already considered in the proof of the previous lemma, when we proved that λ1(J̇2) > max{λ1(J̇3), λ1(J̇4)}, as desired. The previous results lead to the following theorem. Theorem 4.4. The first 4 connected unbalanced signed graphs with n ≥ 6 vertices ordered by their indices are: M. Brunetti and Z. Stanić: Ordering signed graphs with large index 605 U̇1 U̇2 U̇3 U̇4 5 Complete signed graphs As before, let (Ċ1, Ċ1, . . . , Ċc) be the subsequence of S containing complete signed graphs. Clearly, the complete signed graph with the largest index switches to the one with the all positive signature. The next one contains exactly one negative edge. There are 2 candidates for Ċ3, both with 2 negative edges. By Lemma 2.4(i), Ċ3 is the one in which negative edges are adjacent. In what follows we set n ≥ 10. Lemma 5.1. Ċ4 is . Proof. The candidates are: L̇1 L̇2 L̇3 2 1 3 4 The latter two are eliminated by Lemma 2.4(i). By inserting the largest eigenvalue of the quotient matrix QL̇3 into the characteristic polynomial ℓ2, we get ℓ2 ( 1 2 (n− 6 + √ n(n+ 8)− 32) ) = 2(−n− 4 + √ n(n+ 8)− 32) < 0 as n(n + 8) − 32 < (n − 4)2. The latter inequality implies that the largest root of ℓ2 is larger than the largest eigenvalue of QL̇3 , i.e. λ1(L̇2) > λ1(L̇3). If the vertices of L̇2 are labelled as above then the λ1-eigenvector has the form b = b ( (λ1 + 1)(λ1 − 5) λ1(λ1 + 2)− 11 , λ21 + 1 λ1(λ1 + 2)− 11 , λ21 + 1 λ1(λ1 + 2)− 11 , 1, 1, . . . , 1 )⊺ , for b > 0. Now, L̇1 is obtained by reversing the sign of edges 12, 13 and 23, and thus we have λ1(L̇1)− λ1(L̇2) ≥ b⊺(AL̇1 −AL̇2)b = 4(λ21 + 1)b 2 λ1(λ1 + 2)− 11 (2(λ1 + 1)(λ1 − 5) λ1(λ1 + 2)− 11 − λ 2 1 + 1 λ1(λ1 + 2)− 11 ) = 4(λ21 + 1)b 2 λ1(λ1 + 2)− 11 · λ 2 1 − 8λ1 − 9 λ1(λ1 + 2)− 11 > 0 for λ1 > 9. We compute λ1(L̇2) > 9 for n = 11, and then by eigenvalue interlacing we have the same inequality for n ≥ 12. For n = 10, the inequality λ1(L̇1) > λ1(L̇2) is confirmed directly, and we are done. Lemma 5.2. Ċ5 is . 606 Ars Math. Contemp. 22 (2022) #P4.05 / 595–608 Proof. Apart from the signed graphs faced in the proof of the previous lemma, there is exactly one additional candidate: it contains exactly 3 non-adjacent negative edges. This candidate is eliminated on the basis of Lemma 2.4(i), while the remaining ones are already considered in the previous proof. In particular, we know that λ1(L̇2) > λ1(L̇3), and the proof is completed. Lemma 5.3. Ċ6 is . Proof. The only critical case is the comparison of the indices of L̇3 and the signed graph, say L̇, containing 4 negative edges that share the same vertex. Computing the λ1-eigenvector for L̇ and following the proof of Lemma 5.1, we get λ1(L̇3) > λ1(L̇) for λ21−12λ1−13 > 0, i.e. for λ1 = λ1(L̇) > 13. This proves this lemma for n ≥ 15 (as there λ1(L̇) > 13). The case 10 ≤ n ≤ 14 is considered directly, and we are done. We arrive at the following result. Theorem 5.4. The first 6 complete signed graphs with n ≥ 10 vertices ordered by their indices are: Ċ1 Ċ2 Ċ3 Ċ4 Ċ5 Ċ6 Remark 5.5. With a slight modification in which a full line represents a positive edge and an unpictured line represents a non-edge, the result of Theorem 5.4 remains valid for the ordering of unsigned graphs by the index of the Seidel matrix. Indeed, the Seidel matrix of an unsigned graph G coincides with the adjacency matrix of the complete signed graph in which negative edges are induced by the edges of G. 6 Further computations We complete the results of Sections 4 and 5 by determining the 6 signed graphs with largest indices for every order that is not covered by Theorem 4.4 and the 7 signed graphs with largest indices for every order that is not covered by Theorem 5.4. There is exactly one connected unbalanced signed graph with 3 vertices (the unbalanced triangle), while the ordering for n ∈ {4, 5} is given in the first part of Figure 3. We note that there are exactly 6 connected unbalanced signed graphs for n = 4, so in this case the given list is complete. There are exactly 3 complete signed graphs with 4 vertices and their ordering does not deviate from the general case considered in Theorem 5.4. For 5 ≤ n ≤ 9 the one with the largest index switches to the signed graph with all positive signature, while the remaining 6 are given in the second part of Figure 3. Again, for n = 5 the list is complete. In this paper our idea was to give a general method for the ordering by the index and to demonstrate its use by determining the lists of the first few signed graphs as reported in the previous sections. Of course, these results can be extended, but the theoretical approach is becoming more complicated as the number of candidates increases and comparison of their indices requires more sophisticated methods. However, it occurs that the list of Theo- rem 3.9 continues with: M. Brunetti and Z. Stanić: Ordering signed graphs with large index 607 Ġ6 Ġ7 Ġ8 n = 4 unbalanced n = 5 unbalanced n = 5 complete n = 6 complete n = 7 complete n = 8 complete n = 9 complete 2. 2.00001. 2.2361 3. 1.5616 4. 1.4812 5. 1.4142 6. 1.0000 2. 3.10281. 3.3723 5. 2.9173 6. 2.77843-4. 3.0000 3. 3.00002. 3.3723 4. 2.5616 5. 2.3723 6. 2.2361 7. 1.0000 3. 4.06422. 4.4641 4. 3.8284 5. 3.6056 6. 3.4940 7. 3.3871 3. 5.15542. 5.5311 6. 4.7720 7. 4.68424-5. 5.0000 3. 6.23612. 6.5826 4. 6.1231 5. 6.0283 6. 5.8990 7. 5.8284 3. 7.30392. 7.6235 4. 7.2170 5. 7.0813 6-7. 7.0000 Figure 3: Orderings of small signed graphs that are uncovered by Theorem 4.4 or Theo- rem 5.4. 608 Ars Math. Contemp. 22 (2022) #P4.05 / 595–608 We skip the details and note that the proof relies on an intensive algebraic computation that basically does not deviate from those of the previous sections. ORCID iDs Maurizio Brunetti https://orcid.org/0000-0002-2742-1919 Zoran Stanić https://orcid.org/0000-0002-4949-4203 References [1] S. Akbari, F. Belardo, F. Heydari, M. Maghasedi and M. Souri, On the largest eigenvalue of signed unicyclic graphs, Linear Algebra Appl. 581 (2019), 145–162, doi:10.1016/j.laa.2019. 06.016. [2] S. Akbari, S. Dalvandi, F. Heydari and M. Maghasedi, Signed complete graphs with maximum index, Discuss. Math. Graph Theory 40 (2020), 393–403, doi:10.7151/dmgt.2276. [3] F. Belardo, M. Brunetti and A. Ciampella, Signed bicyclic graphs minimizing the least Lapla- cian eigenvalue, Linear Algebra Appl. 557 (2018), 201–233, doi:10.1016/j.laa.2018.07.026. [4] F. Belardo, M. Brunetti and A. Ciampella, Unbalanced unicyclic and bicyclic graphs with extremal spectral radius, Czechoslovak Math. J. 71(146) (2021), 417–433, doi:10.21136/cmj. 2020.0403-19. [5] F. Belardo and Y. Zhou, Signed graphs with extremal least Laplacian eigenvalue, Linear Alge- bra Appl. 497 (2016), 167–180, doi:10.1016/j.laa.2016.02.028. [6] M. Brunetti and Z. Stanić, Unbalanced signed graphs with extremal spectral radius or index, Comput. Appl. Math. 41 (2022), Paper No. 118, 13, doi:10.1007/s40314-022-01814-5. [7] D. M. Cvetković, The main part of the spectrum, divisors and switching of graphs, Publ. Inst. Math. (Beograd) (N.S.) 23(37) (1978), 31–38, http://eudml.org/doc/257490. [8] C. He, Y. Li, H. Shan and W. Wang, On the index of unbalanced signed bicyclic graphs, Com- put. Appl. Math. 40 (2021), Paper No. 124, 14, doi:10.1007/s40314-021-01498-3. [9] T. Koledin and Z. Stanić, Connected signed graphs of fixed order, size, and number of negative edges with maximal index, Linear Multilinear Algebra 65 (2017), 2187–2198, doi:10.1080/ 03081087.2016.1265480. [10] M. Souri, F. Heydari and M. Maghasedi, Maximizing the largest eigenvalues of signed unicyclic graphs, Discrete Math. Algorithms Appl. 12 (2020), 2050016, 8, doi:10.1142/ s1793830920500160. [11] Z. Stanić, Signed graphs of small order, {www.matf.bg.ac.rs/˜zstanic/siggr. htm}. [12] Z. Stanić, Inequalities for Graph Eigenvalues, volume 423 of London Mathematical So- ciety Lecture Note Series, Cambridge University Press, Cambridge, 2015, doi:10.1017/ cbo9781316341308. [13] Z. Stanić, Perturbations in a signed graph and its index, Discuss. Math. Graph Theory 38 (2018), 841–852, doi:10.7151/dmgt.2035. [14] Z. Stanić, Integral regular net-balanced signed graphs with vertex degree at most four, Ars Math. Contemp. 17 (2019), 103–114, doi:10.26493/1855-3974.1740.803. [15] T. Zaslavsky, Matrices in the theory of signed simple graphs, in: Advances in Discrete Mathematics and Applications: Mysore, 2008, Ramanujan Math. Soc., Mysore, volume 13 of Ramanujan Math. Soc. Lect. Notes Ser., pp. 207–229, 2010, http://people.math. binghamton.edu/zaslav/Tpapers/index.html. ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P4.06 / 609–616 https://doi.org/10.26493/1855-3974.2614.44c (Also available at http://amc-journal.eu) The diameter of products of finite simple groups Daniele Dona * Einstein Institute of Mathematics, Edmond J. Safra Campus Givat Ram, The Hebrew University of Jerusalem, 9190401 Jerusalem, Israel Received 27 April 2021, accepted 14 January 2022, published online 11 August 2022 Abstract Following partially a suggestion by Pyber, we prove that the diameter of a product of non-abelian finite simple groups is bounded linearly by the maximum diameter of its factors. For completeness, we include the case of abelian factors and give explicit constants in all bounds. Keywords: Finite simple groups, diameter. Math. Subj. Class. (2020): 20F69, 20D06 1 Introduction An important area of research in finite group theory in the last decades has been the produc- tion of upper bounds for the diameter of Cayley graphs of such groups. For any finite group G, the maximum diameter over all Cayley graphs defined by symmetric sets of generators of G (i.e. sets S with S = S−1 and e ∈ S) is called the diameter of G. Arguably the best known conjecture in the area is Babai’s conjecture [1]: every non-abelian finite simple group G has diameter ≤ logk |G|, where k is an absolute constant; the conjecture is still open, despite great progress towards a solution both for alternating groups and for groups of Lie type. A more modest question is that of producing bounds for the diameter of direct products of finite simple groups, depending on the diameter of their factors. This is not an idle question, for bounds of this sort have been used more than once as intermediate steps towards the proof of bounds for simple groups themselves: Babai and Seress have done so in [2, Lemma 5.4], as well as Helfgott more than two decades later in [5, Lemma 4.13]. We improve on both results in the following theorem, which also features explicit constants. *The author was partially supported by the European Research Council under Programme H2020-EU.1.1., ERC Grant ID: 648329 (codename GRANT). He was also supported by the Israel Science Foundation Grant No. 686/17 of A. Shalev, and by the Emily Erskine Endowment Fund. E-mail address: daniele.dona@mail.huji.ac.il (Daniele Dona) cb This work is licensed under https://creativecommons.org/licenses/by/4.0/ 610 Ars Math. Contemp. 22 (2022) #P4.06 / 609–616 Theorem 1.1. Let n ≥ 2. Let G = ∏n i=1 Ti, where the Ti are finite simple groups. (a) If the Ti are all abelian (say G = ∏s j=1(Z/pjZ)ej , where the pj are distinct primes and ej ≥ 1), then: diam(G) < 2 3 max{ej |1 ≤ j ≤ s} s∏ j=1 pj . (b) If the Ti are all non-abelian, call d = max{diam(Ti)|1 ≤ i ≤ n}; then: diam(G) < 196 243 n3 max{CA, CL, CS}(4d+ 1) + d, where: CA =  max { 3, ⌊ m 2 ⌋} if there are alternating groups among the Ti and where m is their maximum degree, 0 if there are no alternating groups among the Ti, CL =  8(5r + 7) if there are groups of Lie type among the Ti and where r is their maximum untwisted rank, 0 if there are no groups of Lie type among the Ti, CS = { 6 if there are sporadic or Tits groups among the Ti, 0 if there are no sporadic or Tits groups among the Ti. (c) If there are abelian and non-abelian Ti, write G = GA × GNA, where GA collects the abelian factors and GNA collects the non-abelian ones; then: diam(G) ≤ dA + 4dNA, where dA = diam(GA), dNA = diam(GNA). The result of part (a) is known and elementary: see [2, Lemma 5.2], where the con- stant is marginally worse only due to the fact that sets of generators are not required to be symmetric (cfr. also [5, Lemma 4.14], which treats the case of G = (Z/pZ)e under this as- sumption). Part (c) is quite natural, given the different (in some sense, opposite) behaviour of abelian and non-abelian factors, as it can be readily observed in its short proof. Part (b) is where the novelty of the result resides. Dependence on the maximum of the diameter of the components, instead of dependence on their product as Schreier’s lemma (see Lemma 2.1) would naturally give us, was already established in [2, Lemma 5.4]: in that case, the diameter was bounded as O(d2), where the dependence of the constant on n was polynomial as in our statement. This result was improved in [5, Lemma 4.13] to O(d), but only in the case of alternating groups: this was done in part to fix a mistake in the use of the previously available result in Babai-Seress, which is why only alternating groups were considered, as permutation subgroups were the sole concern in both papers; a suggestion by Pyber, reported in Helfgott’s paper, points at the results by Liebeck and Shalev [8] as a way to prove a bound of O(d) for a product of arbitrary non-abelian finite simple groups. D. Dona: The diameter of products of finite simple groups 611 Indeed, the general approach that we follow in our proof owes its validity to [8, Theo- rem 1.6], although we do not explicitly use the statement of that theorem: rather, we closely follow the proof of [5, Lemma 4.13] and show that the same reasoning applies to groups of Lie type as well. The way that the lemma is related to Liebeck-Shalev is through the use of the fact that every element in Alt(m) is a commutator ([5, Lemma 4.12], first proved in [9, Theorem I]), which is essentially [8, Theorem 1.6] with w = xyx−1y−1 and a c that is just equal to 1 for Alt(m); the same can be said for all non-abelian finite simple groups (i.e., c = 1 in general) since Ore’s conjecture [10] was established to be true in [7], a fact yet unproved at the time of [8]. 2 Preliminaries Before we turn to the proof of Theorem 1.1, we will need a certain number of group- theoretic results. Lemma 2.1 (Schreier’s Lemma). Let G be a finite group, let N ⊴G, and let S be a set of generators of G with e ∈ S = S−1. Then S2d+1∩N generates N , where d = diam(G/N). Proof. This is a standard result dating back to Schreier [11], written in various fashions across the literature according to the needs of the user; let us prove here the present version. Calling π : G → G/N the natural projection, by definition we have π(S)d = G/N ; this equality means that Sd contains at least one representative for each coset gN in G. For any coset gN , choose a representative τ(g) ∈ Sd. Then, for any h ∈ N and any way to write h as a product of elements si ∈ S, we have: h = s1s2 . . . sk = (s1τ(s1) −1) · (τ(s1)s2τ(τ(s1)s2)−1) · · · (τ(τ(τ(. . .)sk−2)sk−1)sk). Each element of the form τ(x)siτ(τ(x)si)−1 is contained in S2d+1∩N , so the same can be said about the last element of the form τ(x)sk (since h itself is in N ); therefore S2d+1 ∩N is a generating set of N . Proposition 2.2 (Ore’s Conjecture). Let G be a finite non-abelian simple group. Then, for any g ∈ G, there exist g1, g2 ∈ G such that g = [g1, g2]. Proof. See [7], for references to previously known results and for the proof of the final case. Notice that, for any finite non-abelian simple group G, any nontrivial conjugacy class C must generate the whole G (because ⟨C⟩ would be a normal subgroup). This observation justifies the following definition. Definition 2.3. Let G be a finite non-abelian simple group. The conjugacy diameter cd(G) is the smallest m such that (C ∪C−1 ∪ {e})m = G for all nontrivial conjugacy classes C. We will need to have bounds for cd(G). Proposition 2.4. Let G be a finite non-abelian simple group. (a) If G is an alternating group of degree m, then cd(G) ≤ max { 3, ⌊ m 2 ⌋} . (b) If G is a group of Lie type of untwisted rank r, then cd(G) ≤ 8(5r + 7). 612 Ars Math. Contemp. 22 (2022) #P4.06 / 609–616 (c) If G is a sporadic group or the Tits group, then cd(G) ≤ 6. Proof. First of all, cd(G) is trivially bounded by definition by the covering number of G, which is defined as cn(G) = min{m|∀C ̸= {e}(Cm = G)}; therefore it suffices to give bounds for cn(G). For (a), see [4, Theorem 9.1] (our specific result is credited therein to a manuscript by J. Stavi). For (b), see [6, Theorem 1]. To prove (c), the sporadic groups all satisfy cn(G) ≤ 6: this inequality can be checked directly from [13, Table 1]; if G = 2F4(2)′ is the Tits group, we can show the same inequality using [13, Lemma 3] and the character values reported in the ATLAS of Finite Groups [3]. Let us also perform a side computation separately from the proof of the main theorem, so as not to bog down the exposition there. Lemma 2.5. Let n ≥ 2. Then: n−1∑ i=1 4⌈log2 i⌉ < 196 243 n3. Proof. Call m = ⌈log2(n − 1)⌉, and write n − 1 = 2m−1 + l, where 1 ≤ l ≤ 2m−1; ⌈log2 i⌉ = j for all i ∈ (2j−1, 2j ], hence we can rewrite the sum in the statement as: n−1∑ i=1 4⌈log2 i⌉ = 1 + m−1∑ j=1 4j2j−1 + 4ml = 1 2 + 1 2 8m − 1 7 + 4m ( 2log2(n−1) − 2m−1 ) = 3 7 + 4m2log2(n−1) − 3 7 8m = 3 7 + 22m ′ ( 1− 3 7 2m ′ ) (n− 1)3, where m′ = m − log2(n − 1) ∈ [0, 1). We have x2 ( 1− 37x ) ≤ 196243 for x ∈ [1, 2), and 3 7 < 196 243 (3n 2 − 3n+ 1) for all n ≥ 2, so the result is proved. 3 Proof of the main theorem Proof of Theorem 1.1(a). Let G = (Z/p1Z)e1×(Z/p2Z)e2×. . .×(Z/psZ)es , with primes p1 < p2 < . . . < ps; we have: G = A1A2 . . . As (3.1) (we are using multiplicative notation even if G is abelian) where the Ai are any sets such that: Ai,i =(Z/piZ)ei Ai,j =(0)ej (∀j < i) (3.2) where Ai,j is the projection of Ai to the j-th component of G. Let S be a set of generators of G with e ∈ S = S−1: {tp1...pi−1 |t ∈ S} ⊆ Sp1...pi−1 has elements that are all 0 on the first i− 1 components of G and that still generate the i-th one since (p1 . . . pi−1, pi) = 1; from now on, let us focus exclusively on the i-th component. (Z/piZ)ei is also a vector space over Z/piZ, so there must be ei generators that also form a basis: any element of the space can be written as a linear combination of those generators with coefficients in [ − ⌊ pi 2 ⌋ , ⌊ pi 2 ⌋] , which corresponds to a word of length ≤ ei ⌊ pi 2 ⌋ ; thus, D. Dona: The diameter of products of finite simple groups 613 each set Ai with the properties in (3.2) is covered in ei ⌊ pi 2 ⌋ p1 . . . pi−1 steps. This fact and (3.1) imply that G has diameter bounded by: s∑ i=1 ei ⌊pi 2 ⌋ i−1∏ j=1 pj  ≤ 1 2 max{ej |1 ≤ j ≤ s} s∏ j=1 pj · s∑ i=1  s∏ j=i+1 1 pj  . (3.3) The sum in (3.3) is maximized when each pj is the j-th prime number: for s = 1 the sum is 1 and for s = 2 it is bounded by 43 ; for s ≥ 3, we use ps ≥ 5 and pj ≥ 3 for all 1 < j < s, so that the sum is bounded by 1 + 15 1 1− 13 = 1310 . The result follows. Proof of Theorem 1.1(b). Calling Gj = ∏j i=1 Ti, we have natural projections πj : G = Gn → Gj and ρj1,j2 : Gj1 → Tj2 for any j1 ≥ j2. As in (3.1), we write G as a product of subsets Ai with ρn,i(Ai) = Ti and ρn,j(Ai) = {e} for all j < i, and our aim is to cover each one of them. Suppose that we have two subsets X1, X2 of G for which ρn,i(X1) = ρn,i(X2) = Ti for some fixed i ∈ {1, . . . , n} and that have ρn,j1(X1) = {e} = ρn,j2(X2) for all j1 ∈ I1, j2 ∈ I2, where I1, I2 are two subsets of indices in {1, . . . , n} \ {i}: then, the set X = {[x1, x2]|x1 ∈ X1, x2 ∈ X2} has ρn,i(X) = Ti by Proposition 2.2 (Ore’s conjecture) and ρn,j(X) = {e} for all j ∈ I1∪ I2. Now consider the set of indices I = {1, . . . , i−1}: if |I| > 1 we can partition I into two parts of size ⌊ |I| 2 ⌋ , ⌈ |I| 2 ⌉ , then partition each part I ′ with |I ′| > 1 into two new parts again of size ⌊ |I′| 2 ⌋ , ⌈ |I′| 2 ⌉ , and continue until we reach a subdivision where all sets have size 1; the tree of partitions that we constructed to reach this subdivision will have exactly ⌈log2 |I|⌉ layers. Notice that, given any two parts I1, I2 inside the tree, if we have two subsets X1, X2 (as described before) that are covered by a certain Sa, the resulting set X will be covered by S4a: this observation, together with the information about the layers, tells us that if we can cover sets Xi,j with ρn,i(Xi,j) = Ti and ρn,j(Xi,j) = {e} in a steps (for a fixed i > 1 and all j < i) then we are able to cover a set Ai defined as at the beginning of the proof in 4⌈log2(i−1)⌉a steps as well. Let us start now with a generating set S with e ∈ S = S−1 and fix two indices i ≥ j: πi(S) is a set of generators for Gi, and the set πi(S)2d+1 contains generators for the whole T1 × . . . × Tj−1 × {e} × Tj+1 × . . . × Ti = Gi ∩ ker(ρi,j) by Lemma 2.1 (Schreier’s lemma), where d is as in the statement. In particular, there is an element x ∈ S2d+1 with ρn,i(x) ̸= e and ρn,j(x) = e; by hypothesis ρn,i(Sd) = Ti, which means that there is a set S′ = {yxy−1|y ∈ Sd} ∪ {yx−1y−1|y ∈ Sd} ∪ {e} ⊆ S4d+1 with ρn,i(S′) = C ∪ C−1 ∪ {e} and ρn,j(S′) = {e}, where C is the conjugacy class of ρn,i(x). By Proposition 2.4, ρn,i(S′max{3,⌊ mi 2 ⌋}) = Ti if Ti = Alt(mi), ρn,i(S′8(5ri+7)) = Ti if Ti is of Lie type of untwisted rank ri, and ρn,i(S′6) = Ti otherwise; in all three cases, the projection to Tj is still {e}, therefore we managed to cover a set Xi,j of the aforementioned form. A set A1 is reached in d steps, hence the final count for the whole G following the reasoning above is: diam(G) ≤ d+ n∑ i=2 4⌈log2(i−1)⌉xi(4d+ 1), where xi is either max { 3, ⌊ mi 2 ⌋} , 8(5ri + 7) or 6, accordingly. The result follows by Lemma 2.5. 614 Ars Math. Contemp. 22 (2022) #P4.06 / 609–616 A note on the connection between the proof given above and [8]. As mentioned before, Pyber pointed at [8] as a way to prove linear dependence on d for products of arbitrary non- abelian finite simple groups. In particular, [8, Theorem 1.6] seems to fit the bill: it states that for any word w that is not a law in a finite simple group T there is cw ∈ N, depending on w but not on T , such that any element of T can be written as a product of at most cw values of w. We use this property, in disguise, when we want to pass from two subsets being indentically e at indices I1, I2 and filling an entire component Ti to a third subset that also fills the same component and is e for the whole I1 ∪ I2: the creation of the new subset is made possible by taking cw values of a word w, so that Ti remains filled, where w has two distinct letters x1, x2 and presents the same number of xi and x−1i for i ∈ {1, 2}, so that when any one xi is equal to e on a given factor of the product G the result is e on that factor; in our case, w was the shortest nontrivial word with these characteristics, namely the commutator [x1, x2] = x1x2x−11 x −1 2 (not a law for any non-abelian group), and cw = 1 by Ore’s conjecture. In this sense w = [x1, x2] is also computationally the best word we can expect, for it yields the lowest possible value of |w|cw, the 4 that we find in Lemma 2.5. Proof of Theorem 1.1(c). Define the two projections πA, πNA in the obvious way; for any generating set S of G, by definition there is a subset XA ⊆ SdA with πA(XA) = GA and there is a subset XNA ⊆ SdNA with πNA(XNA) = GNA, and then: G = XA[XNA, XNA] ⊆ SdA+4dNA , again by the fact that [T, T ] = T for non-abelian finite simple groups by Ore’s conjecture and [T, T ] = {e} for abelian groups. 4 Concluding remarks One could wonder how tight the inequalities in Theorem 1.1 are. The results are essen- tially in line with what is generally expected from the behaviour of the diameter of fi- nite groups. The abelian case is tight up to constant: for the group G(x) = ∏ p≤x Z/pZ (nontrivial for x ≥ 2) one generator s = (1, 1, . . . , 1) is enough, and then the diameter of Cay(G(x), {s, s−1, e}) is 12 |G(x)|; the fact that abelian groups behave in the worst possible way, i.e. linearly in the size of the group, should not be a surprise for anyone. The non-abelian bound of case (b) also matches what is anticipated in general. Babai’s conjecture posits a polylogarithmic bound on the diameter of finite simple groups: the natural extension to direct products of such groups would suggest a bound of the form nkd, which is exactly what we have obtained. Case (c) also fits into the same idea, as a product |G| = |GA||GNA| becomes a sum of the corresponding diameters. The dependence on d in Theorem 1.1(b) is almost best possible by definition (we cannot drop the “almost”, as m, r are not independent from d). It would be more interesting to understand which power of n is the correct one: here we have proved Om,r,d(n3), and we can quickly show that the bound is Ωm,r,d(n), as illustrated in the following example. Example 4.1. If G = (Alt(m))n then diam(G) = Ω(m2n). We prove it for m ≥ 5 odd and n even, but the proof is analogous for the general case. Consider the two permutations σ = (1 2 3 . . . m) and τ = (1 2 3 . . . m − 2); they D. Dona: The diameter of products of finite simple groups 615 generate Alt(m), and the elements: s0 = (σ, σ, . . . , σ, σ), s1 = (τ, σ, . . . , σ, σ), s2 = (σ, τ, . . . , σ, σ), . . . sn = (σ, σ, . . . , σ, τ) generate G. Let S = {e} ∪ {si, s−1i }0≤i≤n: to prove the lower bound on the diameter of G, we construct a function f : G → N such that there are two elements g1, g2 ∈ G with |f(g1)− f(g2)| large and such that |f(g)− f(gs)| is small for any g ∈ G, s ∈ S; this is a known technique to prove lower bounds for the diameter of Sym(m), as shown for instance in [12, Proposition 3.6]. Call c(g, i, j) = (g(i))(j) the image of j ∈ {1, . . . ,m} under the i-th component of g ∈ G, for 1 ≤ i ≤ n; define: f(g) = m∑ j=1 n∑ i=1 ||c(g, i+ 1, j)− c(g, i, j)||Z/mZ, where ||a||Z/mZ = min{a,m−a} (in the case i = n, c(g, n+1, j) means c(g, 1, j)). First, f(e) = 0; also, if we call em the identity element in Alt(m) and η = ( 1 m+12 ) ( 2 m+32 ) · · ·( m−1 2 m− 1 ) , for g ∈ G that has em at all odd components and η at all even ones we have f(g) = 12 (m − 1) 2n. Finally, notice that σ simply adds 1 modulo m to all the elements of {1, . . . ,m}, so that f(g) = f(gs±10 ), while τ is defined so that it adds 1 for m − 3 elements, adds 3 (modulo m) for one element and fixes two elements, which means that |f(g)− f(gs±1i )| ≤ 10; these facts taken together imply that diam(G,S) ≥ 120 (m− 1) 2n. The correct (or even expected) order of magnitude for a bound of the form diam(G) = Om,r(n kd) for a generic product G is not known to the author, besides knowing that 1 ≤ k ≤ 3 by Theorem 1.1 and Example 4.1. ORCID iDs Daniele Dona https://orcid.org/0000-0001-7966-3357 References [1] L. Babai and A. Seress, On the diameter of Cayley graphs of the symmetric group, J. Comb. Theory Ser. A 49 (1988), 175–179, doi:10.1016/0097-3165(88)90033-7. [2] L. Babai and A. Seress, On the diameter of permutation groups, Eur. J. Comb. 13 (1992), 231– 243, doi:10.1016/s0195-6698(05)80029-0. [3] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups, Clarendon Press, Oxford (UK), 1985. [4] Y. Dvir, Covering properties of permutation groups, in: Z. Arad and M. Herzog (eds.), Products of Conjugacy Classes in Groups, Springer-Verlag, Berlin (Germany), pp. 197–221, 1973, doi: 10.1007/bfb0072288. 616 Ars Math. Contemp. 22 (2022) #P4.06 / 609–616 [5] H. A. Helfgott, Growth in linear algebraic groups and permutation groups: towards a uni- fied perspective, in: C. M. Campbell, C. W. Parker, M. R. Quick, E. F. Robertson and C. M. Roney-Dougal (eds.), Groups St Andrews 2017 in Birmingham, Cambridge University Press, Cambridge, volume 455 of London Mathematical Society Lecture Note Series, pp. 300–345, 2019. [6] R. Lawther and M. W. Liebeck, On the diameter of a Cayley graph of a simple group of Lie type based on a conjugacy class, J. Combin. Theory Ser. A 83 (1998), 118–137, doi:10.1006/ jcta.1998.2869. [7] M. W. Liebeck, E. A. O’Brien, A. Shalev and P. H. Tiep, The Ore conjecture, J. Eur. Math. Soc. (JEMS) 12 (2010), 939–1008, doi:10.4171/jems/220. [8] M. W. Liebeck and A. Shalev, Diameters of finite simple groups: sharp bounds and applica- tions, Ann. of Math. (2) 154 (2001), 383–406, doi:10.2307/3062101. [9] G. A. Miller, On the commutators of a given group, Bull. Am. Math. Soc. 6 (1899), 105–109, doi:10.1090/s0002-9904-1899-00683-9. [10] O. Ore, Some remarks on commutators, Proc. Am. Math. Soc. 2 (1951), 307–314, doi:10.2307/ 2032506. [11] O. Schreier, Die Untergruppen der freien Gruppen, Abh. Math. Semin. Univ. Hambg. 5 (1927), 161–183, doi:10.1007/bf02952517. [12] Y. S. Tan, On the diameter of Cayley graphs of finite groups, 2011, {http://www.math. uchicago.edu/˜may/VIGRE/VIGREREU2011.html}. [13] I. Zisser, The covering numbers of the sporadic simple groups, Israel J. Math. 67 (1989), 217– 224, doi:10.1007/bf02937296. ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P4.07 / 617–636 https://doi.org/10.26493/1855-3974.2740.7ab (Also available at http://amc-journal.eu) S2 coverings by isosceles and scalene triangles – adjacency case II* Catarina P. Avelino † Center of Mathematics of the University of Minho – UTAD Pole (CMAT-UTAD), University of Trás-os-Montes e Alto Douro, Vila Real, Portugal and Center for Computational and Stochastic Mathematics (CEMAT), University of Lisbon (IST-UL), Portugal Altino F. Santos Center of Mathematics of the University of Minho – UTAD Pole (CMAT-UTAD), University of Trás-os-Montes e Alto Douro, Vila Real, Portugal Received 4 April 2019, accepted 6 January 2022, published online 11 August 2022 Abstract The aim of this paper is to complete the study and classification of spherical f-tilings by scalene triangles T and isosceles triangles T ′ within a subclass defined by the adjacency of the lower side of T and the longest side of T ′. It consists of eight families of f-tilings (two families with one continuous parameter, one family with one discrete parameter and one continuous parameter, and five families with one discrete parameter). We also analyze the combinatorial structure of all these families of f-tilings, as well as the group of symmetries of each tiling; the transitivity classes of isogonality are included. Keywords: Dihedral f-tilings, combinatorial properties, spherical trigonometry. Math. Subj. Class. (2020): 52C20, 52B05, 20B35 *This research was partially financed by Portuguese Funds through FCT (Fundação para a Ciência e a Tec- nologia) within the projects UIDB/00013/2020 and UIDP/00013/2020 of CMAT-UTAD, Center of Mathematics of University of Minho, and projects UIDB/04621/2020 and UIDP/04621/2020 of CEMAT/IST-ID, Center for Computational and Stochastic Mathematics, Instituto Superior Técnico, University of Lisbon. †Corresponding author. E-mail addresses: cavelino@utad.pt (Catarina P. Avelino), afolgado@utad.pt (Altino F. Santos) cb This work is licensed under https://creativecommons.org/licenses/by/4.0/ 618 Ars Math. Contemp. 22 (2022) #P4.07 / 617–636 1 Introduction A folding tessellation or folding tiling (f-tiling, for short) of the sphere S2 is an edge-to- edge finite polygonal tiling τ of S2 such that all vertices of τ satisfy the angle-folding relation, i.e., each vertex is of even valency and the sums of alternating angles around each vertex are equal to π. F-tilings are intrinsically related to the theory of isometric foldings of Riemannian man- ifolds, introduced by Robertson [10] in 1977. In several situations (beyond the scope of this paper), the edge-complex associated to a spherical f-tiling is the set of singularities of some spherical isometric folding. The classification of f-tilings was initiated by Breda [2], with a complete classification of all spherical monohedral (triangular) f-tilings. Afterwards, in 2002, Ueno and Agaoka [11] have established the complete classification of all triangular monohedral tilings of the sphere (without any restrictions on angles). Curiously, the triangular tilings of even valency at any vertex are necessarily f-tilings. Dawson has also been interested in special classes of spherical tilings, see [3, 4, 5], for instance. Spherical f-tilings by two noncongruent classes of isosceles triangles have recently been obtained [6, 7]. The study of dihedral triangular f-tilings involving scalene triangles is clearly more unwieldy and was initiated in [1]. In this paper we complete the classification of spherical f-tilings by scalene triangles T and isosceles triangles T ′ resulting from the adjacency of the lower side of T and the longest side of T ′. From now on, (i) T denotes a spherical scalene triangle with internal angles α > β > γ and side lengths a > b > c; (ii) T ′ denotes a spherical isosceles triangle with internal angles (δ, δ, ε), δ ̸= ε, and side lengths (d, d, e), as illustrated in Figure 1. d e e T g b c a b T ‘ a d d d Figure 1: A spherical scalene triangle, T , and a spherical isosceles triangle, T ′. We shall denote by Ω(T, T ′) the set, up to isomorphism, of all dihedral folding tilings of S2 whose prototiles are T and T ′ in which the lower side of T is equal to the longest side T ′. Taking into account the area of the prototiles T and T ′, we have α+ β + γ > π and 2δ + ε > π. As α > β > γ, we also have α > π3 . In [8] it was established that any τ ∈ Ω (T, T ′) has necessarily vertices of valency four. C. P. Avelino and A. F. Santos: S2 coverings by isosceles and scalene triangles . . . 619 We begin by pointing out that any element of Ω (T, T ′) has at least two cells congruent to T and T ′, respectively, such that they are in adjacent positions and in one and only one of the situations illustrated in Figure 2. 1 T b g a I. e d dT ‘ 2 T b g a I. e d dT ‘d c= d c= T b g a II e d d T ‘e c= Case :I d > e Case :II d < e Figure 2: Distinct cases of adjacency. In this paper we will consider the second case of adjacency. The next section con- tains the main result of this paper. In Section 2 we describe the eight families of spherical f-tilings that we may obtain from the second case of adjacency (Figure 2-II). The combi- natorial structure of these tilings, the classification of the group of symmetries and also the transitivity classes of isogonality are presented. The proof of the main result consists of a long and exhaustive method and it is presented in Section 3. 2 Main result - Elements of Ω (T, T ′) in the case of Adjacency II Theorem 2.1. Let T and T ′ be a spherical scalene triangle and a spherical isosceles triangle, respectively, such that they are in adjacent positions as illustrated in Figure 2-II. Within this case, the f-tilings of Ω (T, T ′) are Lβ , Dkε (k ≥ 4), Mγ , N k (k ≥ 6), Pk (k ≥ 3), Qk (k ≥ 4), Rk (k ≥ 6) and Sk (k ≥ 7), that satisfy, respectively: (i) α+ δ + β = π, ε = π2 , γ = π 3 , where α and β satisfy sin2(α+ β) (1 + 2 cos(α− β)) = 2 sinα sinβ and β ∈ ( π 3 , arccos √ 6 6 ) ; (ii) α+ δ = π, δ + β + ε = π, kγ = π, δ = δ1k(ε), ε ∈ ( εmin, (k−1)π k ) ,k ≥ 4, where δ1k(ε) = arctan 2 sin ε cos2 ε2 cos πk − cos2 ε and εmin = arccos √ 1 + 8 cos πk − 1 4 ; (iii) α+ δ = π, ε = π2 , β + δ + γ = π, δ = γ and γ ∈ ( π 4 , π 3 ) ; 620 Ars Math. Contemp. 22 (2022) #P4.07 / 617–636 (iv) α+ δ = π, ε = π2 , β + 3δ = π, kγ = π and δ = δ 2 k = arccos √ 1 2 cos π k , k ≥ 6; (v) α+ δ = π, ε = π2 , 2β + 2δ = π, β + δ + kγ = π and δ = δ 3 k = arctan ( sec π2k ) , k ≥ 3; (vi) α+ δ = π, ε = π2 , 2β + 2δ + γ = π, β + δ + kγ = π, δ = δ 4 k, k ≥ 4, where δ4k = arctan ( sin (k − 1)π 2k − 1 sec π 2k − 1 ) ; (vii) α + δ = π, ε = π2 , 2β + 2δ = π, kγ = π and α = α 2 k = 2arctan(cos π k +√ 1 + cos2 πk ), k ≥ 6; (viii) α+ δ = π, ε = π2 , 2β + 2δ + γ = π, kγ = π and δ = δ 5 k, k ≥ 7, where δ5k = arctan ( sin (k − 1)π 2k sec π k ) . For each family of f-tilings we present the distinct classes of congruent vertices in Figure 3 (including the respective number of vertices in each tiling). Particularizing suitable values for the parameters involved in each case, the correspond- ing 3D representations of these families of f-tilings are given in Figure 4. In each case, we present two perspectives in order to provide a more effective visualization of each f-tiling’s combinatorial structure. Regarding the f-tiling P k, k ≥ 3, it can be observed that, if we consider the great circle that contains the four vertices surrounded by (β, δ, δ, β, γ, γ, ..., γ) (marked in red) as the equator line and rotating the southern hemisphere 90 degrees (around the “vertical” axis) we obtain the f-tiling R2k. Also, it is interesting to relate the mono- hedral edge-to-edge tilings TI16n+8 and I8n described by Ueno and Agaoka in [11] with the families of f-tilings Qk, k ≥ 4, and Sk, k ≥ 7, obtained by subdividing the prototypes in the monohedral tilings into two triangles satisfying the conditions of Figure 2-II. Seeing from another perspective, we obtain TI16n+8 and I8n eliminating the vertices surrounded by (α, α, δ, δ) (marked in green) and two suitable edges emanating from those vertices of Qk, k ≥ 4, and Sk, k ≥ 7, respectively. C. P. Avelino and A. F. Santos: S2 coverings by isosceles and scalene triangles . . . 621 g g ee d a g g g g g e e g g g b g g b 6 8 2 2 D k e M P k Q k R k e e aa d d d a d d b b d b g d g b g g 24 2k 2k 2 dd b db d d d 4 2 2k k Lb g N k ee e e ee e e ee e e ee e e ee e e aa d d 4 aa d d 2k aa d d 4k aa d d 2k aa d d g g g g g g g g g g g g g g g g b d d g g b b d b bb d d d 4 b g g g d d b g g g g g d d b b d d g g b b g g g g g g g g g g g gb d b bb d d d 4 2k 2k 2( 1)-k k 2(2 1)-k 4( 1)-k4(2 1)-k 2S k ee e e aa d d 4k g g g g g g g g g g g g 2k 2k d d b b d d g g b b Figure 3: Distinct classes of congruent vertices. 622 Ars Math. Contemp. 22 (2022) #P4.07 / 617–636 a Lβ b Mγ c D4ε d D5ε e N 6 f N 7 g P3 h P4 Figure 4: Elements of Ω(T, T ′) in the case of adjacency II. C. P. Avelino and A. F. Santos: S2 coverings by isosceles and scalene triangles . . . 623 i Q4 j Q5 k R6 l R7 m S7 n S8 Figure 4: Elements of Ω(T, T ′) in the case of adjacency II. The combinatorial structure of the classes of spherical f-tilings mentioned in Theo- rem 2.1, including the symmetry groups, is summarized in Table 1 (the analysis of the symmetry groups is similar to that applied in previous articles, e.g. [9]). Our notation is as follows: • |V | is the number of distinct classes of congruent vertices; • N1 and N2 are, respectively, the number of triangles congruent to T and T ′, respec- tively; • G(τ) is the symmetry group of each tiling τ ∈ Ω (T, T ′) and the index of isogonality for the symmetry group is denoted by #isog.; • Cn is the cyclic group of order n; • V ≃ C2 × C2 is the Klein group; • Dn is the nth dihedral group (it consists of n rotations and n reflections); • O is the chiral group with 24 elements; 624 Ars Math. Contemp. 22 (2022) #P4.07 / 617–636 • i(k) = { 3k 2 + 1 if k even 3k+1 2 + 1 if k odd. f-tiling α β γ δ ε |V | N1 N2 G(τ) #isog. Lβ α(β) ( π 3 , arccos √ 6 6 ) π 3 π − α − β π 2 3 48 24 O 3 Dkε , k ≥ 4 π − δ π − δ − ε πk δ 1 k(ε) (εmin, εmax) 3 4k 4k D2k 3 Mγ π − γ π − 2γ ( π 4 , π 3 ) γ π2 3 8 8 V 3 Nk, k ≥ 6 π − δ π − 3δ πk δ 2 k π 2 4 4k 8k D2k 4 Pk, k ≥ 3 π − δ π2 − δ π 2k δ 3 k π 2 4 8k 8k C2 × C2 × C2 i(k) Qk, k ≥ 4 π − δ (k−1)π2k−1 − δ π 2k−1 δ 4 k π 2 4 14k 14k V 4k − 2 Rk, k ≥ 6 α2k π2 − δ π k π − α π 2 4 4k 4k C2 × Dk 4 Sk, k ≥ 7 π − δ (k−1)π2k − δ π k δ 5 k π 2 4 8k 8k D2k 4 Table 1: Combinatorial structure of the dihedral f-tilings of S2 by scalene triangles T and isosceles triangles T ′ performed by the lower side of T and the longest side of T ′ in the case of adjacency II. 3 Proof of Theorem 2.1 In order to better understand the structure of each tiling and due to the complexity of a global planar representation, in the following proof some f-tilings τ are illustrated only by a fundamental region F that generates τ by successive reflections and rotations of F . Com- paring the fundamental region F with its associated f-tiling τ (in Figure 4), it becomes clear how it is generated. In two of the situations (tilings Qk and Sk), instead of a fundamental region, we illustrate planar representations that correspond to a half of the f-tilings. In the case of adjacency II, any element of Ω (T, T ′) has at least two cells congruent to T and T ′, respectively, such that they are in adjacent positions and in one and only one of the situations illustrated in Figure 2. After certain initial assumptions are made, it is usually possible to deduce sequentially the nature and orientation of most of the other tiles. Eventually, either a complete tiling or an impossible configuration proving that the hypothetical tiling fails to exist is reached. In the diagrams that follow, the order in which these deductions can be made is indicated by the numbering of the tiles. For j ≥ 2, the location of tiling j can be deduced directly from the configurations of tiles (1, 2, . . . , j−1) and from the hypothesis that the configuration is part of a complete tiling, except where otherwise indicated. Observe that we have ε > π3 (since we are considering the case of adjacency II). Also, as e = c and using spherical trigonometric formulas, we get cos γ + cosα cosβ sinα sinβ = cos ε+ cos2 δ sin2 δ . (3.1) Proof of Theorem 2.1. Suppose that any element of Ω (T, T ′) has at least two cells con- gruent, respectively, to T and T ′, such that they are in adjacent positions as illustrated in Figure 2-II. With the labeling of Figure 5a, we have θ1 ∈ {ε, δ}. It is easy to verify that θ1 must be δ. In fact, if θ1 = ε, v1 cannot have valency four (see side lengths), α + ε + ρ > π, C. P. Avelino and A. F. Santos: S2 coverings by isosceles and scalene triangles . . . 625 b g a e1 2 d d v 1 q1 a b g a e1 2 d d v 1 q2 d e 3dq1 b Figure 5: Local configurations. ∀ρ ∈ {α, β, δ, ε}, and if α + ε+ kγ = π, k ≥ 1, an incompatibility between sides cannot be avoided. Now, at vertex v1 (see Figure 5b) we must have α+ δ < π or α+ δ = π. 1. Suppose firstly that α + δ < π. If θ2 = δ and ε + δ = π (Figure 6a), we reach a contradiction at vertex v2, as ε+ β + ρ > π, for all ρ ∈ {α, β, γ, δ, ε}. In fact, taking into account the side lengths, v2 cannot have valency four and also observe that ε + β + ρ1 ≥ α+ β + γ, ρ1 ∈ {α, β, γ}, and ε+ β + ρ2 > ε+ δ = π, ρ2 ∈ {δ, ε}. b g a e1 2 d d v 1 d e 3 dq2 e d 4d d 5 d e v 2 a b g a e1 2 d d v 1 d e 3 q2 e d 4d d 5 d v 2 e d g a 10 9 6 g b v 3 a 8 7 a g a b g b g b a b b Figure 6: Local configurations. On the other hand, if θ2 = δ and ε + δ < π, we must have ε + δ + ρ ≤ π, for some ρ ∈ {α, β, γ}. If ρ = α, we get ε > δ > α > β > γ; but then ε+ δ+α > α+β+ γ > π, which is not possible. If ρ = β, we obtain δ > β and α > ε, which implies α+ δ+ ρ̄ > π, ∀ρ̄, which is a contradiction. Finally, due to an incompatibility between sides, it is not possible to have ε+ δ + kγ = π, k ≥ 1. Therefore, θ2 = ε and, due to the side lengths, we must have ε+ ε = π, and obviously α > δ, with δ ∈ ( π 4 , π 2 ) . 1.1 If α ≥ ε, at vertex v1 (Figure 5b) we must have α + δ + kγ = π, with k ≥ 1, and α > β > δ > γ. The last configuration extends to the one illustrated in Figure 6b. If, at vertices v2 and v3, we have (i) β + δ + δ = π, we reach a vertex surrounded by six angles δ, implying δ = π3 = β, which is not possible as β > δ; 626 Ars Math. Contemp. 22 (2022) #P4.07 / 617–636 (ii) β + δ + β = π, we obtain the configuration illustrated in Figure 7a. Taking into account the edge lengths and the fact that β > δ > π4 , at vertex v4 we reach a contradiction. Note that it is easy to conclude that is not possible to include angles γ in the previous sums. b g a e1 2 d d v 1 d e 3 q2 e d 4d d 5 d e v 2 e d g a 10 d 9 6 g b v 3 a 8 7 a g a b g b g b a b b b b b b 11 12 13 14 a a g g a a g g 15 16 v 4g g a a b b 17 18 a g b g a d 19 20 21 22d d d d d d e e ee 23 24 d d d d e a b g a e1 2 d d v 1 q3 d e 3dq1 q2 e d 4 d 5 de d v 2 b Figure 7: Local configurations. 1.2 Suppose now that α < ε. 1.2.1 If δ ≥ γ, with the labeling of Figure 7b, we have θ3 ∈ {δ, γ}. Additionally is important to note that ε = π2 > α > β > δ ≥ γ, δ > π 4 and β + γ > π 2 . If θ3 = δ, we obtain the configuration of Figure 8a. Observe that θ4 cannot be δ, as δ + δ + δ < δ + δ + α = π and δ + δ + δ + ρ > π, with ρ ∈ {α, β, δ, ε}; ρ2 cannot be γ due to an incompatibility between sides. Moreover, θ4 cannot be β, as δ + δ + β < π and δ + δ + β + γ > π. b g a e1 2 d d d e 3dq1 q2 e d 4 d 5 de d v 1 g a 9 6 b 8 7 a g a g b b dq3 e d d d e 10 ee d d d d 11 12 q4 v 2 a b g a e1 2 d d d e 3dq1 q2 e d 4 d 5 de d v 1 g a 9 6 b 8 7 a g a g b b dq3 e d d d e 10 ee d d d d 11 12 q4 v 2 15 13 14 b b b g a a g g a 16 g b a v 3 b Figure 8: Local configurations. Now, at vertex v2 we have necessarily β + δ + β = π or β + δ + kγ = π, with k ≥ 2. These cases lead to the configurations illustrated in Figure 8b and Figure 9a, respectively. In both cases, at vertex v3 we reach a contradiction. In fact, due to the edge and angles lengths there is no way to satisfy the angle-folding relation around this vertex. C. P. Avelino and A. F. Santos: S2 coverings by isosceles and scalene triangles . . . 627 b g a e1 2 d d d e 3dq1 q2 e d 4 d 5 de d v 1 g a 9 6 b 8 7 a g a g b b dq3 e d d d e 10 ee d d d d 11 12 q4 v 2 1513 14 b a a g a 18 g b a v 3 g b g g g b b b a a 17 16 a b g a e1 2 d d d e 3dq1 q2 e d 4 d 5 de d v 1 g 12 6 b 8 7 a g a g b b q3 16 e d d 14 13 15 g g b g a a g b b b a a a v 2 10 9 b b b g a a g g 11 a a a b b g 17g 18 d 19 a g b 20 e d v 3 b Figure 9: Local configurations. If θ3 = γ (Figure 7b), at vertex v2 we have necessarily β+δ+β = π or β+δ+δ = π. These cases lead to the configurations illustrated in Figure 9b and Figure 10a, respectively. In the first case, at vertex v3 we have β + δ + δ < π and β + δ + δ + ρ > π, for all ρ ∈ {α, β, γ, δ, ε}. In the last case, at vertex v3 we also reach a contradiction, as δ = π3 implies β = π3 and, due to the edge and angles lengths, it is not possible that this vertex has valency greater than three. b g a e1 2 d d d e 3dq1 q2 e d 4 d 5 de d v 1 g 12 6 8 7 a b q3 d 14 13g g a a g b b b a v 2 9 10 b g a 11 e dd d d d ee e d d v 3 a b g a e1 2 d d d e 3dq1 q2 e d 4 d 5 de d v 1 g 6 8 7 a b a g b b v 2 a g b Figure 10: Local configurations. 1.2.2 If δ < γ, then ε = π2 > α > β > γ > δ > π 4 . At vertex v1 (Figure 7b) we must have one of the following situations: (i) α+ δ+α = π; in this case (Figure 10b), there is no way to satisfy the angle-folding relation around vertex v2. (ii) α+ δ+ δ = π; as we can observe in Figure 11a, an incompatibility between sides at vertex v3 cannot be avoided. (iii) α+ δ+ γ = π; in this case (Figure 11b), there is no way to satisfy the angle-folding relation around vertex v4. 628 Ars Math. Contemp. 22 (2022) #P4.07 / 617–636 b g a e1 2 d d d e 3dq1 q2 e d 4 d 5 de d v 1 g 6 8 7 a b v 2 dd d d ee 13 b g a 10 e e d d d d 9 a a b b g g 11 12 v 3 a b g a e1 2 d d v 1 d e 3 q2 e d 4d d 5 d v 2 e d g a 10 d 15 6 g b v 3 a 8 7 a g a b g b g b a b b b b b b 9 11 14 16 a a g g a a g g 12 13 v 4g g a a b b 21 18 a g b g a d 20 22 17 19d d d d d d e e ee b Figure 11: Local configurations. (iv) α+δ+β = π; in this situation, the last configuration extends to the one illustrated in Figure 12a. Now, at vertex v4 we have necessarily γ+γ+ρ = π, with ρ ∈ {α, β, γ}. It is easy to verify that the two first cases lead to impossibilities. The last case (ρ = γ) leads to a continuous family of f-tilings formed by 72 tiles. Due to the large dimension of the corresponding planar representation, we only illustrate its eighth fundamental region in Figure 12b. b g a e1 2 d d v 1 d e 3 q2 e d 4d d 5 d v 2 e d g a 10 15 6 g b v 3 a 8 7 a g a b g b g b a b b b b b 9 11 14 16 a a g g a a g g 12 13 v 4 g g a a b b a b g e d de d d d d a ab b b b g g a a a a g gg e b b A eighth fundamental region of Lβ Figure 12: Local configurations. We denote this continuous family of f-tilings by Lβ , where α+ δ + β = π, 2ε = π and 3γ = π. Using Equation (3.1), we get sin2(α+ β) (1 + 2 cos(α− β)) = 2 sinα sinβ, with π 3 < β < arccos √ 6 6 . 3D representations of Lβ are illustrated in Figure 4a. C. P. Avelino and A. F. Santos: S2 coverings by isosceles and scalene triangles . . . 629 2. Suppose now that α + δ = π. We have α > δ and α > π2 . In fact, if α ≤ δ, we would have ε > δ ≥ α > β > γ, with δ ≥ π2 , and consequently ε+ θ2 > π, θ2 ∈ {ε, δ}. 2.1 If θ2 = δ (Figure 5b), then it is a straightforward exercise to prove that γ = πk , for some k ≥ 4, and the complete planar representation derives uniquely as illustrated in Figure 13. b g a e 1 2 d d d d e 3 a b g 4 d e 5 d d e 6 v 2 a g 7 8 g a d d d d e 9 a g b 10 13 d 11 a g a g b 12 v 3 d d 14 g a b 15 d e a 16 g b 17 q2 d d d b 18 20 d d g b a 21 22 g 19 g a g ba q1 d eb b e 25 e 26 e ed d d d e b g a g b a b g a a b b 28 29 23 27 d e d d d e d e e d d 24 30 31 Figure 13: Planar representation of Dkε , k ≥ 4. This family of f-tilings is denoted by Dkε , where α+ δ = π, δ+β+ε = π and kγ = π, with k ≥ 4. Using (3.1), we get δ = δk(ε) = arctan 2 sin ε cos2 ε2 cos πk − cos2 ε , k ≥ 4, with ε ∈ ( εmin, (k−1)π k ) , where εmin = arccos √ 1+8 cos πk −1 4 . 3D representations of D 4 ε and D5ε are given in Figures 4c – 4d. 2.2 If θ2 = ε, we have β ≥ δ or β < δ. 2.2.1 If β ≥ δ, we have α > π2 = ε > δ > π 4 and the last configuration extends to the one illustrated in Figure 14a. Now, we have θ3 ∈ {β, δ, γ}. 2.2.1.1 If θ3 = β, at vertex v2 we must have δ + β + β + kγ = π, with k ≥ 0. It is easy to verify that k has to be zero, giving rise to the configuration of Figure 14b. At vertex v3 we obtain α + k̄γ = π, with k̄ ≥ 2. Taking into account Equation (3.1) and the relations between angles, we get 2 cos δk = cos δ csc δ 2 . Consequently, we obtain sin δ ≤ cos δ and δ ≤ π4 , which is not possible. 2.2.1.2 If θ3 = δ, at vertex v2 we have δ + β + δ + kγ = π, with k ≥ 0. It is a straightforward exercise to show that (i) if k = 0, although a complete configuration is achieved, it leads to β = γ = π3 , which is not possible; (ii) if k = 1, again a complete configuration is achieved, with δ = π3 = β + γ, which is a contradiction; (iii) the case k > 1 leads to an incompatibility between sides. 2.2.1.3 If θ3 = γ, at vertex v2 we must have one of the following situations: 630 Ars Math. Contemp. 22 (2022) #P4.07 / 617–636 b g a e1 2 d d d e 3dq1 q2 e d 5 d 6 de d v 1 g 4 a b q3 v 2 a b g a e1 2 d d d e 3dq1 q2 e d 5 d 6 de d v 1 g 4 a b q3 7 b g g a b b a a g 9 8 11 v 2 g a b 10 a b g v 3 b Figure 14: Local configurations. (i) β+δ+kγ = π, k ≥ 1; the case k > 1 leads to δ = β = π3 , which is not possible, and so k = 1. In this case we obtain the planar representation of Figure 15. We denote this family of f-tilings by Mγ , where α + δ = π, β + δ + γ = π and γ ∈ ( π 4 , π 3 ) . Using Equation (3.1), we get δ = γ. b g a e1 2 d d d e 3dq1 q2 e d 5 d 6 de d v 1 g 4 a b q3 7 g g a b b a a g 9 811 v 2 g a b 10 a b gv 3 b d d e 12 d d e 13 e d d 14 e d d 15 g b a Figure 15: Planar representation of Mγ . 3D representations of Mγ , γ ∈ ( π 4 , π 3 ) , are illustrated in Figure 4b. (ii) β+δ+β+kγ = π, k ≥ 1; as we can observe in Figure 16a, we reach an impossibility as there is no way to complete the sum of alternate angles around vertex v3. b g a e1 2 d d d e 3dq1 q2 e d 5 d 6 de d v 1 g 4 a b q3 7 g g a b b a a g 9 8 11 v 2 a b 10 a b g v 3 b 12 13 b g g a a g b a b g a e1 2 d d d e 3dq1 q2 e d 5 d 6 de d v 1 g 4 a b q3 b Figure 16: Local configurations. C. P. Avelino and A. F. Santos: S2 coverings by isosceles and scalene triangles . . . 631 (iii) β + δ + δ + kγ = π, k ≥ 1; in this case there is no way to satisfy the angle-folding relation around vertex v2. 2.2.2 If β < δ (Figure 5b), it is easy to conclude that α > π2 = ε > δ > β > γ. Now, with the labeling of Figure 16b we have θ3 ∈ {δ, β}. 2.2.2.1 If θ3 = δ, we obtain the configuration illustrated in Figure 17a. Note that θ4 cannot be ε, otherwise there is no way to satisfy the angle-folding relation around vertex v2. Also, θ5 cannot be ε, as it implies θ6 = α. Now, at vertex v3, we have necessarily 3δ+β = π. In fact, if 3δ = π, at vertex v2 we obtain δ+δ+β < π and δ+δ+β+ρ > π, ρ ∈ {β, γ}, as β + γ > δ. Then, the last configuration extends uniquely to a complete b g a e1 2 d d d e 3dq1 q2 e d 5 d 6 de d v 1 g 4 a b dq3 d e 7 dq4 e d 8 dq5 e d9 e 11 dq6 d e 10 d d v 3 v 2 a g g g aa bb d d e a d d d a ee b e e g b e e d d d e d d d d d d d d b A kth fundamental region of N k , k ≥ 6. Figure 17: Local configurations. planar representation formed by 12k tiles. Due to its large dimension, we only illustrate the kth fundamental region in Figure 17b. As β > γ, using Equation (3.1), we must have kγ = π, with k ≥ 6. We denote this family of f-tilings by N k, where α+δ = π, 3δ+β = π and kγ = π, k ≥ 6. Moreover, δ = δk = arccos √ 1 2 cos π k , k ≥ 6. 3D representations of N k, for k = 6, 7, are illustrated in Figures 4e – 4f. 2.2.2.2 If θ3 = β, we obtain the configuration of Figure 18a. It is a straightforward exercise to prove that if vertex v2 has valency six, we obtain α+ kγ = π, k ≥ 2, or β + δ + γ = π, and in either cases Equation (3.1) has no solution. Moreover, this equation also has no solution if there is a vertex with a sum of alternate angles of the form β + δ + δ = π. Now, we consider separately the cases θ4 = θ5 = γ, θ4 = θ5 = β, and θ4 = β and θ5 = γ. 2.2.2.2.1 If θ4 = θ5 = γ, vertex v2 must have valency greater than eight. In fact, valency eight implies the existence of a vertex with a sum of alternate angles of the form β+δ+δ = π. Now, with the labeling of Figure 18b, if v2 has valency greater or equal to ten and • there is an additional angle β in the sum of alternate angles (note that is not possible to have an additional angle δ, as 2δ + 2(β + γ) > π), the last configuration extends 632 Ars Math. Contemp. 22 (2022) #P4.07 / 617–636 b g a e 1 2 d d de 3 dq1 q2 e d 5 d 6 d e d v 1 g 4 a b bq3 78 aa b g g q4 q5 v 2 a b g a e 1 2 d d de 3 dq1 q2 e d 5 d 6 d e d v 1 g 4 a b bq3 78 aa b g g gq4 g b b a a 9 11 12 10 gq5 g b b a a v 2 b Figure 18: Local configurations. to the one illustrated in Figure 19a and there is no way to satisfy the angle-folding relation around vertex v3. b g a e 1 2 d d de 3 dq1 q2 e d 5 d 6 d e d v 1 g 4 a b bq3 78 aa b g g gq4 g b b a a 9 11 12 10 13 14 d d d e e 15 d gq5 g b b a a v 2 a b g a 16 g v 3 b a b g a e 1 2 d d de 3 dq1 q2 e d 5 d 6 d e d v 1 g 4 a b bq3 78 aa b g g gq4 g b b a a 9 11 12 10 13 14 d d d d d e e 15 e e d d gq5 g b b a a v 2 a b b g a 16 17 18 d 20 19 d d d d e e g v 3 v 4 b Figure 19: Local configurations. • β + δ + kγ = π, with k ≥ 3, we obtain the configuration illustrated in Figure 19b. At vertex v3 we must have one of the following situations: (i) β + δ + ε = π; this condition leads to a sum of alternate angles at vertex v4 containing ε+ δ + β + γ > π, which is not possible; (ii) β + δ + δ + β = π; in this case we obtain a complete planar representation formed by 16k tiles. Due to its dimension, we only illustrate one octant of the sphere (fundamental region) in Figure 20. Observe that one of the hemispheres is obtained from the other through a 90 degree rotation. Note that if θ6 = δ, we would obtain β + 3δ = π and consequently no solution would exist for Equation (3.1). We have δ = arctan ( sec π2k ) , β = π2 − δ, γ = π 2k , α = C. P. Avelino and A. F. Santos: S2 coverings by isosceles and scalene triangles . . . 633 π − δ, ε = π2 and k ≥ 3. We denote this f-tiling by P k, k ≥ 3, whose 3D representations, for k = 3, 4, are presented in Figures 4g – 4h. b d e e d d d d d a a a b g g eg b Figure 20: A eighth fundamental region of Pk, k ≥ 3. (iii) β + δ + δ + β + β = π; in this case we have necessarily k ≥ 4 and it gives rise to a sum of alternate angles of the form α + k̄γ = π, with k̄ ≥ 2. Due to the angles relations, we have k̄ = 2. Nevertheless, under these conditions, Equation (3.1) has no solution. (iv) β + δ + δ + β + γ = π; in this case we also have k ≥ 4 and we obtain a com- plete planar representation formed by 28k tiles. Due to its large dimension, we only illustrate one hemisphere in Figure 21. The other hemisphere is ob- tained through a 180 degree rotation along the x axis and a reflection. We have δ = arctan ( sin (k−1)π2k−1 sec π 2k−1 ) , β = (k−1)π2k−1 − δ, γ = π 2k−1 , α = π − δ and ε = π2 . We denote this f-tiling by Q k, k ≥ 4, whose 3D representations, for k = 4, 5, are presented in Figures 4i – 4j. g a g b g e d g g g g e e e e e e e e e e e e e e e e e e e e e e e ee e e d d d d d d d dd d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d a a a a a a a a a a a a a a a a a a a a b b b b b b b g b b b b b b g a a a a a a a bb g g g g g g b b g b b g g b b g g g b g g b bg g b d d b g g g b Figure 21: One hemisphere of Qk, k ≥ 4. (v) β + δ + δ + δ = π; under this condition, it is easy to verify that we achieve at vertex v4 (see Figure 19b) a sum of alternate angles containing δ + δ + β + γ, but δ + δ + β + γ < δ + δ + β + δ = π and δ + δ + β + γ + ρ > π, for all ρ ∈ {α, β, δ, ε}. 2.2.2.2.2 If θ4 = θ5 = β (Figure 18a), it is easy to observe that vertex v2 cannot be surrounded by six consecutive angles β, as we obtain a vertex with a sum of alternate 634 Ars Math. Contemp. 22 (2022) #P4.07 / 617–636 angles of the form α+ γ+ ρ, with ρ ∈ {α, β, δ}, which is not possible. Moreover, it is not possible to have angles γ surrounding v2, as it gives rise to a vertex with a sum of alternate angles containing α and β. Taking into account these restrictions and analyzing the angles relations and side lengths, at vertex v2 we must have one of the following cases: (i) β+ δ+ β+ δ = π; in this case we obtain the configuration illustrated in Figure 22a. b g a e 1 2 d d de 3 dq1 q2 e d 5 d 6 d e d v 1 g 4 a b bq3 78 aa b g g q4 b a a 9 11 12 10 13 14 d 15 d d q5 a v 2 b g a 16 d d e e b g g b d d d e e g v 3 v 4 a b g a e 1 2 d d de 3 dq1 q2 e d 5 d 6 d e d v 1 g 4 a b bq3 78 aa b g g q4 b a a 9 11 12 10 13 14 d 15 d d q5 a v 2 b g a 16 17 18 20 19 d d e e b g g b d d d e e g v 3 v 4 b b d d d d e e a a e e b b d d d d a a g g 21 23 22 24 e e e e d d d d b b a a g g g g g g d d d d a a 31 29 30 32 27 25 26 28 b b v 5 v 6 b Figure 22: Local configurations. Given the sums of alternate angles S1 : β+ δ+β+ δ = π and S2 : β+ δ+ kγ = π, k ≥ 3, it is a straightforward exercise to prove that at vertices v3 and v4 we must have only S1 or a combination of S1 and S2 (note that we have symmetry, so order does not matter). If we have a combination of S1 and S2, we obtain a complete representation of f-tiling Pk, k ≥ 3, previously achieved. On the other hand, if we have only S1, the last configuration extends to the one illustrated in Figure 22b. At vertices v5 and v6 we must have only S1 or S2. In the last case, as before we obtain the f-tiling Pk, with k ≥ 4. If S1 is the sum of alternate angles at vertices v5 and v6, then we obtain a complete representation formed by 8k tiles. A fundamental region is illustrated in Figure 23. For each k ≥ 6, we have α = 2arctan ( cos πk + √ 1 + cos2 πk ) , β = π2 − δ, γ = π k , δ = π − α and ε = π 2 . We denote this f-tiling by R k, k ≥ 6, whose 3D representations, for k = 6, 7, are presented in Figures 4k – 4l. b d d a dee g d a g b Figure 23: A 2kth fundamental region of Rk, k ≥ 6. (ii) β + δ+ β + δ+ β = π; this case leads to the following additional relations between angles: α+γ+γ = π and kγ = π, with k ≥ 8. Nevertheless, under these conditions, (3.1) has no solution. C. P. Avelino and A. F. Santos: S2 coverings by isosceles and scalene triangles . . . 635 2.2.2.2.3 If θ4 = β and θ5 = γ (Figure 18a), it is easy to observe that vertex v2 cannot be surrounded by the sequence (. . . , β, β, γ, γ, . . .), as we achieve a vertex with a sum of alternate angles containing α + β, which is not possible as α + β + ρ > π, for all ρ. As the sum of alternate angles surrounding v2 must contain at least one angle γ, taking into account the previous restriction and analyzing angles relations and side lengths, at vertex v2 we have necessarily β + δ + β + δ + γ = π, as illustrated in Figure 24. b g a e 1 2 d d de 3 dq1 q2 e d 5 d 6 d e d v 1 g 4 a b bq3 78 aa b g g q4 b a a 9 11 12 10 13 14 d 15d q5 a v 2 b g a 16 d eb g g g d d d e g v 3 b b a e e d b 17 d d 18 a g d d d e e 19 20 e e d d d d 22 21 bq6 g a a g b 23 24 Figure 24: Local configuration. Note that θ6 must be β (tile 23), as θ6 = δ immediately leads to an impossibility. It is a straightforward exercise to verify that at vertex v3 we must have β + δ + kγ = π, with k ≥ 4, or β + δ + β + δ + γ = π. In the first case, analyzing the symmetry of the figure and all possible combinations of angles surrounding specific vertices, we obtain the f-tiling Qk, k ≥ 4, formerly achieved. In the last case, beside this family of f-tilings, we also obtain a complete planar representation formed by 16k tiles. Due to its dimension, we only illustrate one hemisphere in Figure 25. the other hemisphere is obtained through a 180 degree rotation along the x axis and a reflection. g a g b g e d g g g g e e e e e e e e e e e e e e e e e e e e e e e ee e e d d d d d d d dd d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d a a a a a a a a a a a a a a a a a a a a b b b b b b b g b b b b b b g a a a a a a a bb g g g g g g b b g b b g g b b g g g b g g b bg g b d d b g g g b Figure 25: One hemisphere of Sk, k ≥ 7. We have δ = arctan ( sin (k−1)π2k sec π 2k ) , β = (k−1)π2k − δ, γ = π k , α = π − δ and 636 Ars Math. Contemp. 22 (2022) #P4.07 / 617–636 ε = π2 . We denote this f-tiling by S k, k ≥ 7, whose 3D representations, for k = 7, 8, are presented in Figures 4m – 4n. ORCID iDs Catarina P. Avelino https://orcid.org/ 0000-0003-4335-0185 Altino F. Santos https://orcid.org/ 0000-0002-8638-4644 References [1] C. P. Avelino and A. F. Santos, S2 coverings by isosceles and scalene triangles—adjacency case I, Ars Math. Contemp. 16 (2019), 419–443, doi:10.26493/1855-3974.1401.7d9. [2] A. M. R. Azevédo Breda, A class of tilings of S2, Geom. Dedicata 44 (1992), 241–253, doi: 10.1007/bf00181393. [3] R. J. M. Dawson, Tilings of the sphere with isosceles triangles, Discrete Comput. Geom. 30 (2003), 467–487, doi:10.1007/s00454-003-2846-4. [4] R. J. M. Dawson and B. Doyle, Tilings of the sphere with right triangles. I. The asymptotically right families, Electron. J. Comb. 13 (2006), Research Paper 48, 31, doi:10.37236/1074. [5] R. J. M. Dawson and B. Doyle, Tilings of the sphere with right triangles. II. The (1, 3, 2), (0, 2, n) subfamily, Electron. J. Comb. 13 (2006), Research Paper 49, 22, doi: 10.37236/1075. [6] A. M. R. d’Azevedo Breda and P. S. Ribeiro, Spherical f -tilings by two non-congruent classes of isosceles triangles—I, Math. Commun. 17 (2012), 127–149. [7] A. M. R. d’Azevedo Breda and P. S. Ribeiro, Spherical f -tilings by two non-congruent classes of isosceles triangles—II, Acta Math. Sin. Engl. Ser. 30 (2014), 1435–1464, doi: 10.1007/s10114-014-3302-5. [8] A. M. R. d’Azevedo Breda and A. F. Santos, Dihedral f-tilings of the sphere by spherical triangles and equiangular well-centered quadrangles, Beiträge Algebra Geom. 45 (2004), 447– 461. [9] A. M. R. d’Azevedo Breda and A. F. Santos, Symmetry groups of a class of spherical foldings tilings, Appl. Math. Inf. Sci. 3 (2009), 123–134. [10] S. A. Robertson, Isometric folding of Riemannian manifolds, Proc. Roy. Soc. Edinburgh Sect. A 79 (1977/78), 275–284, doi:10.1017/S0308210500019788. [11] Y. Ueno and Y. Agaoka, Classification of tilings of the 2-dimensional sphere by congruent trian- gles, Hiroshima Math. J. 32 (2002), 463–540, http://projecteuclid.org/euclid. hmj/1151007492. ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P4.08 / 637–648 https://doi.org/10.26493/1855-3974.2559.e4f (Also available at http://amc-journal.eu) Configured polytopes and extremal configurations Tibor Bisztriczky Department of Mathematics and Statistics, University of Calgary, Canada Gyivan Lopez-Campos , Deborah Oliveros * Instituto de Matemáticas, Universidad Nacional Autónoma de México, Boulevard Juriquilla 3001, Juriquilla, Querétaro, 076230 Received 17 February 2021, accepted 7 January 2022, published online 19 August 2022 Abstract We examine a class of involutory self-dual convex polytopes with a specified sets of diameters, compare their vertex sets to extremal Lenz configurations, and present some of their realizations. Keywords: Involutory self-dual polytopes, configured polytopes, Lenz configurations, extremal con- figurations. Math. Subj. Class. (2020): 52-01, 52A15, 52B11 1 Introduction We describe points in Rd by standard coordinates (x1, x2, . . . , xd). For 3 ≤ i ≤ d, let Hi(bi) denote the hyperplane xi = bi, and Le(be+1, . . . , bd) = ∩di=e+1Hi(bi), e = 2, . . . , d − 1. Le(be+1, . . . , bd) is an e-flat, and denote the (e − 1)-sphere with centre c and radius t in Le(be+1, . . . , bd) by Se−1(c, t). We denote the origin of Rd by cd, and let (λw, p) := λw + (0, . . . , 0, p), for a point w ∈ Hd(0) = Ld−1(0) and {λ, p} ⊂ R. Let Y be a set of points in Rd. Then conv(Y ) and aff(Y ) denote, respectively, the con- vex hull and the affine hull of Y . For sets Y1, Y2, . . . Yn, let [Y1, Y2, . . . Yn] = conv (∪ni=1Yi) and ⟨Y1, Y2, . . . Yn⟩ = aff (∪ni=1Yi). If Y = {y1, y2, . . . , yn} is finite, we let [y1, y2, . . . , yn] = conv(Y ) and ⟨y1, y2, . . . , yn⟩ = aff(Y ). *Corresponding author. Supported by Proyecto PAPIIT IG100721, 106318 and CONACYT Ciencia Básica 282280. E-mail addresses: tbisztri@ucalgary.ca (Tibor Bisztriczky), gyivan.lopez@im.unam.mx (Gyivan Lopez-Campos), doliveros@im.unam.mx (Deborah Oliveros) cb This work is licensed under https://creativecommons.org/licenses/by/4.0/ 638 Ars Math. Contemp. 22 (2022) #P4.08 / 637–648 Let P ⊂ Rd denote a convex d−polytope with L(P ) and Fi(P ), 0 ≤ i ≤ d−1, denot- ing the face lattice and the set of i−faces of P . We let fi(P ) = |Fi(P )|, V (P ) = F0(P ) and F(P) = Fd−1(P ), assume familiarity with the basic notions of convex polytopes, and refer to [3, 6] and [18] for basic terminology and definitions. Specifically, two polytopes P1 and P2 are combinatorially equivalent (P1 ∼= P2) if there is an isomorphism (inclusion preserving) from L(P1) to L(P2), and are dual if there is an anti-isomorphism (inclusion reversing) from L(P1) to L(P2). If there is an anti-isomorphism Φ from L(P ) to L(P ) then P is self-dual, moreover, if Φ2 = id then P is involutory self-dual. Let P ⊂ Rd be involutory self-dual via the anti-isomorphism on L(P ) induced by the map v → v∗ with v ∈ V (P ), v∗ ∈ F(P ) and v /∈ v∗. A segment [v, w], with end-points v and w, both vertices of P and with w ∈ v∗, is called a principal diagonal of P and let D(P ) denote the set of principal diagonals of P . Finally, we say that P is configured if each principal diagonal in P has length diam(P ), and that P is strictly configured if it is configured and only principal diagonals of P have length diam(P ). We note that odd regular polygons are strictly configured. Let Xn ⊂ Rd be a set of n > d ≥ 2 points and Md(Xn) be the number of pairs {x, y} ⊂ Xn such that diam(Xn) = ∥x− y∥, the distance between x and y. Let M(d, n) be the maximum of Md(Xn) over all Xn ⊂ Rd. Then Xn is an extremal configuration if Md(Xn) = M(d, n). The problem of determining M(d, n) is due to Erdős in [4]. We list contributions to the problem in the References, with specific mention of [11, 12] and [17] and the following results: (1) M(2, n) = n, and Xn ⊂ R2 is extremal if and only if V (P ) ⊆ Xn ⊆ bd(P ) for some Reuleaux polygon P . (2) M(3, n) = 2n − 2 and Xn ⊂ R3 is extremal if and only if Xn is the vertex set of certain types of polytopal (Reuleaux) ball polytopes. (3) M(d, n), d ≥ 4, grows quadratically with n, and extremal Xn are attained only by Lenz Constructions. In this last regard, we note (cf. [17]) that an (even dimensional) Lenz Configuration in Rd, d = 2p ≥ 2, is any translate of a finite subset of ∪pi=1Ci where Ci is a circle with centre at the origin O and radius ri, so that r2j + r 2 k = 1 for all j, k and the subspaces Ui, spanned by Ci, yield the orthogonal decomposition Rd = U1 ⊕ U2 ⊕ ... ⊕ Up. For odd dimensions d = 2p+ 1, C1 is replaced by a 2-sphere with centre O and radius r = 1√2 . Theorem 1.1 (K. Swanepoel). For each d ≥ 4, there exists a number N(d) such that all extremal configurations Xn, with n ≥ N(d), are Lenz Configurations. We note that in [17], Swanepoel also determines M(d, n) for sufficiently large n. Our interests in this paper are realizations (constructions) of strictly configured d- polytopes P , d ≥ 3, and values of Md(P ) (number of principal diagonals of P ). In Section 2, we will show that for strictly configured 4-polytopes there is a formula similar to 1) and 2) that depends on the number of vertices and edges; furthermore we show the convex hull of vertices of an extremal Lenz configuration is never a configured d-polytope. The former raises the question of whether in dimension d ≥ 4 the situation for M(d, n) may have at least another possible scenario, if the points are not in Lenz configurations. In T. Bisztriczky et al.: Configured polytopes and extremal configurations 639 Section 3 we will give constructions of configured d-polytopes P for d ≥ 3 such that for d = 4, M4(P ) ≤ 4n. These constructions consist of two steps: determining self-dual poly- topes so that all principal diagonals have length (say 1), and then showing that the diameter of the polytope is 1. 2 Principal diagonals In this section, we assume that P ⊂ Rd is an involutory self-dual d-polytope via the anti- isomorphism on L(P ) induced by v ∈ V (P )→ v∗ ∈ F(P ), and recall that D(P ) denotes the set of principal diagonals of P . Theorem 2.1. Let P ⊂ R3 be a configured 3-polytope. Then P is strictly configured and extremal, that is, |D(P )| = 2f0(P )− 2. Proof. Since P is self-dual, we have that f0(P ) = f2(P ) and so, f1(P ) = 2f0(P )− 2 by Euler’s Theorem. Let v ∈ V (P ). Then v∗ ∈ F2(P ) is a polygon and f0(v∗) = f1(v∗). On the one hand, f0(v ∗) = |{g ∈ D(P ) | v ∈ g}| by definition. On the other hand, v ∈ e ∈ F1(P ) iff e∗ ∈ F1(v∗), and so, f1(v∗) = |{e ∈ F1(P ) | v ∈ e}|. Thus |{g ∈ D(P ) | v ∈ g}| = |{e ∈ F1(P ) | v ∈ e}| and |D(P )| = |F1(P )|. Theorem 2.2. Let P ⊂ R4 be a strictly configured 4-polytope. Then |D(P )| ≤ 2f1(P )− 2f0(P ). Proof. Let V (P ) = {v1, ..., vn} and F1(P ) = {e1, ..., em}. Then F2(P ) = {e∗1, ..., e∗m} and F(P ) = {v∗1 , ..., v∗n} by the self-duality of P . We recall from [1] that fjk(P ), 0 ≤ j < k ≤ 3, is the number of pairs of j-faces Gj and k-faces Gk such that Gj ⊂ Gk, and that f02(P ) ≤ 6f1(P ) − 6f0(P ). By the self-duality of P , we have also that n∑ i=1 f1(v ∗ i ) = f13(P ) = f02(P ), n∑ i=1 f2(v ∗ i ) = f23(P ) = f01(P ) and f01(P ) = m∑ j=1 f0(ej) = 2f1(P ) Finally, let v ∈ V (P ) and e ∈ D(P ) of a configured P ⊂ R4. Then v ∈ e if, and only if, e = [v, w] and w ∈ F0(v∗). Thus, f0(v∗) is the number of principal diagonals of P that contain v, and ∑n i=1 f0(v ∗ i ) = 2|D(P )|. Then by Euler’s Theorem, |D(P )| = 1 2 n∑ i=1 (2 + f1(v ∗ i )− f2(v∗i )) = n+ 1 2 n∑ i=1 f1(v ∗ i )− 1 2 n∑ i=1 f2(v ∗ i ) = f0(P ) + 1 2 f02(P )− 1 2 f01(P ) ≤ f0(P ) + [3f1(P )− 3f0(P )]− f1(P ). (2.1) 640 Ars Math. Contemp. 22 (2022) #P4.08 / 637–648 End of Theorem 2.2. We let Md(Q) = Md(V (Q)) for a d−polytope Q, and observe that if P ⊂ R4 is strictly configured then M4(P ) is linear in f1(P ) and f0(P ). This raises the following question: Is there a set of n vertices of a strictly configured polytope in Lenz Configuration? We show below that the answer is no if f0(P ) > 5; in fact, we present in Section 3 a subfamily of such P ⊂ R4 with f1(P ) ≤ 3f0(P ) and M4(P ) ≤ 4f0(P ). If n = 5 and d = 4, it is easy to prove that the polytope with vertices (0, 0, √ 6 12 , √ 10 4 ), (0, 0, √ 2 3 , 0), 1√ 3 (cos π3 , sin π 3 , 0, 0), 1√ 3 (cos 2π3 , sin 2π 3 , 0, 0) and 1√ 3 (1, 0, 0, 0) is a Lenz Configuration and that it is strictly configured. This is the only case with d = 4 where the vertices of a strictly configured polytope is a Lenz Configuration. Theorem 2.3. Let X ⊂ R4 be a 4-dimensional extremal Lenz Configuration with |X| ≥ 6. Then P = conv(X) is not configured. Proof. We assume X ⊂ R4 is a 4-dimensional Lenz Configuration with X ⊂ C1 ∪ C2, Ci ⊂ Ui, where R4 = U1 ⊕ U2. It is clear that P is a 4-polytope with V (P ) = X and diameter 1. Let X ∩ C1 = {w1, ..., wa}, X ∩ C2 = {z1, ..., zb} and note that for i = 1, 2, Gi := Ui ∩ P ∈ F2(P ). From [17], we have that M4(X) = M(4, n) with |X∩C1| = ⌈n2 ⌉ and |X∩C2| = ⌊ n 2 ⌋, say. Furthermore, M(4, 6) = t2(6) + 4, M(4, 7) = t2(7) + 4 and M(4, n) ≤ t2(n) + ⌈n2 ⌉+ 1 for n ≥ 8 where t2(n) is the number of pairs {wj , zk} such that ∥wj − zk∥ = 1. Accordingly, there are M(4, n) − t2(n) diameters of X that have end points in either C1 or C2. We suppose that P is configured via the anti-isomorphism induced by v → v∗, v /∈ v∗, and seek a contradiction. Then a ≥ 3, b ≥ 3, v /∈ v∗ and F(P ) = {w∗1 , ..., w∗a, z∗1 , ..., z∗b } yield that v∗ ∩ C1 ̸= ∅ ̸= v∗ ∩ C2 for v ∈ X ∩ C1, and G1 = z∗1 ∩ z∗2 and G2 = w∗1 ∩ w∗2 say: Thus, w∗j ∩G1 ∈ F1(w∗j ) and z∗k ∩G2 ∈ F1(z∗k) for 3 ≤ j ≤ a and 3 ≤ k ≤ b. It now follows that the number of principal diagonals of P in G1 and G2 is: • two through each wj and zk with j ≥ 3, k ≥ 3 and • at least one through each of w1, w2, z1 and z2; that is, at least 12 (2(a−2)+2(b−2)+4) = a+b−2 = n−2 and n−2 ≤ ⌈ n 2 ⌉ ≠ 1. Then n = 6, w∗3 ∩G1 = [w1, w2] and so, w3 ∈ w∗1 ∩ w∗2 , [G1, w3] ⊂ w∗1 ∩ w∗2 , and w∗1 = w∗2 ; a contradiction. We note that the arguments and the result in Theorem 2.3 extend to d ≥ 5 for extremal Lenz Configuration X with sufficiently large |X|. This raises the issue of how to realize configured polytopes with a large number of vertices in higher dimensions. 3 Constructions of strictly configured polytopes In this section, we present realizations of strictly configured polytopes that are (d − 2)- fold d-pyramids or “stratified” d-polytopes. We note that configured polytopes play an important part in the study of, among others, graphs, hypergraphs, and bodies of constant width. T. Bisztriczky et al.: Configured polytopes and extremal configurations 641 3.1 Prismoids Let m ≥ d ≥ 3 and Q ⊂ Hd(0) be a (d − 1)-polytope with V (Q) = {w1, w2, . . . , wm} and cd as a relative interior point. We consider translated homothetic copies (homotheties) Qjm of Q. For k ≥ 2 and 1 ≤ j ≤ k, let Qjm = [yj1, yj2, . . . , yjm] with yjr = (λjrwr, pj), pk < pk−1 < · · · < p1 and λj > 0. We let Rkm = [Q1m,Q2m, . . . ,Qkm], and say that Rkm is a k-layered d-prismoid if |V (Rkm)| = km and for r = 1, . . . ,m, [y(j−1)r, yjr] are the edges of Rkm that intersect Q(j−1)m and Qjm. Then [Qim,Qjm] is a d-prismoid for 1 ≤ i ≤ j ≤ m, {Q1m,Qkm} ⊂ F(Rkm) and we let Pkm = [y00, Rkm] for some point y00 = (0, . . . , 0, q) ∈ Rd. We say that Pkm is a stratified d-polytope if y00 is beyond either Q1m or Qkm, and beneath all other facets of Rkm (cf. [6] p. 78), and hence, |V (Pkm)| = km+ 1. In what follows, we assume that Pkm = [y00, Rkm] ⊂ Rd is stratified with Rkm as above and y00 beyond exactly Q1m. It is clear that Pkm is dependent upon the (d − 1)- polytope Q = [w1, w2, . . . , wm] ⊂ Hd(0), and we examine properties of Pkm that are inherited from Q. As a point of reference, P2m ⊂ R3 is called an apexed 3-prism in [11]. 3.1.1 Let Q = [w1, w2, . . . , wm] ⊂ Hd(0) be involutary self-dual via the anti-isomorphism on L(Q) induced by wr → w̃r ∈ F(Q). Then F(Q) = {w̃1, w̃2, . . . , w̃m} and we have that • Qjm is involutary self-dual via the anti-isomorphism of L(Qjm) that sends yjr → ỹjr, and yjs ∈ ỹjr if, and only if, ws ∈ w̃r, • F(Qjm) = {ỹj1, ỹj2, . . . , ỹjm}, • F(Rkm) = {Q1m,Qkm} ∪ {ỹ(j−1)r, ỹjr}|2 ≤ j ≤ k, 1 ≤ r ≤ m} and • F(Pkm) = (F(Rkm) \ {Q1m}) ∪ {[y00, ỹ1r]|1 ≤ r ≤ m}. Then (cf. [2], Theorem 2.1) Pkm is involutary self-dual via the anti-isomorphism on L(Pkm) induced by the map yjr → Yjr with Y00 = Qkm, Ykr = [y00, ỹ1r] and Yjr = [ỹ(k−j)r, ỹ(k−j+1)r] for j = 1, . . . , k − 1 and r = 1, . . . ,m. □ 3.1.2 With Q as in 3.1.1, let V (Q) ⊂ Sd−2(cd, t) ⊂ Hd(0) and ∥wr − ws∥ = 1 for each wr ∈ V (Q) and ws ∈ w̃r. We say that Pkm is metrically embedded in Rd if ∥y − y′∥ = 1 for every {y, y′} ⊂ V (Pkm) such that [y, y′] is a principal diagonal of Pkm. Thus, a metrically embedded Pkm of diameter 1 is configured. From Theorem 4.1 in [2]; if y00 = (0, 0, . . . , 0, q), then there are 0 < λk ≤ λ1 < · · · < λj ≤ λk−j < · · · < λ[ k+12 ] = 1 that yield 0 = pk < pk−1 < · · · < p1 < q so that for every yjr ∈ V (Pkm): if yis ∈ Yjr then ∥yjr−yis∥ = 1. Specifically, we note that q2 = 1−λ2kt2, p21 = 1− ∥λkwr − λ1ws∥2 and pk−1 = p1 − √ β with β = 1− ∥λk−1wr − λ1ws∥2. □ Our present interest is to determine involutary self-dual Pkm ⊂ Rd of, say, diameter 1 and then to characterize its diameters. To that end, we seek involutary self-dualQ ⊂ Hd(0) of diameter 1 and with vertices on a (d− 2)-sphere. 642 Ars Math. Contemp. 22 (2022) #P4.08 / 637–648 3.2 Pyramids with polygonal bases With the ai’s to be specified, let d ≥ 3 and Q ⊂ L2(−a3, . . . ,−ad) be a regular m-gon with cyclically labeled vertices w1, w2, . . . , wm, the circumradius t, the diameter 1 and m = 2u + 1 ≥ 3. Then it is well known that 1 = ∥wr − wr+u∥ = ∥wr − wr+u+1∥ for each wr, and that Q has 2m diameters. As a simplification, we write wr = (x1, x2,−a3, . . . ,−ad) as wr = (x1, x2) in relation to the plane L2(−a3, . . . ,−ad). 3.2.1 With θ = 2πm and wr = t(cos(rθ), sin(rθ)) for r = 1, . . . ,m, we note that wm = (t, 0), wm+u = wu and 1 = ∥wm − wu∥2 = 2t2(1 − cos(uθ)) = 2t2(1 + cos( πm )) from m = 2u+ 1. 3.2.2 With m = 2u + 1 ≥ 5 and λ > 0, we claim that ∥λwr − wj∥ < ∥λwr − wr+u∥ for wj ∈ V (Q) \ {wr, wr+u, wr+u+1}. With coordinates as in 3.2.1, we may assume that wr = wm and that wj is in the upper half-plane. Then 0 < jθ < uθ < π and cos(uθ) < cos(jθ) and ∥λwm − wu∥2 − ∥λwm − wj∥2 = 2λt2(cos(jθ)− cos(uθ)). □ 3.2.3 For λ > µ > 0 and ws ∈ {wr+u, wr+u+1}, we have that [λwr, µwr, µws, λws] is an isosceles trapezoid of side lengths λ, µ and (λ−µ)t and ∥λwr−µws∥2 = λµ+(λ−µ)2t2 = ∥λws − µwr∥2. □ 3.2.4 From 1 = ∥wm − wu∥2 = 2t2(1 + cos( πm )) and m ≥ 3, we obtain that 1 4 < t 2 ≤ 13 and 1 3 < 1 4(1−t2) ≤ 3 8 . We let t2 = t, t 2 d = 1 4(1−t2d−1) for d ≥ 3 and note that 13 < t 2 3 ≤ 38 < t24 ≤ 25 < t 2 5 ≤ 512 < t 3 6 ≤ 37 < t 2 7 ≤ 716 < t 2 d < 1 2 with d ≥ 8. □ 3.2.5 With d ≥ 4 and Q ⊂ L2(−a3, . . . ,−ad) ⊂ L3(−a4, . . . ,−ad) as above, we write wr = (t2 cos(rθ), t2 sin(rθ),−a3) in relation to L3(−a4, . . . ,−ad). We consider the 2-sphere S2 := S2 ((0, 0, 0), t3) ⊂ L3(−a4, . . . ,−ad) with t23 = 14(1−t22) , and let a3 = √ t23 − t22. Then V (Q) ⊂ S2 and with wm+1 = (0, 0, t3), we claim that ∥wm+1 − wr∥ = 1 for r = 1, 2, . . . ,m. AsQ is symmetric about the x3-axis, we verify the claim with wr = wm = (t2, 0,−a3). From t23 = ∥wm∥2 = t22 + a23 and t22 = 4t23−1 4t33 , it follows that ∥wm+1 − wm∥2 = t22 + (t3 + a3) 2 = 2t23 + 2t3 √ t23 − t22 = 2t23 + 2t3 ( (1−2t23) 2 4t23 ) 1 2 = 1. T. Bisztriczky et al.: Configured polytopes and extremal configurations 643 Theorem 3.1. Let d ≥ 3 and Q2 = [w1, . . . , wm] ⊂ L2(−a3, . . . ,−ad) be a regular m-gon of diameter 1 and circumradius t2; m = 2u + 1 ≥ 3. Then for e = 3, . . . , d, t2e = 1 4(1−t2e−1) , a2e = t 2 e − t2e−1 and ce = (0, . . . ,−ae+1, . . . ,−ad) if e ̸= d, there is an involutary self-dual (e− 2)-fold e-pyramid Qe = [w1, . . . , wm, . . . , wm+e−2] of diameter 1 and basis Q2 such that (i) Qe ⊂ Le(−ae+1, . . . ,−ad) if e ̸= d, (ii) V (Qe) ⊂ Se−1(ce, te) and (iii) Qe is strictly configured. Proof. With reference to Subsections 3.2.1, 3.2.2, 3.2.3, 3.2.4 and 3.2.5, we let: • wi = (t2 cos(iθ), t2 sin(iθ),−a3, . . . ,−ad) for i = 1, . . . ,m • wm+i = (0, . . . , 0, ti+2,−ai+3, . . . ,−ad) for i = 1, . . . , d− 3 and • wm+d−2 = (0, . . . , 0, td). We observe first that for 2 ≤ i < j ≤ d, t2i +a2i+1 = t2i+1 and so, t2i +a2a+1+ · · ·+a2j = t2j . From this it follows that ∥wi − ce∥2 = t22 + a23 + · · · + a2e = t2e for wi ∈ V (Q2), 3 ≤ e ≤ d ∥wm+i − ce∥2 = t2i+2 + a2i+3 + · · · + a2e = t2e for i + 2 ≤ e ≤ d − 1 and ∥wj − cd∥2 = ∥wj∥2 = t2d for wj ∈ V (Qd). Next, with wr = (t2 cos(rθ), t2 sin(rθ),−a3, . . . ,−ad) and w′r = (t2 cos(r + u)θ, t2 sin(r + u)θ,−a3, · · · − ad), we note that Q2 is involutary self-dual via the anti-isomorphism of L(Q2) induced by wr → w̄r = [w′r, w′r+1]. Then for e = 3, . . . , d, F(Qe) = {[w̄r, wm+1, . . . wm+e−2]|r = 1, . . .m}∪{[V (Qe)\{wr}]|r = m+1, . . . , m+ e− 2} andQe is involutary self-dual via the anti-isomorphism onL(Qe) induced by wr → w̃r where w̃r = { [w̄r, wm+1, . . . wm+e−2], r = 1, . . . ,m; [V (Qe) \ {wr}], r = m+ 1, . . . ,m+ e− 2. Finally, we observe that for 1 ≤ j ≤ m+ i, ∥wm+i − wj∥2 = t2i+1 + (ti+2 + ai+2)2. Then, as in 3.2.5, t2i+1 = 4t2i+2−1 4t2 yields that ∥wm+i − wj∥ = 1. From this and t 2 2 = 1 2(1+cos( πm ) , we obtain that ∥wr − ws∥ = 1 for ws ∈ w̃r; furthermore, if {wr, wz} ⊂ V (Q2) and wz /∈ w̃r then ∥wr − wz∥ < 1. We note that Me(Qe) = 2M2(Q2) + ∑m+e−3 m+1 j = (e − 1)m + ( e−2 2 ) and that Q3 is extremal. Theorem 3.2. Let d ≥ 3, m = 2u + 1, n = m + d − 3 and k ∈ {2, 3}. Then there is an involutary self-dual stratified Pkn = [y00, Rkn] ⊂ Rd of diameter 1 that is strictly configured. Proof. With reference to Subsection 3.1 and Theorem 3.1 with e = d− 1 and ad = 0, we consider Pkn with the property that: • y00 is beyond exactly Q1n. 644 Ars Math. Contemp. 22 (2022) #P4.08 / 637–648 • Q = [w1, ..., wn] ⊂ Ld−1(−ad) = Hd(0), • Qd−1 is a involutary self-dual (d−3)-fold (d−1)-pyramid with diameter 1 and basis Q2, and • Q2 = [w1, ..., wm] ⊂ L2(−a3, ...,−ad) is a regular m-gon of diameter 1. Then cd−1 = (0, . . . , 0,−ad) = cd and with t2, . . . td−1 as in 3.2.4, we simplify notation and let t = td−1. We now apply 3.1.2 with y00 = (0, . . . , 0, q) and pk < pk−1 < · · · < p1 < q. Case 1: k = 2 and hence, λ1 = 1 and p2 = 0. With 0 < λ2 < 1: P2n is stratified, Y00 = Q2n, Y1r = [ỹ1r, ỹ2r] and Y2r = [y00, ỹ1r]. With q2 = 1− λ2t2 and p21 = 1−∥λ2wr −ws∥2 = 1− (λ2 + (1− λ2)2t2) (cf. 3.2.3), we have that ∥yjr − yis∥ = 1 for yis ∈ Yjr. With λ2 = 12 ; we have q 2 = 4−t 2 4 , p 2 1 = 2−t2 4 and claim that ∥yjr − yiz∥ < 1 for yiz /∈ Yjr. From 13 < t 2 < 12 , we obtain that ∥y00 − y1r∥2 = ∥(0, q)− (wr, p1)∥2 = ∥wr∥2 + (q − p1)2 = t2 + q2 + p21 − 2qp1 = 1 4 (6− 2t2 − 2 √ 4− t2 √ 2− t2) ≤ 1 4 ( 6 + 2 ( 1 2 ) − 2 √ 4− 1 3 √ 2− 1 3 ) < 1 (3.1) Let yiz ̸= y00 ̸= yjr and yiz /∈ Yjr. Then yiz = (λiwz, pi), yjr = (λjwr, pj) and wz /∈ w̃r (cf. 3.1.1). Since Q1n and Q2n are homothets of Q, we may assume by Theorem 3.1(iii) that j = 1 and i = 2, say. Since wz /∈ w̃r, it follows as in the proof of Theorem 3.1 that wz = wr or {wz, wr} ⊂ V (Q2). If wz = wr, then ∥y1r − y2r∥2 = t 2 4 + p 2 1 = 1 2 . If {wz, wr} ⊂ V (Q2), then it follows from 3.2.2 that ∥wr − 12wz∥ < ∥wr − 1 2ws∥ with ws ∈ w̃r ∩ V (Q2). Hence, ∥y1r − y2z∥ < ∥y1r − y2s∥ = 1. Case 2: k = 3 and hence, λ2 = 1 and p3 = 0. Let Y00 = Q3n, Y1r = [ỹ2r, ỹ3r], Y2r = [ỹ1r, ỹ2r] and Y3r = [y00, ỹ1r]. With λ = λ1 = λ3 = 1 2 and q 2 = 1−λt2 = 4−t4 , p 2 1 = 1−∥λwr−λws∥2 = 1−λ2 = 34 (cf. 3.1.2 and 3.2.3), β = 1− ∥λ2wr − λ1ws∥2 = 1− ∥wr − λws∥2 = 1− λ+ (1− λ)2t2 = 2−t 2 4 and p2 = p1 − √ β, we obtain that ∥yjr − yis∥ = 1 for yis ∈ Yjr. Let yiz /∈ Yjr. We claim that ∥yjr − yiz∥ < 1 and then it follows that each Yjr is a facet of P3n; that is, R3n is a 3-layered prismoid and P3n is stratified. We observe that if a < t2 ≤ b then ∥y00 − y2r∥2 = ∥(0, q)− (wr, p2)∥2 = ∥wr∥2 + (q − p2)2 = t2 + q2 + p21 + β + 2q √ β − 2p1 ( q + √ β ) = 1 4 ( 9 + 2t2 + 2 √ (4− t2)(2− t2)− 2 √ 3( √ 4− t2 + √ 2− t2 ) < 1 4 ( 9 + 2b+ 2 √ (4− a)(2− a)− 2 √ 3( √ 4− b+ √ 2− b ) (3.2) T. Bisztriczky et al.: Configured polytopes and extremal configurations 645 and ∥y00 − y2r∥ < 1 for (a, b) ∈ {( 13 , 3 8 ), ( 3 8 , 2 5 ), ( 2 5 , 5 12 ), ( 5 12 , 3 7 ), ( 3 7 , 7 16 ), ( 7 16 , 1 2 )}, that is, for each d ≥ 3 (cf. 3.2.4). It is clear that ∥y00 − y1r∥ < ∥y00 − y2r∥, and hence, we may assume that yiz = (λiwz, pi), yjr = (λjwr, pj) and wz /∈ w̃r. Then ∥wr − wz∥ < ∥wr − ws∥ for ws /∈ w̃r, and ∥y1r − y3z∥ < ∥y1r − y3s∥ = 1 for y3s ∈ ỹ1r ⊂ Y1r. From t2 < 12 , we obtain that β > 3 16 = p21 4 , p2 = p1 − √ β < p12 and p2 < p1 − p2. Thus, ∥y3r − y2z∥ < ∥y1r − y2z∥ and we argue as above that ∥y1r − y2z∥ < 1. In summary; ∥yjr − y′∥ < 1 for {yjr, y′} ⊂ {y00} ∪ {yjr|j = 1, . . . , k and r = 1, . . . , n}, and with equality if and only if y′ ∈ Yjr. Thus F(Pkn) = {Y00} ∪ {Yjr|j = 1, . . . , k, r = 1, . . . , n}, V (Pkn) = {y00} ∪ {yjr|j = 1, . . . , k, r = 1, . . . , n} and Pkn is involutary self-dual under the anti-isomorphism on L(Pkn) induced by yjr → Yjr. Theorem 3.3. Let Pkm ⊂ R3 be an involutary self-dual stratified 3-polytope that is con- figured with diameter 1; k ≥ 2 and m = 2u+ 1 ≥ 3. Then there is an involutary self-dual stratified P(k+1)m ⊂ R3 that is configured with diameter 1. Proof. We let l = k + 1 and denote Pkm as in 3.1.1 and 3.1.2 with d = 3. Specifically, • Q = [w1, . . . , wm] ⊂ H3(0) is a regular m-gon of diameter 1 and circumcentre c3 = (0, 0, 0) as in 3.2.1, • Qjm = [yj1, . . . , yjm] with yjr = (λjwr, pj) and 0 < λk ≤ λ1 < · · · < λj ≤ λk−j < · · · < λ[ l2 ] = 1, 0 < pk < pk−1 < · · · < p1 < q ≤ 1 and y00 = (0, 0, q), • the anti-isomorphism on L(Pkm) is induced by yjr → Yjr with Y00 = Qkm, Ykm = [y00, ỹ1r], Yjr = [ỹ(k−j)r, ỹ(l−j)r], 1 ≤ j ≤ k − 1, and ỹjr = [yj(r+u), yj(r+u+1)], and • ∥yjr − yis∥ = 1 if, and only if, yis ∈ Yjr. Let S(y) := S2(y, 1) for y ∈ R3, and consider the homothets Q0m = [y01, . . . , yom] of Q with y0r = (λ0wr, p0), 0 < λ0 < λ1 and p1 < p0 < q. From [yk(r+u), yk(r+u+1)] = ỹkr = Y00∩Y1r, it follows that ∥y00−yks∥ = 1 = ∥y1r−yrs∥ for s ∈ {r+u, r+u+1}, and so, {y00, y1r} ⊂ Ckr := S(yk(r+u)) ∩ S(yk(r+u+1)), a circle with centre 12 (yk(r+u) + yk(r+u+1)). It is now clear that (i) for each p1 < p0 < q, there is 0 < λ0 < λ1 such that y0r ∈ Ckr. In fact, y0r ∈ αkr, the shorter arc of Ckr with end points y00 and y1r. We note also that V (Q0m) ∩ V (Pkm) = ∅ for each such p0. Let V = V (Pkm), B(y) = [S(y)] and B(V ) = ∩y∈V B(y). Since diam(Pkm) = 1, it follows that (ii) αkr ⊂ bd(B(V )) for r = 1, . . .m. Since Pkm is involutary self-dual with no fixed points, it follows from Theorem 3.2 of [13] that B(V ) is polytopal and the face polyhedral structure of B(V ) is a lattice 646 Ars Math. Contemp. 22 (2022) #P4.08 / 637–648 isomorphic to L(Pkm). Accordingly, B(V ) is similarly self-dual and from Theo- rem 4.1 of [13], any surface Φ ⊂ R3 obtained from bd(B(V )) (by performing their surgery on one edge-arc of each pair of dual edge-arcs of bd(B(V ))) is the boundary of a body of constant width. In this case, V ⊂ Φ and diam(V ) = 1 yield Φ is of constant width 1. We note that dual edge-arcs of bd(B(V )) correspond to dual edges of L(Pkm). Thus, the duality [y00, y1r]←→ Y00 ∩ Y1r = ỹkr yields that αkr is dual to the shorter edge-arc in S(y00) ∩ S(y1r) with end point yk(r+u) and yk(r+u+1). We consider those Φ that contain each of αk1, αk2, . . . , αkm. Then the symmetry of Pkm about the x3–axis and i) yield that (iii) V ′ = V ∪ V (Q0m) ⊂ Φ and diam(V ′) = 1, (iv) S(y00)∩V ′ = V (Qkm) and the spherical region S(y00)∩Φ is not empty and bounded in H3(0) by the circumcircle of Qkm, and (v) y′00 = (0, 0, q − 1) ∈ S(y00) ∩ Φ. From diam(V ) = 1, |V | = km + 1, M(3, km + 1) = 2km and Theorem 2.1, we have that M3(V ) = 2km+ 1. From diam(V ′) = 1, |V ′| = lm+ 1 and i), we have that M3(V ′) ≥M3(V ) + 2m = 2lm. Thus, M3(V ′) = 2lm and (vi) ∥y0r − y∥ < 1 for y0r ∈ V (Q0m) and y ∈ V \ {yk(r+u), yk(r+u+1)}. Let V ′′ = V ′ ∪ {y′00}. Then diam(V ′′) = 1, |V ′′| = lm+ 2, ∥y00 − y′00∥ = 1 and 2|V ′′| − 2 = 2lm+ 2 ≥M3(V ′′) ≥ 2lm+ 1. From the rotational symmetry of V ′′ and S(y′00) about the x3–axis, it follows that (vii) ∥y′00 − y∥ < 1 for y ∈ V ′ \ {y0}, and (viii) ∥yϵ−y∥ < 1 for y ∈ V ′ \{y0} for sufficiently small ϵ > 0 and yϵ = (0, 0, q−1−ϵ). Let p0 = q−ϵ and µ be the radius of the circle H3(p0)∩S(y′00). Then {(0, 0, p0)} = H3(p0)∩S(yϵ) ⊂ Q0m ⊂ [H3(p0)∩S(y′00)] and with λ0 chosen so that 0 < λ0 < λ1 and y0r ∈ αkr, we have that 0 < λ0t ≤ µ. Accordingly, there is a point z00 ∈ [y′00, yϵ] such that λ0t is the radius of H3(p0) ∩ S(z00); that is, (ix) ∥z00 − y0r∥ = 1 for r = 1, 2, . . . ,m. Finally, let zjr = y(l−j)r,z̃jr = ỹ(l−j)r and Q′jm = Q(l−j)m for j = 1, 2, . . . , l and r = 1, 2, . . . ,m. In addition, let Z00 = Q′lm = Q0m, Zlr = [z00, z̃1r] = [z00, ỹkr] and Zjr = [z̃(l−j)r, z̃(l−j+1)r] = [ỹjr, ỹ(j−1)r]. From the preceding, we have that Plm = [z00, Q ′ 1m, . . . , Q ′ lm] is involutary self-dual via zjr → Zjr, stratified and configured with diameter 1. Finally, we show that if a set of n points are the vertices of a configured 4-polytope P such as in Theorem 3.2 then M4(P ) ≤ 4n. Theorem 3.4. Let Pkm = [y00, Rkm] ⊂ R4 be a configured stratified 4-polytope, with n = km+ 1 vertices. Then number of principal diagonals of Pkm is at most 4n. Proof. By Theorem 2.2, it is sufficient to prove that f1(P ) ≤ 3n for every configured stratified 4-polytope. By construction, Rkm = [Q1m,Q2m, . . .Qkm] where each copy T. Bisztriczky et al.: Configured polytopes and extremal configurations 647 Qim is self-dual and contains m vertices, and thus, f1(Qim) = 2m−2 by Euler’s Theorem and self-duality. Finally, there are m edges through y00 and m(k−1) edges connecting the k homothets Qim, and so, f1(Pkm) = k(2m− 2)+m(k− 1)+m = 3km− 2k ≤ 3km+3 = 3n. ORCID iDs Tibor Bisztriczky https://orcid.org/0000-0001-7949-4338 Gyivan Lopez-Campos https://orcid.org/0000-0003-2005-8210 Deborah Oliveros https://orcid.org/0000-0002-3330-3230 References [1] M. Bayer, The extended f -vectors of 4-polytopes, J. Comb. Theory Ser. A 44 (1987), 141–151, doi:10.1016/0097-3165(87)90066-5. [2] T. Bisztriczky and D. Oliveros, d-dimensional self-dual polytopes and Meissner polytopes, in: Polytopes and discrete geometry, Amer. Math. Soc., [Providence], RI, volume 764 of Contemp. Math., pp. 21–30, [2021] ©2021, doi:10.1090/conm/764/15357. [3] A. Brøndsted, An Introduction to Convex Polytopes, volume 90 of Graduate Texts in Mathe- matics, Springer-Verlag, New York-Berlin, 1983, doi:10.1007/978-1-4612-1148-8. [4] P. Erdös, On sets of distances of n points, Am. Math. Mon. 53 (1946), 248–250, doi:10.2307/ 2305092. [5] B. Grünbaum, A proof of Vazonyi’s conjecture, Bull. Res. Council Israel. Sect. A 6 (1956), 77–78. [6] B. Grünbaum, Convex Polytopes, volume 221 of Graduate Texts in Mathematics, Springer- Verlag, New York, 2nd edition, 2003, doi:10.1007/978-1-4613-0019-9. [7] B. Grünbaum and G. C. Shephard, Is selfduality involutory?, Am. Math. Mon. 95 (1988), 729– 733, doi:10.2307/2322253. [8] A. Heppes, Beweis einer Vermutung von A. Vázsonyi, Acta Math. Acad. Sci. Hungar. 7 (1956), 463–466, doi:10.1007/bf02020540. [9] H. Hopf and E. Pannwitz, Aufgabe nr. 167, Jber. Deutsch. Math. 43 (1934). [10] S. Jendroł, A non-involutory selfduality, Discrete Math. 74 (1989), 325–326, doi:10.1016/ 0012-365x(89)90144-1. [11] Y. S. Kupitz, H. Martini and M. A. Perles, Finite sets in Rd with many diameters – a survey, in: Proceedings of the International Conference on Mathematics and its Applications, ICMA-MU 2005, Bangkok, Thailand, December 15–17, 2005, Mahidol University, Bangkok, pp. 91–112, 2005. [12] H. Martini, L. Montejano and D. Oliveros, Bodies of Constant Width. An Introduction to Convex Geometry with Applications, Birkhäuser,Cham, 2019, doi:10.1007/978-3-030-03868-7. [13] L. Montejano and E. Roldán-Pensado, Meissner polyhedra, Acta Math. Hungar. 151 (2017), 482–494, doi:10.1007/s10474-017-0697-3. [14] C. C. Neaderhouser and G. B. Purdy, On finite sets in Ekin which the diameter is frequently achieved, Period. Math. Hung. 13 (1982), 253–257, doi:10.1007/bf01847922. [15] B. Servatius and H. Servatius, Self-dual graphs, Discrete Math. 149 (1996), 223–232, doi: 10.1016/0012-365x(94)00351-i. 648 Ars Math. Contemp. 22 (2022) #P4.08 / 637–648 [16] S. Straszewicz, Sur un probleme géométrique de P. Erdös, Bull. Acad. Polon. Sci. Cl. III. 5 (1957), 39–40, IV–V. [17] K. J. Swanepoel, Unit distances and diameters in Euclidean spaces, Discrete Comput. Geom. 41 (2009), 1–27, doi:10.1007/s00454-008-9082-x. [18] G. M. Ziegler, Lectures on Polytopes, volume 152 of Graduate Texts in Mathematics, Springer- Verlag, New York, 1995, doi:10.1007/978-1-4613-8431-1. ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P4.09 / 649–674 https://doi.org/10.26493/1855-3974.1753.52a (Also available at http://amc-journal.eu) Top-heavy phenomena for transformations* Yaokun Wu † , Yinfeng Zhu School of Mathematical Sciences and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China Received 19 July 2018, accepted 5 February 2022, published online 19 August 2022 Abstract Let S be a transformation semigroup acting on a set Ω. The action of S on Ω can be naturally extended to be an action on all subsets of Ω. We say that S is ℓ-homogeneous provided it can send A to B for any two (not necessarily distinct) ℓ-subsets A and B of Ω. On the condition that k ≤ ℓ < k + ℓ ≤ |Ω|, we show that every ℓ-homogeneous transfor- mation semigroup acting on Ω must be k-homogeneous. We report other variants of this result for Boolean semirings and affine/projective geometries. In general, any semigroup action on a poset gives rise to an automaton and we associate some sequences of integers with the phase space of this automaton. When this poset is a geometric lattice, we pro- pose to investigate various possible regularity properties of these sequences, especially the so-called top-heavy property. In the course of this study, we are led to a conjecture about the injectivity of the incidence operator of a geometric lattice, generalizing a conjecture of Kung. Keywords: Incidence operator, kernel space, rank, strong shape, valuated poset, weak shape. Math. Subj. Class. (2020): 05B35, 20M20, 47D03, 52C35 1 Introduction 1.1 Transformation and phase space Let Γ be a digraph, namely a pair consisting of its vertex set V(Γ) and arc set E(Γ) ⊆ V(Γ) × V(Γ). We call Γ symmetric if (u, v) ∈ E(Γ) holds if and only if so does (v, u) ∈ E(Γ). For any A ⊆ V(Γ), we adopt the notation Γ[A] for the subdigraph of Γ induced by *We thank Peter Cameron, Alexander Ivanov, Peter Šemrl, Johannes Siemons and Qing Xiang for useful discussions. Especially, Peter Šemrl reminded us the result of Taussky and Zassenhaus while Johannes Siemons suggested that the isomorphism of two permutation modules can be read from Figure 6. This work has been supported by NSFC (11971305, 11671258) and STCSM (17690740800). †Corresponding author. E-mail addresses: ykwu@sjtu.edu.cn (Yaokun Wu), fengzi@sjtu.edu.cn (Yinfeng Zhu) cb This work is licensed under https://creativecommons.org/licenses/by/4.0/ 650 Ars Math. Contemp. 22 (2022) #P4.09 / 649–674 A which has vertex set A and arc set E(Γ) ∩ (A × A). The number of weakly connected components and the number of strongly connected components of Γ will be dubbed wcc(Γ) and scc(Γ), respectively. For a set Ω, all maps from Ω to itself form the set ΩΩ. For each g ∈ ΩΩ and α ∈ Ω, we write αg for the image of α under the map g. The composition of maps provides an associative product on the set ΩΩ and thus turns it into a monoid, namely a semigroup with a multiplicative unit. We call this monoid the full transformation monoid on Ω and denote it by T(Ω). A subset of T(Ω) which is closed under map composition, whether or not it contains the identity map on Ω, is called a transformation semigroup acting on Ω. Let S be a transformation semigroup on Ω. We say that S is transitive on a set A ⊆ Ω if for every α, β ∈ A we can find g ∈ S such that αg = β; we call S transitive if S is transitive on Ω. If the transformation semigroup S is generated by a set G ⊆ ΩΩ, namely S consists of products of elements of G of positive length, we call (S,G) a deterministic automaton on Ω [67, Section 1]. The phase space of an automaton (S,G) on Ω, denoted by Γ(S,G), is the digraph with vertex set Ω and arc set {(α, αg) : α ∈ Ω, g ∈ G}. When Ω has at least two elements, the claim that S is transitive is equivalent to the claim that Γ(S,G) is strongly connected for any generator setG of S. We write Γ(S, S) simply as Γ(S) and note that each strongly/weakly connected component of Γ(S) coincides with a strongly/weakly connected component of Γ(S,G) for any generator set G of S. For all work in this paper, we can simply focus on Γ(S) instead of considering Γ(S,G) for any specific generator set G. We emphasize Γ(S,G) from the phase space viewpoint here to highlight the connection between semigroup theory and automata theory, and to indicate the role played by the choice of G in some problems related to various distance functions on the phase space, say the Černý conjecture. For any set Ω, a subset of T(Ω) forms a permutation group on Ω whenever it is a transformation semigroup and each element has an inverse in it, namely it is a set of bijective transformations of Ω and is closed under compositions and taking inverses. Permutation groups correspond to reversible deterministic automata. Let Ω be a set. We follow the common practice to use 2Ω for the power set of Ω. For each g ∈ T(Ω), let g be the element in T(2Ω) that sends each A ∈ 2Ω to Ag .= {ag : a ∈ A}. More generally, for each G ⊆ T(Ω), G refers to the set {g : g ∈ G}. For any transformation semigroup S on Ω and any generator set G of S, S, as a semigroup derived from S, is known to be the powerset transformation semigroup of S acting on 2Ω and (S,G) is known to be the powerset automaton of (S,G). It may be interesting to iterate the powerset automaton construction and examine the evolution of the phase spaces of the resulting automata. When discussing transformation semigroups, we may often be more interested in those which preserve some structures, say simplicial maps for simplicial complexes, continuous maps for topological spaces, ordering preserving maps for posets, or adjacency-preserving maps in matrix geometry [52, 66]. Unlike the work on group actions on posets [3, 59] and matroids [19], very little has been done on semigroup actions on these structures [62]. Moving from group actions to semigroup actions is just to consider general deterministic automata instead of reversible ones. 1.2 Valuated poset and its shape For any two sets Ω and Ψ, if they are different or if we do not emphasize that they may be equal, the image of ω ∈ Ω under a map g ∈ ΨΩ is denoted g(ω); note that we often write it as ωg when Ω = Ψ. Y. Wu and Y. Zhu: Top-heavy phenomena for transformations 651 A poset P consists of a set Ω and a binary relation

0 > cj − ck; we call it log-concave if c2i ≥ ci−1ci+1 for all i = 1, . . . , n − 1. When n is finite, we call the sequence real-rooted provided the polynomial c0 + c1x + · · · + cnxn in the unknown x only has real roots and we call it ultra-log-concave provided c0 (n0) , . . . , cn (nn) forms a log- concave sequence. Note that Question 1.2 is about the possible 1-top-heavy property of the strong shape of Gr(Q3) under a matrix semigroup action. In the 1970s, two log-concavity conjectures [61, Conjecture 3] appeared in combina- torics community which claim that the sequences of Whitney numbers of both the first kind and the second kind of a finite matroid are log-concave. The first conjecture was verified by Adiprasito, Huh and Katz [1]. Mason [40] had made variants and stronger versions of the second conjecture; but even the original conjecture is still open. Dowling and Wilson [23] conjectured that the sequence of Whitney numbers of the second kind of a finite ma- troid is top-heavy. When restricted to finite realizable matroids, this top-heavy conjecture was proved by Huh and Wang [28]. The second log-concavity conjecture as described above, which is about the Whitney numbers of the second kind [50], simply says that the shape of every geometric lattice is log-concave. The above-mentioned Dowling-Wilson top-heavy conjecture says that the shape of every finite geometric lattice is top-heavy. On Y. Wu and Y. Zhu: Top-heavy phenomena for transformations 653 the condition that these two conjectures are both true, we know that the shape of a finite geometric lattice is both log-concave (and hence unimodal) and top-heavy. Can we draw this conclusion for the strong/weak shape of some semigroup actions on some geometric lattices? Boolean lattices, partition lattices and projective/affine geometries are some most well- known geometric lattices. It is easy to see that their shapes are all ultra-log-concave (and hence real-rooted) and top-heavy [37]. The main result of this paper, Theorems 2.1 and 2.12, declare the top-heavy property for the strong/weak shape of some semigroups acting on Boolean lattices and projective/affine geometries. The semigroups considered by us are those derived from “simple” transformations. We also report our attempt at tackling the same problem for partition lattices and the Vámos matroid. In Section 2, we will present our main results as well as pertinent problems, examples, and remarks. The first three subsections are devoted to Boolean lattices, partition lattices and projective/affine geometries. The last subsection is a simple discussion in the context of matroids. Before digging into the proofs of the main results, we develop some technical tools in Section 3. In the sequel, we provide in Sections 4 to 7 all the proofs missing from Sections 2.1 to 2.4. We conclude the paper in Section 8 with a brief discussion of the present work and some possible further research. 2 A top-heavy promenade 2.1 Boolean semiring and homogeneity For any set Ω, the set BΩ . = ⋃∞ k=0 ( Ω k ) forms a poset under the inclusion relationship, which is often known as the Boolean semiring over Ω, and the set 2Ω gives rise to the Boolean algebra over Ω. When we view BΩ as a valuated poset, unless stated otherwise, the valuation will be r(A) = |A| for all A ∈ BΩ. If Ω is a finite set, BΩ coincides with 2Ω and is referred to as a Boolean lattice. Let A and Ω be two sets with A ⊆ Ω. For any g ∈ ΩΩ, write g|A for the restriction of g on A. Let S be a transformation semigroup on Ω. For any positive integer k ≤ |Ω|, we name S k-homogeneous if the transformation semigroup S is transitive on ( Ω k ) , that is, scc(Γ(S)[ ( Ω k ) ] = 1. The stabiliser permutation group of (S,A) is the permutation group SA . = {g|A : g ∈ S,Ag = A} acting on A. The relative transformation semigroup of (S,A) is the transformation semigroup S̃A . = {g|A : g ∈ S,Ag ⊆ A} acting on A. Note that the action of S̃A on A may not be transitive even if S acts on A transitively. Theorem 2.1. Let Ω be a set of size n. Let S be a transformation semigroup on Ω and let Γ be the phase space of S. (1) SS(S,BΩ) is 1-top-heavy. (2) Both WS(S,BΩ) and SS(S,BΩ) are top-heavy. (3) Let k and ℓ be two integers such that 0 ≤ k ≤ ℓ ≤ k + ℓ ≤ n+ 1. Let A ∈ ( Ω k ) and B ∈ ( Ω ℓ ) . If n < ∞ and S is ℓ-homogeneous, then scc(Γ(SA)) = wcc(Γ(SA)) ≤ wcc(Γ(SB)) = scc(Γ(SB)). Question 2.2. Take a finite set Ω and two integers k and ℓ such that k ≤ ℓ < k+ℓ ≤ |Ω|+1. Let S be an ℓ-homogeneous transformation semigroup acting on Ω. For any A ∈ ( Ω k ) and B ∈ ( Ω ℓ ) , does it always hold that wcc(Γ(S̃A)) ≤ wcc(Γ(S̃B))? 654 Ars Math. Contemp. 22 (2022) #P4.09 / 649–674 When restricting to permutation groups, the results in Theorem 2.1 are all known more than 40 years ago: Claim (1) for an infinite set Ω was discovered by Brown [12, Corol- lary 1]; Claim (2) for a finite set Ω was derived by Livingstone and Wagner [38, Theo- rem 1]; Claim (3), as well as a positive answer to Question 2.2 for permutation groups, was proved by Cameron [15, Proposition 2.3] under the mild restriction of k + ℓ ≤ |Ω|. Let G be a group acting on a finite set Ω. By Theorem 2.1(2), or more precisely Livingstone- Wagner Theorem [38, Theorem 1], we know that the strong/weak shape of 2Ω under the action of G is a symmetric unimodal distribution. This means that, for any two integers k and ℓ such that k ≤ ℓ < k + ℓ ≤ |Ω|, the number of G-orbits on ( Ω ℓ ) is equal to the sum of a nonnegative integer c plus the number of G-orbits on ( Ω k ) . As an improvement of this fact, Siemons [56, Corollary 4.3] found a natural linear space whose dimension equals this integer c and he [56, Theorem 4.2] even obtained an algorithm to reconstruct the G-orbits on ( Ω k ) from the information on the G-orbits on ( Ω ℓ ) without reference to the group G. Question 2.3. Let Ω be a finite set, and let k and ℓ be two integers such that k ≤ ℓ < k + ℓ ≤ |Ω|. Let S be a transformation semigroup on Ω and let Γ be the phase space of S. (1) Is there a counterpart of [56, Corollary 4.3] which explains the nonnegativity con- straint on the integer wcc(Γ[ ( Ω ℓ ) ])− wcc(Γ[ ( Ω k ) ])? (2) If S is (ℓ + 1)-homogeneous, is there a counterpart of [56, Corollary 4.3] which explains the nonnegativeness of the integer scc(Γ(SB)) − scc(Γ(SA)) for any A ∈( Ω k ) and B ∈ ( Ω ℓ+1 ) ? (3) Is there any algorithm to determine the weakly connected components of Γ[ ( Ω k ) ] from the weakly connected components of Γ[ ( Ω ℓ ) ] without reference to the transformation semigroup S? Example 2.4. Let Ω be a set carrying a linear order ≺. A map g ∈ ΩΩ is order-preserving with respect to ≺ provided αg is not bigger than βg in ≺ whenever α is not bigger than β in ≺. Let S be the monoid consisting of all order-preserving maps on Ω with respect to the given linear order ≺. It is easy to see that S is ℓ-homogeneous for all ℓ ≤ |Ω| but it is even not 2-transitive; by contrast, this phenomenon never happens for permutation groups due to a result of Livingstone and Wagner [38, Theorem 2(b)]. Note that the only permutation contained in S is the identity map in case that Ω is a finite set. This suggests that you may not be able to read Theorem 2.1 or answer Question 2.3 directly from those known facts on permutation groups. Example 2.5. Let Ω = {1, . . . , 6}. Let r and b be two maps in T(Ω) such that r(1) = r(2) = 3, r(3) = r(4) = 5, r(5) = r(6) = 1; b(6) = b(1) = 2, b(2) = b(3) = 4, b(4) = b(5) = 6. Let S = ⟨r, b⟩. On the left of Figure 1, we depict the phase space Γ(S, {r, b}); on the right of Figure 1, we display both the strong shape and the weak shape of 2Ω under the action of S. Both weak shape and strong shape are unimodal and top-heavy. But neither of them is log-concave. Note that the peak of the weak shape does not happen at the middle rank 3. Y. Wu and Y. Zhu: Top-heavy phenomena for transformations 655 Vertex set( Ω 6 ) ( Ω 5 ) ( Ω 4 ) ( Ω 3 ) ( Ω 2 ) ( Ω 1 ) ( Ω 0 ) wcc 1 6 15 7 1 1 1 scc 1 6 15 19 10 1 1 1 2 3 4 5 6 Figure 1: Γ(S, {r, b}) and Γ(S, {r, b})[ ( Ω k ) ], k ∈ {0, 1, . . . , 6}. See Example 2.5. Example 2.6. Let Ω be a set of size n ≥ 3 and let S be a transformation semigroup acting on Ω. If SS(S, 2Ω) is not a sequence of all ones and has at least two ones at the beginning of it, then it cannot be log-concave. This happens when S is the alternating group of order n ≥ 4 and when S is 2-homogeneous but not 3-homogeneous. Example 2.7. Let n and k be two integers such that 1 ≤ k ≤ n. Let Ω be a set of size n and take X ∈ ( Ω k ) . Let S be the set {f ∈ T(Ω) : f |X = Id |X ,Ωf = X}. Note that S is a transformation semigroup on 2Ω satisfying wcc(Γ(S)[ ( Ω i ) ] = { 1, if 0 ≤ i ≤ k;( n i ) , if k + 1 ≤ i ≤ n. This shows that the sequence WS(S, 2Ω) is unimodal and top-heavy and that it is not log- concave when n ≥ 2. Note that SS(S, 2Ω) is a sequence of all ones. Question 2.8. Let S be a transformation semigroup acting on an n-element set Ω. When can we conclude that the strong/weak shape of 2Ω under the action of S is unimodal? Neumann [44] asked whether every λ-homogeneous permutation group is θ-homogene- ous for all cardinals λ > θ ≥ ℵ0. Assuming Martin’s Axiom, Shelah and Thomas [55] gave a negative answer to it. Hajnal [26] supplied an example to show that 2θ-homogeneity does not imply θ-homogeneity. An observation in the same vein by Penttila and Siciliano [47, Remark 4.6] was based upon the Generalized Continuum Hyphothesis. For each statement in Theorem 2.1(3), Question 2.2 and 2.3, it is interesting to see whether or not it holds in the case that Ω is an infinite set. We are also wondering if the rich theory on oligomorphic permutation groups [16] should have a counterpart for transformation semigroups. 656 Ars Math. Contemp. 22 (2022) #P4.09 / 649–674 2.2 Partition lattice Let Ω be a set. For any map s ∈ ΩΩ, we define its kernel map, denoted by s−1, to be the map from 2Ω to 2Ω that sends X ∈ 2Ω to Xs−1 = {y ∈ Ω : ys ∈ X} ∈ 2Ω. To illustrate the definition, we depict the phase space of a map s on the left of Figure 2 and part of the phase space of s−1 in the middle of Figure 2. A partition of Ω is a set of nonempty disjoint subsets of Ω whose union is Ω. We call these elements of a partition its blocks. The rank of a partition π is ∑ B∈π(|B| − 1). Write P(Ω) for the set of all partitions of Ω of finite ranks. When |Ω| < ∞, the set P(Ω) together with the refinement relation forms a geometric lattice, which we call the partition lattice of Ω. Note that the rank of a partition in this geometric lattice is |Ω| minus the number of its blocks. Let Pk(Ω) be the set of rank-k partitions of Ω, namely, those partitions of Ω of size |Ω| − k. Each transformation s ∈ ΩΩ induces a transformation s∗ of 2Ω such that Πs∗ = {πs−1 : π ∈ Π} \ {∅} for all Π ∈ P(Ω). We demonstrate part of the phase space of s∗ on the right of Figure 2 for the map s as shown on the left there. Let S be a transformation semigroup on Ω. We have a derived transformation semigroup S∗ := {s∗ : s ∈ S} on P(Ω), which we call the kernel space of S. We say that S is k-kernel homogeneous if for all Π,Π′ ∈ Pk(Ω) there exists s ∈ S such that Πs∗ = Π′, which surely implies scc(Γ(S∗)[Pk(Ω)]) ≤ 1. 1 2 3 4 5 6 s {2, 3}{4, 5} {1, 6} s−1 123|4|56 16|23|45 s∗ Figure 2: A map, its inverse and the derived action on partitions. Example 2.9. On the left of Figure 3, we depict the so-called Černý automaton C4 = Γ(S,G), whereG = {a, b} consists of two transformations on a four-element set Ω. On the right of Figure 3, we depict the automaton Γ(S∗, G∗) where S∗ is acting on P(Ω). Observe that WS(S∗,P(Ω)) = (1, 1, 1, 1) and SS(S∗,P(Ω)) = (1, 2, 2, 1) are both unimodal and top-heavy. For any finite set Ω, A ∈ 2Ω, π ∈ P(Ω) and s ∈ ΩΩ, it holds r(A) ≥ r(As) and r(π) ≤ r(πs∗). This difference between Boolean lattice and partition lattice somehow hints at our difficulty of turning the following conjecture into a result like Theorem 2.1. Conjecture 2.10. Let Ω be a finite set and let S be a semigroup acting on Ω. Then both WS(S∗,P(Ω)) and SS(S∗,P(Ω)) are top-heavy. For each set Ω and each positive integer k ≤ |Ω|, we use P(Ω, k) for the set of partitions of Ω into k blocks. Y. Wu and Y. Zhu: Top-heavy phenomena for transformations 657 0 1 23 a b a b a b a b 0 1 2 3 0 3 1 2 2 3 0 1 0 1 2 3 1 2 0 3 0 1 2 3 0 3 1 2 1 3 0 2 0 2 1 3 0 1 3 2 0 1 2 3 0 2 3 1 1 2 3 0 0 2 1 3 0 1 2 3 b∗ a∗ a∗ b∗ a∗ b∗ a∗ b∗ a∗ b∗ a∗ b∗ a∗ b∗ a∗ b∗ a∗ b∗ a∗ b∗ a∗ b∗ a∗ b∗ a∗, b∗ a∗ b∗ a∗ Figure 3: Černý automaton C4 and its kernel space. See Example 2.9. Question 2.11. (1) Take two positive integers k and ℓ with k < ℓ. Let Ω be an infinite set and let S be a semigroup S acting on Ω. If S∗ is transitive on P(Ω, ℓ), is it true that S∗ is transitive on P(Ω, k)? (2) The shapes of all Dowling lattices, which include all partition lattices, are real-rooted [8]. What about the top-heavy property of the (strong/weak) shapes of Dowling lattices under a semigroup action? There has been an active study of those permutation groups which are transitive on the set of all ordered or unordered partitions of a set of a given shape [2, 21, 39, 43]. But even when confining our attention to permutation groups, we are not aware of any work related to Conjecture 2.10 and Question 2.11. 2.3 Subspace lattice Let Ω be a possibly infinite set of size n, let k be a nonnegative integer with k ≤ n and let F be a finite field. We mention that Gr(k, FΩ) is a q-analogue of ( Ω k ) and their relationship is like the one between Johnson graphs and Grassmann graphs [45]. For each prime power q, we write Fq for the q-element finite field. Write Matn(Fq) for the multiplicative semigroup of all Ω by Ω matrices over Fq each row/column of which have finitely many nonzeros; and write Affn(Fq) for the semigroup of all affine linear transformation on Fnq equipped with the associated product of composition. We regard the empty set as the dimension-(−1) affine/linear subspace. The set of all nonempty finite linear subspaces of Fnq is denoted by Pq,n . = Gr(Fnq ) and the set of all dimension-k linear subspaces of Fnq is denoted by Pkq,n . = Gr(k,Fnq ). By a finite affine subspace of a linear space V , we mean a translate of a finite linear subspace of V . The set of all finite affine subspaces of Fnq is denoted by Aq,n and the set of all dimension-k affine subspaces of Fnq is denoted by Ak+1q,n . Note that Pq,n and Aq,n are known as projective geometry and affine geometry over the field Fq , respectively. For each nonnegative integer k, we put the rank of each element in Pkq,n and the rank of each element in Akq,n to be k, thus getting two valuated posets Pq,n and Aq,n, 658 Ars Math. Contemp. 22 (2022) #P4.09 / 649–674 which are geometric lattices when n <∞. We are ready to display Theorem 2.12, a q-analogue of Theorem 2.1. Kantor [31, Theorem 1] deduced a q-analogue of the aforementioned result of Livingstone and Wagner [38, Theorem 1]. If the semigroup S ≤ Matn(Fq) is a subgroup of the general linear group GLn(Fq), Stanley [59, Corollary 9.9] found that SS(SP ,Pq,n) and WS(SP ,Pq,n) are both symmetric and unimodal for finite n. Penttila and Siciliano [47, Theorem 4.4(ii), (iii)] generalized this result of Stanley for groups to the case that n is infinite. Theorem 2.12. Let n be the size of a nonempty set, and let q be a prime power. (1) Let S ≤ Matn(Fq) be a linear transformation semigroup acting on Fnq . For each g ∈ S, write gP for g|Pq,n . Let SP be the transformation semigroup {gP : g ∈ S} acting on Pq,n. Then SS(SP ,Pq,n) and WS(SP ,Pq,n) are both top-heavy. (2) Let T ≤ Affn−1(Fq) be an affine linear transformation semigroup acting on Fn−1q . For each g ∈ T , write gA for g|Aq,n−1 . Let TA be the transformation semigroup {gA : g ∈ T} acting on Aq,n−1. Then SS(TA,Aq,n−1) and WS(TA,Aq,n−1) are both top-heavy. Remark 2.13. When n is infinite, Theorems 2.1 and 2.12 in the original version of this paper, submitted on 19 July 2018, contains weaker results. Following the proof presented by Bercov and Hobby for [9, Corollary 1] and also the proof of Roy for [51, Theorem], we used the existence of Ramsey number [49, Theorem A] to derive Theorem 2.1(1) for infinite n. A similar argument based on Ramsey number shows that both SS(SP ,Pq,n) and SS(TA,Aq,n−1) are 1-top-heavy for infinite n in the setting of Theorem 2.12. After the acceptance of this paper in 2022, we notice the work of Penttila and Siciliano [47, Lemma 3.1], which was submitted on 30 April 2019 and published in 2021, and thus arrive at the corresponding strengthening in Theorem 2.1(2) and Theorem 2.12 via an application of their idea. See Lemma 3.6. Remark 2.14. Kantor [32, Theorem 2] determined all the ordered-basis-transitive finite geometric lattices of rank at least three: Roughly speaking, they are Boolean lattices, pro- jective (affine) geometries, and four sporadic designs. Kantor’s classification theorem along with Theorems 2.1 and 2.12 may be a basis for getting homogeneity results about ordered- basis-transitive matroids. Question 2.15. A general projective geometry is defined to be a modular combinatorial geometry that is connected in the sense that the point set cannot be expressed as the union of two proper flats [64, page 313]. Can we establish a counterpart of Theorem 2.12 for general projective geometries? In mathematics we encounter quite some nice duality phenomena, say Chow’s Theorem [45, Corollary 3.1] and many duality concepts for matroids [13]. For projectie geometry, we have the following duality result of Stanley [59, Corollary 9.9]. Theorem 2.16 (Stanley). Let F be a finite field and let k and n be two positive integers with k < n. For any subgroup G of GL(n, F ), the number of orbits of the action of G on Gr(k, Fn) must be the same with the number of orbits of G acting on Gr(n− k, Fn). Question 2.17. If n is the size of an infinite set, does Theorem 2.16 still hold? Here, we should first of all choose a good definition for infinite Grassmannians [46]. Y. Wu and Y. Zhu: Top-heavy phenomena for transformations 659 2.4 A glimpse of matroid In previous subsections, we discuss those poset endomorphisms which are derived from either set transformations or linear transformations. Since finite geometric lattices just en- code information of finite matroids, it is natural to ask why not directly consider matroids and morphisms among matroids, namely those transformations which preserve “indepen- dence structure”. Let M1 and M2 be two matroids and let f be a map from EM1 to EM2 . We call f a weak map from M1 to M2 provided rM1(A) ≥ rM2(Af) holds for all A ⊆ EM1 , and we call f a strong map from M1 to M2 provided the preimage of any flat in M2 is a flat of M1 [33, 35, 57]. It is known that all strong maps must be weak maps. Let M be a matroid on the ground set EM = Ω. Let TM (Ω) (T∗M (Ω)) be the monoid consisting of all elements of T(Ω) which are weak (strong) maps from M to itself. If we know that S is a subsemigroup of TM (Ω) (T∗M (Ω)) acting on Ω, we can define a digraph ΓM,t(S) on Ft(M) as follows: for any X,Y ∈ Ft(M), there is an arc from X to Y if and only if there is g ∈ S such that the minimum flat containing Xg in M is Y. What is the relationship between the connectivity of ΓM,t(S) and ΓM,r(S) for different t and r? We can ask the same question by imposing the extra condition that every element f ∈ S is a bijection on Ω. If the matroid is a very special uniform matroid, namely a matroid in which all sets are independent, one can see that what is discussed in Section 1.3 becomes a very special case of this general setting. Vámos matroid, also known as Vámos cube, is a famous non-algebraic matroid [5, 22, 42, 54]; see [24, Example 6.30] for a description of this rank-4 matroid over a ground set of size eight. Example 2.18. Let M be the Vámos matroid and let S be a subsemigroup of T∗M (EM ). It holds wcc(ΓM,1(S)) ≤ wcc(ΓM,2(S)) ≤ wcc(ΓM,3(S)) and scc(ΓM,1(S)) ≤ scc(ΓM,2(S)) ≤ scc(ΓM,3(S)). Remark 2.19. Compared with the Fundamental Theorem of Projective (Affine) Geometry [17, 48], we think that weak/strong maps and bijective weak/strong maps for matroids are natural extensions of linear transformations and invertible linear transformations for linear spaces. We also mention the well-adopted viewpoint that the full permutation group and the full transformation semigroup can be interpreted as the general linear group and the linear transformation semigroup over the field with one element. 3 Valuated poset and incidence operator 3.1 Hereditary endomorphism and injective incidence operator To prepare for a proof of our main results listed in Section 2, we will introduce a key prop- erty and then present a key lemma for our work. The key property is the so-called hereditary endomorphisms. The key lemma is Lemma 3.2, which gives us some information of the strong/weak shapes of a poset under some semigroup action, provided the semigroup con- sists of hereditary endomorphisms and that some linear map associated with the poset is injective. 660 Ars Math. Contemp. 22 (2022) #P4.09 / 649–674 β βgPℓ Pk ∩ P↓(β) Pℓ Pk ∩ P↓(βg) g g bijective Figure 4: An (ℓ, k)-hereditary endomorphism. Let P be a valuated poset. For any nonnegative integers k ≤ ℓ, we call the poset P (k, ℓ)-finite provided Pk ̸= ∅, Pℓ ̸= ∅ and the set Pℓ ∩ P↑(α) is finite for every α ∈ Pk; we call P (ℓ, k)-finite provided Pk ̸= ∅, Pℓ ̸= ∅ and the set P↓(β) ∩ Pk is finite for every β ∈ Pℓ; we call g ∈ End(P ) a (k, ℓ)-hereditary endomorphism if for all α ∈ Pk which satisfies rP (g(α)) = rP (α) = k it happens that g induces a bijection from the set Pℓ ∩ P↑(α) to Pℓ ∩ P↑(αg); we call g ∈ End(P ) an (ℓ, k)-hereditary endomorphism if for each β ∈ Pℓ, rP (βg) = rP (β) = ℓ ensures that g induces a bijection from the set Pk ∩ P↓(β) to Pk ∩ P↓(βg). See Figure 4 for an illustration. For any k, ℓ ∈ Z≥0, we designate by hEndk,ℓ(P ) the set of all (k, ℓ)-hereditary endomorphisms of the valuated poset P . Let S be a transformation semigroup on a valuated poset P and let G be a generating set of S. For any two nonnegative integers k and ℓ with k ≤ ℓ ≤ r(P ), we set ΠS,G(k, ℓ) to be the digraph with vertex set Pk and arc set {(α, α′) ∈ Pk × Pk : ∃g ∈ G, β ∈ Pℓ s.t. βg ∈ Pℓ, α′ = αg, α ∈ P↓(β)}; we set ΠS,G(ℓ, k) to be the digraph with vertex set Pℓ and arc set {(α, α′) ∈ Pℓ × Pℓ : ∃g ∈ G, β ∈ Pk s.t. βg ∈ Pk, α′ = αg, α ∈ P↑(β)}. We use the shorthand ΠS(k, ℓ) for ΠS,S(k, ℓ). Lemma 3.1. Let P be a valuated poset. Take two nonnegative integers k and ℓ such that k, ℓ ≤ r(P ) and that P is (ℓ, k)-finite. Let S be a sub-semigroup of hEndℓ,k(P ), let G be a generator set of S, and let Γ .= Γ(S,G). Let β ∈ Pℓ and let α ∈ Pk be an element comparable with β. Assume that g and h are two elements of S such that βg ∈ Pℓ and βgh = β. Then there exists f ∈ S such that βgf ∈ Pℓ and αgf = α. Especially, if every weakly connected component of Γ[Pℓ] is strongly connected, then so is ΠS,G(k, ℓ). Proof. The second claim is immediate from the first and so our task is just to prove the first one. Without loss of generality, we assume that k < ℓ. Since β(gh) = β and gh ∈ S ≤ hEndℓ,k(P ), it follows that gh induces a permutation on Pk ∩ P↓(β). But from the assumption that P is (ℓ, k)-finite, we see that Pk ∩ P↓(β) is a finite set, which contains α. This means that there exists a positive integer r such that α(gh)r = α. Accordingly, for f = (hg)r−1h ∈ S it holds (βg)f = (βg)(hg)r−1h = β(gh)r = β ∈ Pℓ and (αg)f = (αg)(hg)r−1h = α(gh)r = α, finishing the proof. For any set Ω, QΩ refers to the linear space of all rational functions on Ω. If P is an (ℓ, k)-finite valuated poset, the incidence operator ζk,ℓP : QPk → QPℓ is the linear operator Y. Wu and Y. Zhu: Top-heavy phenomena for transformations 661 such that for all f ∈ QPk and β ∈ Pℓ, we have (ζk,ℓP (f))(β) = {∑ α∈Pk∩P↓(β) f(α), if k ≤ ℓ;∑ α∈Pk∩P↑(β) f(α), if k > ℓ. (3.1) Lemma 3.2. Let P be a valuated poset. Take two nonnegative integers k and ℓ not ex- ceeding r(P ) such that P is (ℓ, k)-finite, and hence ζk,ℓP is well-defined. Let S be a sub- semigroup of hEndℓ,k(P ) and let Γ stand for Γ(S). Assume that ζ k,ℓ P is an injective linear map from QPk to QPℓ . (1) wcc(Γ[Pk]) ≤ wcc(ΠS(k, ℓ)) ≤ wcc(Γ[Pℓ]). (2) scc(Γ[Pk]) ≤ scc(ΠS(k, ℓ)) ≤ scc(Γ[Pℓ]). Proof. (1) The first inequality is a consequence of the fact that E(ΠS(k, ℓ)) ⊆ E(Γ[Pk]). Let W ⊆ QPℓ be the subspace of all functions which are constant on each weakly connected component of Γ[Pℓ]; let V ⊆ QPk be the subspace of all functions which are constant on each weakly connected component of ΠS(k, ℓ). Note that dim(V ) = wcc(ΠS(k, ℓ)) and dim(W ) = wcc(Γ[Pℓ]) and so it suffices to demonstrate dim(V ) ≤ dim(W ). By symmetry, we only deal with the case of k ≤ ℓ. For every f ∈ V and every arc (β, βg) of Γ[Pℓ], we have (ζk,ℓP (f))(βg) = ∑ α′∈Pk∩P↓(βg) f(α′) = ∑ α∈Pk∩P↓(β) f(αg) (g ∈ hEndℓ,k(P )) = ∑ α∈Pk∩P↓(β) f(α) (f ∈ V ) = (ζk,ℓP (f))(β). This says that ζk,ℓP (f) ∈ W for all f ∈ V . Hence, by the injectivity of ζ k,ℓ P , dim(V ) ≤ dim(W ), as wanted. (2) The first inequality is a consequence of the fact that E(ΠS(k, ℓ)) ⊆ E(Γ[Pk]). Let W ′ ⊆ QPℓ be the subspace of all functions which are constant on each strongly connected component of Γ[Pℓ]; let V ′ ⊆ QPk be the subspace of all functions which are constant on each strongly connected component of ΠS(k, ℓ). Note that dim(V ′) = scc(ΠS(k, ℓ)) and dim(W ′) = scc(Γ[Pℓ]) and so it suffices to demonstrate dim(V ′) ≤ dim(W ′). Take f ∈ V ′. As ζk,ℓP is injective, we aim to show that ζ k,ℓ P (f) ∈W ′. By symmetry, we only deal with the case of k ≤ ℓ. Assume that β and βg are from the same strongly connected component of Γ[Pℓ], where g ∈ S. By the first claim of Lemma 3.1, for every α ∈ Pk ∩ P↓(β), α and αg fall into the same strongly connected component of Γ[Pk] and so, as f ∈ V ′, f(α) = f(αg). (3.2) 662 Ars Math. Contemp. 22 (2022) #P4.09 / 649–674 This allows us to write (ζk,ℓP (f))(βg) = ∑ α′∈Pk∩P↓(βg) f(α′) = ∑ α∈Pk∩P↓(β) f(αg) (g ∈ hEndℓ,k(P )) = ∑ α∈Pk∩P↓(β) f(α) (Equation (3.2)) = (ζk,ℓP (f))(β), proving that ζk,ℓP (V ′) ⊆W ′, as desired. 3.2 Injectivity In order to apply Lemma 3.2, we may need to have some results to guarantee the injectivity of an incidence operator. In this regard, a good understanding of the incidence algebra of a poset may be valuable [36, 68]. We mention that Guiduli [4, Theorem 9.4] established an injectivity result for the so-called rank-regular semi-lattices. It may also be quite useful if the following conjecture [34, Conjecture 1.1] can be verified. Conjecture 3.3 (Kung). Let P be a finite geometric lattice. Let k and ℓ be two positive integers such that k ≤ ℓ ≤ r(P )2 . Then ker(ζ k,ℓ P ) = {0}. We suggest a slight strengthening of Kung’s Conjecture (Conjecture 3.3) as follows. Conjecture 3.4. Let P be a geometric lattice. Let k and ℓ be two nonnegative integers such that k ≤ ℓ ≤ k + ℓ ≤ r(P ). If P is (ℓ, k)-finite, then ζk,ℓP is an injective map. Remark 3.5. Let M be a matroid of rank r. Let S be a subsemigroup of T ∗M (EM ). For every f ∈ S, let f ′ : F(M) → F(M) be the map sending a flat X ∈ F(M) to the minimum flat containing Xf in M . Assume that f ′ ∈ hEndℓ,k(F(M)) for every f ∈ S. In light of Lemma 3.2, if Conjecture 3.4 is valid for the lattice F(M), we will be able to conclude that both the sequence (wcc(ΓM,0(S)), . . . ,wcc(ΓM,r(S))) and the sequence (scc(ΓM,0(S)), . . . , scc(ΓM,r(S))) are top-heavy. Let P be a valuated poset which is (ℓ, k)-finite for all nonnegative integers k ≤ ℓ. We say that P has a top-heavy injective incidence operator provided ζk,ℓP is an injective linear map from QPk to QPℓ for all nonnegative integers k and ℓ satisfying k ≤ ℓ ≤ k+ℓ ≤ r(P ). Penttila and Siciliano [47, Lemma 3.1] pointed out a simple way to establish some injectivity result for linear operators between infinite-dimensional linear spaces whenever they fulfil certain finiteness characteristics. We reformulate their observation below for the convenience of our later usage. Lemma 3.6. Let P be a valuated poset. Let k ≤ ℓ be two nonnegative integers such that P is (ℓ, k)-finite. Assume that for every A ∈ Pk, we can find a finite subset Y of Pk+ℓ such that the ideal generated by Y in P , denoted Y ↓ and with the restriction of rP as its rank function, contains A and possesses a top-heavy injective incidence operator. Then ζk,ℓP is an injective linear map from QPk to QPℓ . Y. Wu and Y. Zhu: Top-heavy phenomena for transformations 663 Proof. Take f ∈ ker ζk,ℓP . Assume, for sake of contradiction, that f(A) ̸= 0 for some A ∈ Pk. Choose Y ⊆ Pk+ℓ such that A ∈ Y ↓ ∩ Pk and Y ↓ possesses a top-heavy injective incidence operator. Let Q represent the resulting valuated poset on Y ↓. Let g be the restriction of f on Qk and let h be the restriction of ζ k,ℓ P (f) = 0 on Y . We have 0 = h = ζk,ℓQ (g) but g(A) = f(A) ̸= 0, violating the assumption that Y ↓ has a top-heavy injective incidence operator. 3.3 Incidence operator as an intertwiner For f ∈ ΨΩ, we sometimes need to talk about f(ω) for ω /∈ Ω. Following the practice of those mathematics with natural multivalued operations [7, 14, 65], we create a universal “don’t care” symbol ⋆ /∈ Ψ and will set f(ω) = ⋆. We often regard ⋆ as all possible values in Ψ and so, whenever we have some addition operation + on Ψ, we extend it to Ψ ∪ {⋆} by setting ⋆+ ψ = ⋆ for all ψ ∈ Ψ ∪ {⋆}. Let P be a valuated poset. Let k and ℓ be two nonnegative integers no greater than r(P ). Let g ∈ PP . For f ∈ QPk , we write fg†,k for the element in ({⋆} ∪ Q)Pk , where ⋆ stands for “don’t care” and can be thought of as the whole set Q, such that the following holds for all β ∈ Pk: fg†,k(β) = { f(βg), if βg ∈ Pk; ⋆, if βg /∈ Pk. Denote by Fix g†,k the set of f ∈ QPk for which fg†,k(β) ∈ {f(β), ⋆} holds for all β ∈ Pk. If g ∈ hEndℓ,k(P ), we say that it is a good (ℓ, k)-hereditary endomorphism of P provided that for any β ∈ Pℓ with βg /∈ Pℓ it holds αg /∈ Pk for some α ∈ Pk which is comparable to β in P . Assuming that g is a good (ℓ, k)-hereditary endomorphism of P , for any β ∈ Pℓ and f ∈ QPk we will have (ζk,ℓP (f)g †,ℓ)(β) = (ζk,ℓP (f))(βg) = ∑ α′∈Pk∩(P↓(βg)∪P↑(βg)) f(α′) = ∑ α∈Pk∩(P↓(β)∪P↑(β)) f(αg) = (ζk,ℓP (fg †,k))(β) whenever βg ∈ Pℓ, and that (ζk,ℓP (f)g †,ℓ)(β) = (ζk,ℓP (f))(βg) = ⋆ = (ζk,ℓP (fg †,k))(β) whenever βg /∈ Pℓ. This observation can be summarized by the commutative diagram in Figure 5, which implies that Fix g†,k is mapped by ζk,ℓP to Fix g †,ℓ for all good (ℓ, k)- hereditary endomorphisms g of P . 664 Ars Math. Contemp. 22 (2022) #P4.09 / 649–674 f ζk,ℓP (f) fg†,k ζk,ℓP (fg †,k) ζk,ℓP g†,k ζk,ℓP g†,ℓ Figure 5: The incidence operator intertwines with every good hereditary endomorphism. Example 3.7. (1) Let Ω be a set of size n. Assume that 2 ≤ k < ℓ ≤ n. Here is an easy observation used often in the study of synchronizing automata: For any g ∈ ΩΩ and any A ∈ ( Ω ℓ ) , we have |Ag| = ℓ if and only if |Bg| = k for all B ∈ ( A k ) . This conclusion is surely not valid any more when k ≤ 1. Note that g is a good (ℓ, k)-hereditary endomorphism of the Boolean lattice 2Ω for each g ∈ ΩΩ. (2) Take integers n, k and ℓ such that 2 ≤ k < ℓ ≤ n and let q be a prime power. Let P = Pq,n or P = Aq,n−1. Similar to the above claim on Boolean lattice, M is a good (ℓ, k)-hereditary endomorphism of P for each M ∈ Matn(Fq) or M ∈ Affn−1(Fq), respectively. 4 Boolean semiring Let Ω be a set and let k and ℓ be two nonnegative integers such that k < ℓ ≤ |Ω|. For the valuated poset P = BΩ, we write the incidence operator ζ k,ℓ P defined in Equation 3.1 as ζk,ℓΩ . That is, (ζk,ℓΩ (f))(B) = ∑ A∈(Bk) f(A) for all f ∈ Q( Ω k) and B ∈ ( Ω ℓ ) . Following a common approach in establishing homogeneity of permutation groups [15, 41] [20, pages 20-22], we will make use of the ensuing result on the rank of the subset inclusion matrix. The result has been discovered independently by many but the earliest ap- pearance of it dates back to the work of Gottlieb [25, Corollary 2]. Among many different proofs of this classical result, we refer the reader to [18, Corollary] and [56, Theorem 2.4]. Note that it gives a positive answer to Conjecture 3.4 for Boolean lattices. Lemma 4.1 (Gottlieb). Let Ω be a nonempty finite set. Then ker ζk,ℓΩ = {0} for any two integers k and ℓ satisfying 0 ≤ k ≤ ℓ ≤ k + ℓ ≤ |Ω|. Let Ω be a set and S be a transformation semigroup on Ω. Let Ω♯ .= {(ω,C) : ω ∈ C ∈ 2Ω} and, for each g ∈ S, let g♯ be the transformation on Ω♯ which sends (ω,C) to (ωg,Cg) for all (ω,C) ∈ Ω♯. Let S♯ stand for the transformation semigroup on Ω♯ consisting of all elements g♯ for g ∈ S. For all positive integers ℓ, we use the following notation: Ω♯ℓ . = {(ω,C) : ω ∈ C ∈ ( Ω ℓ ) } Y. Wu and Y. Zhu: Top-heavy phenomena for transformations 665 and Γ♯ℓ(S) . = Γ(S♯)[Ω♯ℓ]. Here is a result analogous to Lemma 3.1. Lemma 4.2. Let m be a positive integer and let S be an m-homogeneous transforma- tion semigroup acting on a set Ω. Then the digraph Γ♯m(S) is symmetric. Especially, wcc(Γ♯m(S)) = scc(Γ ♯ m(S)). Proof. Take (ω,C) ∈ Ω♯m and g ∈ S such that |Cg| = m.Our task is to show the existence of h ∈ S such that (ωg,Cg)h♯ = (ω,C). As S is m-homogeneous, we can find f ∈ S such that Cgf = (Cg)f = C. Hence, the fact that |C| = m < ∞ allows us to obtain a positive integer r for which (gf)r|C is the identity map on C. This means that we can choose h to be f(gf)r−1. Lemma 4.3. Let Ω be a set, let m be an integer satisfying |Ω| ≥ m > 1, and let S be a transformation semigroup on Ω. For every X ∈ ( Ω m ) , it holds scc(Γ(SX)) = wcc(Γ(SX)) ≤ wcc(Γ♯m(S)) ≤ scc(Γ♯m(S)). (4.1) Moreover, if S is m-homogeneous, then scc(Γ(SX)) = wcc(Γ(SX)) = wcc(Γ ♯ m(S)) = scc(Γ ♯ m(S)). (4.2) Proof. It is trivial to see that wcc(Γ(SX)) = scc(Γ(SX)) and wcc(Γ♯m(S)) ≤ scc(Γ♯m(S)). Let us call each strongly/weakly connected component of Γ(SX) a component. To prove Equation (4.1), let us find an injective map ψ from the set of components of Γ(SX) to the set of weakly connected components of Γ♯m(S). For each γ ∈ X , let the weakly connected component of Γ♯m(S) containing (γ,X) be ψ′(γ). Take γ1, γ2 from the same component of Γ(SX). We may assume that γ1g = γ2 and Xg = X for some g ∈ S. As (γ1, X)g♯ = (γ1g,Xg) = (γ2, X), we see that ψ′(γ1) = ψ ′(γ2). For each component C of Γ(SX), we can now choose any γ ∈ C and get a well-defined map ψ by setting ψ(C) = ψ′(γ). For every weakly connected component C♯ of Γ♯m(S), let ϕ(C ♯) be the set {γ ∈ X : (γ,X) ∈ C♯}. It is routine to check that ϕψ(C) = C for every component C of Γ(SX), proving that ψ is injective, as desired. Assume now S is m-homogeneous. It follows from Lemma 4.2 that wcc(Γ♯m(S)) = scc(Γ♯m(S)). We thus call each strongly/weakly connected component of Γ ♯ m(S) simply a component. Since S is m-homogeneous, for every component C♯ of Γ♯m(S), we have ϕ(C♯) ̸= ∅. This verifies that ϕ and ψ are inverses of each other. We thus get Equation (4.2) and so finish the proof. Proof of Theorem 2.1. (1) This is a special case of (2). (2) This is direct from Lemmas 3.2, 3.6 and 4.1. (3) Since S is ℓ-homogeneous, it follows from Lemma 4.3 that wcc(Γ(SA)) = scc(Γ(SA)) ≤ wcc(Γ♯k(S)) 666 Ars Math. Contemp. 22 (2022) #P4.09 / 649–674 and wcc(Γ(SB)) = scc(Γ(SB)) = wcc(Γ ♯ ℓ(S)). It then remains to prove wcc(Γ♯ℓ(S)) ≥ wcc(Γ ♯ k(S)). We regard Ω♯ as a valuated poset by putting r((α,X)) = |X| and requiring (α,X) < (β, Y ) if and only if α = β ∈ Ω and X ⊊ Y ⊆ Ω. Note that S♯ ⊆ hEndℓ,k(Ω♯). In view of Lemma 3.2(1), it is sufficient to show that ζk,ℓ Ω♯ is injective. For each nonnegative integer m and each α ∈ Ω, let Ω♯m,α . = {(α,A) : (α,A) ∈ Ω♯m}. Corresponding to the partition Ω♯k = ⋃ α∈Ω Ω ♯ k,α and Ω ♯ ℓ = ⋃ β∈Ω Ω ♯ ℓ,β , the Ω ♯ k×Ω ♯ ℓ matrix ζk,ℓ Ω♯ is viewed as a partitioned matrix with blocks ζα,β , which are the submatrices with row index set Ω♯k,α and column index set Ω ♯ ℓ,β , where α, β ∈ Ω. Observe that ζα,β = { ζk−1,ℓ−1Ω\{α} , if α = β; 0, otherwise. Since (k − 1) + (ℓ− 1) ≤ |Ω| − 1, it follows from Lemma 4.1 that ζα,α = ζk−1,ℓ−1Ω\{α} is of full row rank for all α ∈ Ω. This implies that ζk,ℓ Ω♯ is an injective linear map, as desired. Remark 4.4. Let Ω be a set, which is not necessarily finite. Let k and ℓ be two integers with k ≤ ℓ ≤ k + ℓ ≤ |Ω|. For all f ∈ Q( Ω ℓ) and A ∈ ( Ω k ) , we put (ζℓ,kΩ (f))(A) = ∑ A⊆B f(B). Making use of Lemma 4.1, it is easy to see that the linear transformation ζℓ,kΩ : Q (Ωℓ) fin → Q( Ω k) fin is always a surjective map. Unfortunately, we do not see if this observation is helpful for getting a possible counterpart of Theorem 2.1(3) for an infinite set Ω. 5 A graded Möbius algebra Möbius algebra is a semigroup algebra which plays an important role in combinatorics [36, Section 3.6]. Huh and Wang [28] introduced a graded Möbius algebra for geometric lattices. Let L be a finite geometric lattice with rank function (valuation) r. Define a Q- algebra M(L,Q), called the graded Möbius algebra of L [28], to be the linear space with L as a Q-basis together with a multiplication given by xy = { x ∨ y, if r(x) + r(y) = r(x ∨ y) 0, if r(x) + r(y) > r(x ∨ y), and extended by linearity and distributivity. For any nonnegative integers k ≤ ℓ, it is easy to see that the linear map ξk,ℓL as specified below is well-defined: ξk,ℓL : QLk → QLℓ ϕ 7→ ( ∑ x∈L1 x) ℓ−kϕ. We call a finite geometric lattice a realizable lattice if it is the matroid lattice of a finite realizable matroid. Here is the main result of Huh and Wang [28, Theorem 6] in their work Y. Wu and Y. Zhu: Top-heavy phenomena for transformations 667 on solving the realizable case of the top-heavy conjecture of Dowling-Wilson. Huh and Wang [28, Conjecture 7] conjectured that Theorem 5.1 holds without the assumption of realizability. Theorem 5.1 (Huh and Wang). Let L be a finite realizable geometric lattice with rank r. For any integers k and ℓ such that k ≤ ℓ ≤ k + ℓ ≤ r, the linear map ξk,ℓL is injective. Remark 5.2. (1) The partition lattice P(Ω) is isomorphic with the flat lattice of the graphic matroid of the complete graph on Ω. Note that a graphic matroid is regular, namely it is representable over every field. This means that finite partition lattices are realizable. (2) Assume that L is a either a Boolean lattice, or a subspace lattice or a partition lattice. It is easy to see that ξk,ℓL = CL,k,ℓζ k,ℓ L for some positive integer CL,k,ℓ which is determined by L, k and ℓ. Especially, ξk,k+1L = ζ k,k+1 L . An important message here is that, ζk,ℓL and ξ k,ℓ L , as two Q-linear maps, are either both injective or both non-injective. Kung [34, Theorem 1.3] verified Conjecture 3.3 for partition lattices of finite sets. We can improve his result a little bit now. When Ω is finite, Lemma 5.3 claims that Conjec- ture 3.4 holds for partition lattices. Lemma 5.3. Let Ω be a set. Let k and ℓ be two integers such that k ≤ ℓ ≤ k + ℓ ≤ |Ω|. Then ker(ζk,ℓP(Ω)) = {0}. Proof. By Lemma 3.6, Theorem 5.1, and Remark 5.2. Let Ω be a finite set and let k and ℓ be two integers such that 0 ≤ k ≤ ℓ ≤ k+ ℓ ≤ |Ω|. By virtue of Lemma 5.3, ker(ζk,ℓP(Ω)) = {0}. So, to prove Conjecture 2.10 via Lemma 3.2, we want to have s∗ ∈ hEndℓ,k(P(Ω)) for all s ∈ ΩΩ. It is a pity that what we can have instead is s∗ ∈ hEndk,ℓ(P(Ω)) for all s ∈ ΩΩ. For any transformation g on a set Ω, we associate a partition kerΩ(g) of Ω in which two elements α and β fall into the same part provided αg = βg, and we call kerΩ(g) the kernel of g. Note that kerΩ(g1g2) = kerΩ(g2)g∗1 for all g1, g2 ∈ T(Ω). For any transformation semigroup S on Ω, let PS(Ω) stand for the set {kerΩ(s) : s ∈ S} = {kerΩ(IdΩ)s∗ : s ∈ S}, and call it the kernel partition subposet induced by S. It is clear that PS(Ω) is invariant under the action of the kernel space S∗. Inheriting the rank function on PΩ, PS(Ω) is still a valuated poset. For a permutation group, all its elements have the same kernel. For a transformation semigroup, the existence of different kernels may make some arguments for permutation groups invalid. It looks interesting to study the action of the kernel space S∗ on the kernel partition subposet PS(Ω). Example 5.4. Consider the Černý automaton C4 = Γ(S,G) as illustrated in Figure 3, where G = {a, b}. All partitions of {1, 2, 3, 4}, excepting {{0, 2}, {1, 3}} which is dis- played in red in Figure 3, belong to PS(Ω). One can check that WS(S∗|PS(Ω),PS(Ω)) = (1, 1, 1, 1)and SS(S∗|PS(Ω),PS(Ω)) = (1, 2, 1, 1), both of which being unimodal. 668 Ars Math. Contemp. 22 (2022) #P4.09 / 649–674 Example 5.5. Let Ω = {1, . . . , 6} and let S = ⟨r, b⟩ be the transformation semigroup acting on Ω as defined in Example 2.5. Simple calculations shows that PS(Ω) is given by {{{1}, {2}, {3}, {4}, {5}, {6}} , {{1, 2}, {3, 4}, {5, 6}} , {{1, 6}, {2, 3}, {4, 5}}} . One can further check that WS(S∗|PS(Ω),PS(Ω)) = SS(S∗|PS(Ω),PS(Ω)) = (1, 0, 0, 1). If you delete those 0-entries (equivalently, adjusting the rank function for PS(Ω)), the resulting sequence (1, 1) is still unimodal. 6 Linear space 6.1 Top-heavy shape Let n be the size of a nonempty set Ω. Let k and ℓ be two integers satisfying 0 ≤ k ≤ ℓ ≤ n. Let q be a prime power. As q-analogues of the set incidence operator speci- fied in Equation (3.1), we define two linear transformations Mk,ℓq,n : QP k q,n → QP ℓ q,n and Nk,ℓq,n : QA k q,n−1 → QA ℓ q,n−1 as follows: (Mk,ℓq,n(f))(Y ) . = ∑ X≤Y,X∈Pkq,n f(X), and (Nk,ℓq,n(f ′))(Y ′) . = ∑ X′≤Y ′,X′∈Akq,n−1 f(X ′), for all f ∈ QP k q,n , Y ∈ Pℓq,n and f ′ ∈ QA k q,n−1 , Y ′ ∈ Aℓq,n−1. Kantor [30, Theorem] obtained a q-analogue of Gottlieb’s Theorem [25, Corollary 2], which implies that Conjecture 3.4 holds for affine/projective geometries. Lemma 6.1 (Kantor). Let n be a positive integer. Let k and ℓ be two nonnegative integers such that k ≤ ℓ ≤ k + ℓ ≤ n and let q be any prime power. Then both Mk,ℓq,n and N k,ℓ q,n−1 are injective. Proof of Theorem 2.12. Let k and ℓ be two integers such that 0 ≤ k ≤ ℓ ≤ k + ℓ ≤ n. Note that SP ⊆ hEndk,ℓ(Pq,n) and TA ⊆ hEndk,ℓ(Aq,n−1). Since both Pq,n and Aq,n−1 are (ℓ, k)-finite, the result follows readily from Lemmas 3.2, 3.6 and 6.1. 6.2 Duality: A result of Stanley First Proof of Theorem 2.16. Let F be a field and Ω be a set. For each linear subspace U ≤ FΩ, let U⊥ be the subspace of FΩ given by U⊥ . = {f ∈ FΩ : ∑ ω∈Ω f(ω)g(ω) = 0 for all g ∈ U}. Take a matrix A ∈ FΩ×Ω and record its transpose by A⊤. For any f ∈ FΩ, which can be thought of as a row vector indexed by Ω, the image of f under the action of A, written as fA, can be thought of as the product of the row vector f and the matrix A. The matrix A induces a transformation  on Gr(FΩ) such that U ∈ Gr(FΩ) is sent to Y. Wu and Y. Zhu: Top-heavy phenomena for transformations 669 f Mk,ℓq,n(f) fg‡,k Mk,ℓq,n(fg ‡,k) Mk,ℓq,n g‡,k Mk,ℓq,n g‡,ℓ Figure 6: The incidence operator intertwines with every linear isomorphism g. U . = {fA : f ∈ U}. It is easy to see that for any U,W ∈ Gr(V ) we have the implication U =W =⇒W⊥Â⊤ ≤ U⊥; (6.1) especially, when A ∈ GLn(F ) it holds U =W ⇐⇒W⊥Â⊤ = U⊥. (6.2) According to Taussky and Zassenhaus [63, Theorem 1], we can find P ∈ GLn(F ) such that P = P⊤ and A⊤ = PAP−1. This means that Equations (6.1), (6.2) become U =W =⇒ (W⊥P̂ ) ≤ U⊥P̂ and U =W ⇐⇒ (W⊥P̂ ) = U⊥P̂ , (6.3) respectively. It is well-known that q-binomial coefficients (Gaussian coefficients) occur in pairs, namely in any n-dimensional linear space over a finite field, the number of k- dimensional subspaces is equal to the number of (n − k)-dimensional subspaces [24, Proposition 5.31] [60, Section 3]. In general, as a consequence of Equation 6.3, for any A ∈ GLn(F ), the number of k-dimension subspaces of Fn fixed by  equals to the num- ber of (n − k)-dimension subspaces of Fn fixed by Â. If F is a finite field and G is a subgroup of GLn(F ), in view of the Orbit Counting Lemma (also known as Burnside’s Lemma), the above discussion leads to a proof of Theorem 2.16. Second Proof of Theorem 2.16. LetG ≤ GLn(Fq) and let k be a positive integer fulfilling k ≤ n2 . The group G can be seen as a permutation group acting on both Gr(n− k,F n q ) = Pkq,n and Gr(n− k,Fnq ) = Pn−kq,n ; we use Wk and Wn−k for the two permutation modules accordingly. From Lemma 6.1, we see that Mk,n−kq,n is an Fq-linear isomorphism from Pkq,n to Pn−kq,n . From Figure 5 and Example 3.7, we have the commutative diagram in Figure 6 for 2 ≤ k ≤ n2 ; assuming that g comes from the group G, clearly our deduction of Figure 5 shows that Figure 6 is also valid for k = 1. This then shows that Wk and Wn−k are isomorphic permutation modules for G. In particular, the number of orbits of G on Pkq,n and the number of its orbits on Pn−kq,n must be equal. By examining the proofs of Theorem 2.16, we intend to understand the challenge of extending some results on group actions to that on semigroup actions. The above two 670 Ars Math. Contemp. 22 (2022) #P4.09 / 649–674 proofs apply to a set of invertible linear operators over finite linear spaces. If we have a single linear operator A ∈ Matn(F ), by considering its action on the linear space obtained by “collapsing” the eventual kernel of A to zero, we may somehow still say something similar to above. When we have a subsemigroup S of the full linear transformation monoid acting on a finite linear space, different elements of S may have different eventual kernels and that makes it nontrivial to glean global information about the semigroup action. 7 Vámos matroid Proof of Example 2.18. A simple calculation shows that ker(ζk,ℓF(M)) = {0} for (k, ℓ) ∈ {(1, 2), (2, 3)}. Let f ∈ S and let f ′ : F(M) → F(M) be the map sending each flat X ∈ F(M) to the minimum flat containing Xf in M . By Lemma 3.2, we will be done if we can show that f ′ ∈ hEndℓ,k(F(M)) for (k, ℓ) = (1, 2), (2, 3). If we know that f is a bijection or that |EMf | ≤ 2, we can easily check that f ′ ∈ hEndℓ,k(F(M)), as wanted. We intend to find a contradiction under the hypothesis that neither of them holds. By assumption, we can take three distinct elements x, y, z in EMf such that |xf−1| ≥ 2. Let A be the minimum flat containing {x, y, z} and let B = Af−1. Observe that |A| ∈ {3, 4}. Since f is a strong map, B is a flat containing at least four elements and so |B| ∈ {4, 8}. CASE 1: |B| = 8. Take any X ∈ ( A 2 ) . Note that X must be a flat and thus so is Xf−1. Since |EMf | ≥ 3, we deduce that the flat Xf−1 is not equal to EM and so |Xf−1| ≤ 4. Considering that |A| ∈ {3, 4}, we find that |A| = 4 and each element in A has two perimages under f . Note that every element in ( A 2 ) is a flat. It follows that {Xf−1 : X ∈ ( A 2 ) } is a set of six distinct flats and each of them contains four elements, which cannot happen for the Vámos matroid M . CASE 2: |B| = 4. Thanks to the assumption of |B| = 4, we see that C = {x, y} is a flat in M satisfying |Cf−1| = 3. Note that no three-elements subset of any four-elements flat in M can be a flat. This means thatCf−1 is not a flat, violating the assumption that f is a strong map. 8 Concluding remarks We have discussed some top-heavy phenomena for transformation semigroups acting on Boolean semirings, affine/projective geometries, and flat lattice of Vámos matroid; see The- orems 2.1 and 2.12 and Example 2.18. But some problems remain, say Question 2.2, 2.3 and 2.8, Conjecture 2.10 and Question 2.11, and Question 2.15. Our work relies on various injectivity results, say Lemmas 4.1, 5.3 and 6.1, which can all be deduced from Theo- rem 5.1 and Remark 5.2. We may think of Conjecture 3.4 as a natural companion to [28, Conjecture 7]. Since our results on comparing the number of components inside Pk and that of Pℓ for various valuated posets P come from the injectivity of the relevant incidence operators (Lemma 3.2), we indeed have an injective map from components of Pk to that of Pℓ which respects the poset structure. It is noteworthy that we do not find any general results on the unimodality of the strong/weak shape of a semigroup action on a valuated poset. We are wondering if the theory on the Lefschetz properties of commutative graded Y. Wu and Y. Zhu: Top-heavy phenomena for transformations 671 algebras [27] will be useful in this line of research. Penttila and Siciliano [47, Lemma 3.1] suggested a machinery (Lemma 3.6) to remove certain finiteness assumption. But we do not see any way to solve Question 1.2 and 2.17 by this means. Since there are many other approaches to go from finite to infinite [53], it will not be a surprise if Question 1.2 has a positive solution as simple as that for Theorem 2.12. Here is another such question. By our definition, a valuated poset only has nonnegative integers as ranks of its elements. We may allow ranks to be any (not necessarily finite) cardinal number and then examine all the work in this paper again. A few results of this kinds from the literature have been addressed at the end of Section 2.1. ORCID iDs Yaokun Wu https://orcid.org/0000-0002-6811-7067 Yinfeng Zhu https://orcid.org/0000-0003-1724-5250 References [1] K. Adiprasito, J. Huh and E. Katz, Hodge theory for combinatorial geometries, Ann. of Math. (2) 188 (2018), 381–452, doi:10.4007/annals.2018.188.2.1. [2] J. André, J. a. Araújo and P. J. Cameron, The classification of partition homogeneous groups with applications to semigroup theory, J. Algebra 452 (2016), 288–310, doi:10.1016/j.jalgebra. 2015.12.025. [3] C. A. Athanasiadis, The symmetric group action on rank-selected posets of injective words, Order 35 (2018), 47–56, doi:10.1007/s11083-016-9417-9. [4] L. Babai and P. Frankl, Linear Algebra Methods in Combinatorics With Applications to Ge- ometry and Computer Science, Department of Computer Science, The University of Chicago, 1992. [5] A. Bachem and A. Wanka, Separation theorems for oriented matroids, Discrete Math. 70 (1988), 303–310, doi:10.1016/0012-365x(88)90006-4. [6] M. Baker, Hodge theory in combinatorics, Bull. Am. Math. Soc. (N.S.) 55 (2018), 57–80, doi: 10.1090/bull/1599. [7] M. Baker and N. Bowler, Matroids over partial hyperstructures, Adv. Math. 343 (2019), 821– 863, doi:10.1016/j.aim.2018.12.004. [8] M. Benoumhani, Log-concavity of Whitney numbers of Dowling lattices, Adv. in Appl. Math. 22 (1999), 186–189, doi:10.1006/aama.1998.0621. [9] R. D. Bercov and C. R. Hobby, Permutation groups on unordered sets, Math. Z. 115 (1970), 165–168, doi:10.1007/bf01109854. [10] A. Björner and T. Ekedahl, On the shape of Bruhat intervals, Ann. of Math. (2) 170 (2009), 799–817, doi:10.4007/annals.2009.170.799. [11] F. Brenti, Log-concave and unimodal sequences in algebra, combinatorics, and geometry: an update, in: Jerusalem Combinatorics ’93, Amer. Math. Soc., Providence, RI, volume 178 of Contemp. Math., pp. 71–89, 1994, doi:10.1090/conm/178/01893. [12] M. Brown, Weak n-homogeneity implies weak (n− 1)-homogeneity, Proc. Am. Math. Soc. 10 (1959), 644–647, doi:10.2307/2033668. [13] H. Bruhn, R. Diestel, M. Kriesell, R. Pendavingh and P. Wollan, Axioms for infinite matroids, Adv. Math. 239 (2013), 18–46, doi:10.1016/j.aim.2013.01.011. 672 Ars Math. Contemp. 22 (2022) #P4.09 / 649–674 [14] V. M. Bukhshtaber, A. M. Vershik, S. A. Evdokimov and I. N. Ponomarenko, Combinatorial algebras and multivalued involutive groups, Funct. Anal. Its Appl. 30 (1996), 158–162, doi: 10.1007/bf02509502. [15] P. J. Cameron, Transitivity of permutation groups on unordered sets, Math. Z. 148 (1976), 127–139, doi:10.1007/bf01214702. [16] P. J. Cameron, Oligomorphic permutation groups, in: Perspectives in mathematical sciences. II, World Sci. Publ., Hackensack, NJ, volume 8 of Stat. Sci. Interdiscip. Res., pp. 37–61, 2009, doi:10.1142/9789814273657\ 0003. [17] A. Chubarev and I. Pinelis, Fundamental theorem of geometry without the 1-to-1 assumption, Proc. Am. Math. Soc. 127 (1999), 2735–2744, doi:10.1090/s0002-9939-99-05280-6. [18] D. de Caen, A note on the ranks of set-inclusion matrices, Electron. J. Comb. 8 (2001), Note 5, 2, doi:10.37236/1590. [19] E. Delucchi and S. Riedel, Group actions on semimatroids, Adv. in Appl. Math. 95 (2018), 199–270, doi:10.1016/j.aam.2017.11.001. [20] P. Dembowski, Finite Geometries: Reprint of the 1968 Edition, Springer Berlin, Heidelberg, 2012, doi:10.1007/978-3-642-62012-6. [21] E. Dobson and A. Malnič, Groups that are transitive on all partitions of a given shape, J. Alge- braic Comb. 42 (2015), 605–617, doi:10.1007/s10801-015-0593-2. [22] R. Dougherty, C. Freiling and K. Zeger, Networks, matroids, and non-Shannon information inequalities, IEEE Trans. Inform. Theory 53 (2007), 1949–1969, doi:10.1109/tit.2007.896862. [23] T. A. Dowling and R. M. Wilson, Whitney number inequalities for geometric lattices, Proc. Am. Math. Soc. 47 (1975), 504–512, doi:10.2307/2039773. [24] G. Gordon and J. McNulty, Matroids: A Geometric Introduction, Cambridge University Press, Cambridge, 2012, doi:10.1017/cbo9781139049443. [25] D. H. Gottlieb, A certain class of incidence matrices, Proc. Am. Math. Soc. 17 (1966), 1233– 1237, doi:10.2307/2035716. [26] A. Hajnal, A remark on the homogeneity of infinite permutation groups, Bull. London Math. Soc. 22 (1990), 529–532, doi:10.1112/blms/22.6.529. [27] T. Harima, T. Maeno, H. Morita, Y. Numata, A. Wachi and J. Watanabe, The Lefschetz properties, volume 2080 of Lecture Notes in Mathematics, Springer, Heidelberg, 2013, doi: 10.1007/978-3-642-38206-2. [28] J. Huh and B. Wang, Enumeration of points, lines, planes, etc, Acta Math. 218 (2017), 297–317, doi:10.4310/acta.2017.v218.n2.a2. [29] O. Johnson, I. Kontoyiannis and M. Madiman, Log-concavity, ultra-log-concavity, and a max- imum entropy property of discrete compound Poisson measures, Discrete Appl. Math. 161 (2013), 1232–1250, doi:10.1016/j.dam.2011.08.025. [30] W. M. Kantor, On incidence matrices of finite projective and affine spaces, Math. Z. 124 (1972), 315–318, doi:10.1007/bf01113923. [31] W. M. Kantor, Line-transitive collineation groups of finite projective spaces, Israel J. Math. 14 (1973), 229–235, doi:10.1007/bf02764881. [32] W. M. Kantor, Homogeneous designs and geometric lattices, J. Comb. Theory Ser. A 38 (1985), 66–74, doi:10.1016/0097-3165(85)90022-6. [33] J. P. S. Kung, Strong maps, in: Theory of matroids, Cambridge Univ. Press, Cambridge, vol- ume 26 of Encyclopedia Math. Appl., pp. 224–253, 1986, doi:10.1017/cbo9780511629563. 011. Y. Wu and Y. Zhu: Top-heavy phenomena for transformations 673 [34] J. P. S. Kung, The Radon transforms of a combinatorial geometry. II. Partition lattices, Adv. Math. 101 (1993), 114–132, doi:10.1006/aima.1993.1044. [35] J. P. S. Kung and H. Q. Nguyen, Weak maps, in: Theory of matroids, Cambridge Univ. Press, Cambridge, volume 26 of Encyclopedia Math. Appl., pp. 254–271, 1986, doi:10.1017/ cbo9780511629563.012. [36] J. P. S. Kung, G.-C. Rota and C. H. Yan, Combinatorics: the Rota way, Cambridge Mathemat- ical Library, Cambridge University Press, Cambridge, 2009, doi:10.1017/cbo9780511803895. [37] E. H. Lieb, Concavity properties and a generating function for Stirling numbers, J. Comb. Theory. 5 (1968), 203–206, doi:10.1016/s0021-9800(68)80057-2. [38] D. Livingstone and A. Wagner, Transitivity of finite permutation groups on unordered sets, Math. Z. 90 (1965), 393–403, doi:10.1007/bf01112361. [39] W. J. Martin and B. E. Sagan, A new notion of transitivity for groups and sets of permutations, J. London Math. Soc. (2) 73 (2006), 1–13, doi:10.1112/s0024610705022441. [40] J. H. Mason, Matroids: unimodal conjectures and Motzkin’s theorem, in: Combinatorics (Proc. Conf. Combinatorial Math., Math. Inst., Oxford, 1972), 1972 pp. 207–220. [41] V. B. Mnukhin and I. J. Siemons, On the Livingstone-Wagner theorem, Electron. J. Comb. 11 (2004), Research Paper 29, 8, doi:10.37236/1782. [42] M. M. Mukhopadhyay and E. Vassiliev, On the Vámos matroid, homogeneous pregeometries and dense pairs, Australas. J. Comb. 75 (2019), 158–170, https://ajc.maths.uq.edu. au/?page=get_volumes&volume=75. [43] Y. Nakashima, A partial generalization of the Livingstone-Wagner theorem, Ars Math. Con- temp. 2 (2009), 207–215, doi:10.26493/1855-3974.92.46f. [44] P. M. Neumann, Homogeneity of infinite permutation groups, Bull. London Math. Soc. 20 (1988), 305–312, doi:10.1112/blms/20.4.305. [45] M. Pankov, Geometry of Semilinear Embeddings: Relations to Graphs and Codes, World Sci- entific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015, doi:10.1142/9465. [46] M. Pankov, Apartments preserving transformations of Grassmannians of infinite-dimensional vector spaces, Linear Algebra Appl. 531 (2017), 498–509, doi:10.1016/j.laa.2017.06.016. [47] T. Penttila and A. Siciliano, On the incidence map of incidence structures, Ars Math. Contemp. 20 (2021), 51–68, doi:10.26493/1855-3974.1996.db7. [48] A. Putman, The fundamental theorem of projective geometry, 2010, https://www3.nd. edu/˜andyp/notes/. [49] F. P. Ramsey, On a Problem of Formal Logic, Proc. London Math. Soc. (2) 30 (1929), 264–286, doi:10.1112/plms/s2-30.1.264. [50] G.-C. Rota, Combinatorial theory, old and new, in: Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 3, Gauthier-Villars, Paris, pp. 229–233, 1971. [51] P. Roy, Another proof that weak n-homogeneity implies weak (n−1)-homogeneity, Proc. Am. Math. Soc. 49 (1975), 515–516, doi:10.2307/2040675. [52] P. Šemrl, The Optimal Version of Hua’s Fundamental Theorem of Geometry of Rectangu- lar Matrices, Mem. Am. Math. Soc. 232 (2014), vi+74, https://bookstore.ams.org/ memo-232-1089. [53] J.-P. Serre, How to use finite fields for problems concerning infinite fields, in: Arithmetic, Geometry, Cryptography and Coding Theory, Amer. Math. Soc., Providence, RI, volume 487 of Contemp. Math., pp. 183–193, 2009, doi:10.1090/conm/487/09532. 674 Ars Math. Contemp. 22 (2022) #P4.09 / 649–674 [54] P. D. Seymour, On secret-sharing matroids, J. Comb. Theory Ser. B 56 (1992), 69–73, doi: 10.1016/0095-8956(92)90007-k. [55] S. Shelah and S. Thomas, Homogeneity of infinite permutation groups, Arch. Math. Logic 28 (1989), 143–147, doi:10.1007/bf01633987. [56] J. Siemons, On partitions and permutation groups on unordered sets, Arch. Math. (Basel) 38 (1982), 391–403, doi:10.1007/bf01304806. [57] M. T. Stamps, Topological representations of matroid maps, J. Algebraic Comb. 37 (2013), 265–287, doi:10.1007/s10801-012-0366-0. [58] R. P. Stanley, Two combinatorial applications of the Aleksandrov-Fenchel inequalities, J. Comb. Theory Ser. A 31 (1981), 56–65, doi:10.1016/0097-3165(81)90053-4. [59] R. P. Stanley, Some aspects of groups acting on finite posets, J. Comb. Theory Ser. A 32 (1982), 132–161, doi:10.1016/0097-3165(82)90017-6. [60] R. P. Stanley, GL(n,C) for combinatorialists, in: Surveys in combinatorics (Southampton, 1983), Cambridge Univ. Press, Cambridge, volume 82 of London Math. Soc. Lecture Note Ser., pp. 187–199, 1983, https://math.mit.edu/˜rstan/pubs/. [61] R. P. Stanley, Log-concave and unimodal sequences in algebra, combinatorics, and geometry, in: Graph theory and its applications: East and West (Jinan, 1986), New York Acad. Sci., New York, volume 576 of Ann. New York Acad. Sci., pp. 500–535, 1989, doi:10.1111/j.1749-6632. 1989.tb16434.x. [62] T. Stokes, Semigroup actions on posets and preimage quasi-orders, Semigroup Forum 85 (2012), 540–558, doi:10.1007/s00233-012-9436-9. [63] O. Taussky and H. Zassenhaus, On the similarity transformation between a matrix and its transpose, Pacific J. Math. 9 (1959), 893–896, http://projecteuclid.org/euclid. pjm/1103039127. [64] J. H. van Lint and R. M. Wilson, A Course in Combinatorics, Cambridge University Press, Cambridge, 2nd edition, 2001, doi:10.1017/cbo9780511987045. [65] O. Y. Viro, On basic concepts of tropical geometry, Proc. Steklov Inst. Math. 273 (2011), 252– 282, doi:10.1134/s0081543811040134. [66] Z.-X. Wan, Geometry of Matrices, World Scientific Publishing Co., Inc., River Edge, NJ, 1996, doi:10.1142/9789812830234. [67] Y. Wu, Z. Xu and Y. Zhu, A five-element transformation monoid on labelled trees, European J. Comb. 80 (2019), 401–415, doi:10.1016/j.ejc.2018.07.014. [68] Y. Wu and S. Zhao, Incidence matrix and cover matrix of nested interval orders, Electron. J. Linear Algebra 23 (2012), 43–65, doi:10.13001/1081-3810.1504. ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P4.10 / 675–686 https://doi.org/10.26493/1855-3974.2029.01d (Also available at http://amc-journal.eu) The number of rooted forests in circulant graphs* Lilya A. Grunwald , Ilya Mednykh † Sobolev Institute of Mathematics, 630090, Novosibirsk, Russia Novosibirsk State University, 630090, Novosibirsk, Russia Received 28 June 2019, accepted 28 February 2022, published online 26 August 2022 Abstract In this paper, we develop a new method to produce explicit formulas for the number fG(n) of rooted spanning forests in the circulant graphs G = Cn(s1, s2, . . . , sk) and G = C2n(s1, s2, . . . , sk, n). These formulas are expressed through Chebyshev polynomials. We prove that in both cases the number of rooted spanning forests can be represented in the form fG(n) = p a(n)2, where a(n) is an integer sequence and p is a certain natural number depending on the parity of n. Finally, we find an asymptotic formula for fG(n) through the Mahler measure of the associated Laurent polynomial P (z) = 2k+1− ∑k i=1(z si+z−si). Keywords: Rooted tree, spanning forest, circulant graph, Laplacian matrix, Chebyshev polynomial, Mahler measure. Math. Subj. Class. (2020): 05C30, 39A12 1 Introduction The famous Kirchhoff’s Matrix Tree Theorem [15] states that the number of spanning trees in a graph can be expressed as the product of its non-zero Laplacian eigenvalues divided by the number of vertices. Since then, a lot of papers on enumeration of spanning trees for various classes of graphs were published. In particular, explicit formulae were derived for complete multipartite graphs [1, 5], almost complete graphs [33], wheels [3], fans [12], prisms [2], ladders [26], Moebius ladders [27], lattices [28], anti-prisms [31], *The authors are grateful to all the three anonymous referees for careful reading of manuscript and valuable remarks and suggestions. The authors were supported by the Mathematical Center in Akademgorodok, agreement no. 075-15-2019-1613 with the Ministry of Science and Higher Education of the Russian Federation. †Corresponding author. E-mail addresses: lfb o@yahoo.co.uk (Lilya A. Grunwald), ilyamednykh@mail.ru (Ilya Mednykh) cb This work is licensed under https://creativecommons.org/licenses/by/4.0/ 676 Ars Math. Contemp. 22 (2022) #P4.10 / 675–686 complete prisms [25] and for many other families. For the circulant graphs some explicit and recursive formulae are given in [8, 23, 34, 35]. Along with the number of spanning trees in a given graph one can be interested in the number of rooted spanning forests in the graph. According to the classical result [14] (see also more recent paper [7, 16]) this value can be found with the use of determinant of the matrix det(I + L). Here L is the Laplacian matrix of the graph. At the same time, it is known very little about analytic formulas for the number of spanning forests. In particular, P. Chebotarev [6] enumerated the rooted spanning forests in path and cyclic graphs and O. Knill [16] proved that the number of rooted spanning forests in the complete graph Kn on n vertices is equal to (n+1)n−1. The rooted spanning forests in complete bipartite graphs were enumerated in [13]. Explicit formulas for the number of rooted spanning forests for cyclic, star, line and some other graphs were given by [16]. As for the number of unrooted forests, it has much more complicated structure [4, 19, 32]. Starting with Boesch and Prodinger [3] the idea to apply Chebyshev polynomials for counting various invariants of graphs arose. This idea provided a way to find complexity, that is the number of spanning tress, of circulant graphs and their natural generalisations in [8, 17, 23, 24, 35]. Recently, asymptotical behavior of complexity for some families of graphs was inves- tigated from the point of view of so called Malher measure [11, 29]. Mahler measure of a polynomial P (z), with complex coefficients, is the absolute value of the product of all the roots of P (z) whose modulus is greater than 1 multiplied by the leading coefficient. For general properties of the Mahler measure see survey [30] and monograph [10]. The purpose of this paper is to present new formulas for the number of rooted spanning forests in circulant graphs and investigate their arithmetical properties and asymptotics. We arrange the paper in the following way. First, in Sections 3 and 4, we present new explicit formulas for the number of spanning forests in the undirected circulant graphs Cn(s1, s2, . . . , sk) and C2n(s1, s2, . . . , sk, n) of even and odd valency respectively. They will be given in terms of Chebyshev polynomials. Next, in Section 5, some arithmetic properties of the number of spanning forests are investigated. More precisely, it is shown that the number of spanning forests of the circulant graph G can be represented in the form fG(n) = p a(n) 2, where a(n) is an integer sequence and p is a certain natural number. At last, in Section 6, we use explicit formulas for fG(n) in order to find its asymptotics in terms of Mahler measure of the associated polynomials. For circulant graphs of even valency the associated polynomial is P (z) = 2k + 1 − ∑k j=1(z sj + z−sj ). In this case (Theorem 6.1) we have fG(n) ∼ An, n → ∞, where A is the Mahler measure of P (z). For circulant graphs of odd valency we use the polynomial R(z) = P (z)(P (z) + 2). Then the respective asymptotics (Theorem 6.2) is fG(n) ∼ Kn, n → ∞, where K = M(R). In the last Section 7, we illustrate the obtained results by a series of examples. 2 Basic definitions and preliminary facts Consider a finite and not necessary connected graph G without loops. A rooted tree is a tree with one marked vertex called root. A rooted forest is a graph whose connected components are rooted trees. A rooted spanning forest F in the graph G is a subgraph that is a rooted forest containing all the vertices of G. We denote the vertex and edge set of G by V (G) and E(G), respectively. Given u, v ∈ V (G), we set auv to be equal to the number of edges between vertices u and v. The matrix A = A(G) = (auv)u, v∈V (G) is called the L. A. Grunwaldr and I. Mednykh: The number of rooted forests in circulant graphs 677 adjacency matrix of the graph G. The degree d(v) of a vertex v ∈ V (G) is defined by d(v) = ∑ u∈V (G) auv. Let D = D(G) be the diagonal matrix indexed by the elements of V (G) with dvv = d(v). The matrix L = L(G) = D(G) − A(G) is called the Laplacian matrix, or simply Laplacian, of the graph G. By In we denote the identity matrix of order n. Denote by χG(λ) = det(λIn − L(G)) the characteristic polynomial of the Laplacian matrix of a graph G on n vertices. Its extended form is χG(λ) = c1λ+ . . .+ cn−1λ n−1 + λn. The theorem by Kelmans and Chelnokov [14] states that the absolute value of coefficient ck of χG(λ) coincides with the number of rooted spanning k-forests in the graph G. Since all the Laplacian eigenvalues of G are non-negative, the sequence ck is alternating. So, the number of rooted spanning forests of the graph G can be found by the formula fG(n) = f1 + f2 + . . .+ fn = |c1 − c2 + c3 − . . .+ (−1)n−1| (2.1) = (−1)nχG(−1) = det(In + L(G)). This result was independently obtained by many authors (P. Chebotarev and E. Shamis [7] O. Knill [16] and others). Let s1, s2, . . . , sk be integers such that 1 ≤ s1 < s2 < . . . < sk ≤ n2 . The graph Cn(s1, s2, . . . , sk) with n vertices 0, 1, 2, . . . , n− 1 is called circulant graph if the vertex i, 0 ≤ i ≤ n−1 is adjacent to the vertices i±s1, i±s2, . . . , i±sk (mod n). When sk < n2 all vertices of the graph have even degree 2k. If n is even and sk = n2 , then all vertices have odd degree 2k−1. Two circulant graphs Cn(s1, s2, . . . , sk) and Cn(s̃1, s̃2, . . . , s̃k) of the same order are said to be conjugate by multiplier if there exists an integer r coprime to n such that {s̃1, s̃2, ..., s̃k} = {rs1, rs2, . . . , rsk} as subsets of Zn. In this case, the graphs are isomorphic, with multiplication by the unit r (modn) giving an isomorphism. We call an n× n matrix circulant, and denote it by circ(a0, a1, . . . , an−1) if it is of the form circ(a0, a1, . . . , an−1) =  a0 a1 a2 . . . an−1 an−1 a0 a1 . . . an−2 ... . . . ... a1 a2 a3 . . . a0  It is easy to see that adjacency and Laplacian matrices of the circulant graph are circu- lant matrices. The converse is also true. If the Laplacian matrix of a graph is circulant then the graph is also circulant. Recall [9] that the eigenvalues of matrix C = circ(a0, a1, . . . , an−1) are given by the following simple formulas λj = P (εjn), j = 0, 1, . . . , n − 1, where P (x) = a0 + a1x + . . . + an−1x n−1 and εn is an order n primitive root of unity. Moreover, the cir- culant matrix T = circ(0, 1, 0, . . . , 0) is the matrix representation of the shift operator T : (x0, x1, . . . , xn−2, xn−1) → (x1, x2, . . . , xn−1, x0). Let P (z) = a0zs + . . . + as = a0 ∏s i=1(z − αi) be a nonconstant polynomial with complex coefficients. Then, following Mahler [21] its Mahler measure is defined to be M(P ) := exp( ∫ 1 0 log |P (e2πit)|dt), (2.2) 678 Ars Math. Contemp. 22 (2022) #P4.10 / 675–686 the geometric mean of |P (z)| for z on the unit circle. However, M(P ) had appeared earlier in a paper by Lehmer [18], in an alternative form M(P ) = |a0| ∏ |αi|>1 |αi|. (2.3) See, for example [10], for the proof of equivalence of these definitions. The concept of Mahler measure can be naturally extended to the class of Laurent poly- nomials P (z) = a0zp+s+a1zp+s−1+. . .+as−1zs+1+aszs = a0zp ∏s i=1(z−αi), where a0 ̸= 0, s is a positive integer and p is an arbitrary and not necessarily positive integer. Let Tn(z) = cos(n arccos z) be the Chebyshev polynomial of the first kind. The fol- lowing property of the Chebyshev polynomials is widely used in the paper Tn( 1 2 (z+ z −1)) = 12 (z n + z−n). See [22] for more properties of Chebyshev polynomials. 3 The number of rooted spanning forests of circulant graphs of even valency The aim of this section is to find new formulas for the numbers of rooted spanning forests of circulant graph Cn(s1, s2, . . . , sk) in terms of Chebyshev polynomials. Here and below, we will use G to denote the circulant graph under consideration. Theorem 3.1. The number of rooted spanning forests fG(n) in the circulant graph G = Cn(s1, s2, . . . , sk), 1 ≤ s1 < s2 < . . . < sk < n2 , is given by the formula fG(n) = sk∏ p=1 |2Tn(wp)− 2|, where wp, p = 1, 2, . . . , sk are all the roots of the algebraic equation ∑k j=1(2Tsj (w) − 2) = 1 and Ts(w) is the Chebyshev polynomial of the first kind. Proof. The number of rooted spanning forests of the graph G can be found by the formula fG(n) = det(In + L(G)). The latter value is equal to the product of all eigenvalues of the matrix In + L(G). We denote by T = circ(0, 1, . . . , 0) the n× n cyclic shift operator. Consider the Laurent polynomial P (z) = 2k + 1 − ∑k i=1(z si + z−si). Then the matrix In + L(G) has the following form In + L(G) = P (T ) = (2k + 1)In − k∑ i=1 (T si + T−si). The eigenvalues of circulant matrix T are εjn, j = 0, 1, . . . , n− 1, where εn = e 2πi n . Since all of them are distinct, the matrix T is similar to the matrix T = diag(1, εn, . . . , εn−1n ) with diagonal entries 1, εn, . . . , εn−1n . So the matrix In + L(G) is similar to the diagonal matrix P (T). This essentially simplifies the problem of finding eigenvalues of In + L(G). Indeed, let λ be an eigenvalue of P (T) and x be the corresponding eigenvector. Then we have the following system of linear equations ((2k + 1− λ)In − k∑ i=1 (Tsi + T−si))x = 0. L. A. Grunwaldr and I. Mednykh: The number of rooted forests in circulant graphs 679 Recall that the matrices under consideration are diagonal and the (j + 1, j + 1)-th entry of T is equal to εjn, where εn = e 2πi n . Then, for any j = 0, . . . , n − 1, matrix P (T) has an eigenvalue λj = P (εjn) = 2k + 1− ∑k i=1(ε jsi n + ε −jsi n ). Hence we have fG(n) = n−1∏ j=0 P (εjn). (3.1) To continue the proof of the theorem we need the following lemma. Lemma 3.2. n−1∏ j=0 P (εjn) = sk∏ p=1 |2Tn(wp)− 2|, where wp, p = 1, . . . , sk are all the roots of the algebraic equation ∑k j=1(2Tsj (w)−2) = 1. To prove the above formula we introduce integer polynomial P̃ (z) = −zskP (z). This is a monic polynomial with the same roots as P (z) and its degree is 2sk. As P (z) = P ( 1z ), its roots look like z1, 1z1 , . . . , zsk , 1 zsk . We have ∏n−1 j=0 P (ε j n) = ∏n−1 j=0 (−ε−skjn P̃ (εjn)) = (−1)(sk+1)(n+1)−1 ∏n−1 j=0 P̃ (ε j n). Recall one of the basic properties of resultants Res (P̃ (z), Q̃(z)) = (−1)deg(P̃ ) deg(Q̃)Res (Q̃(z), P̃ (z)), where P̃ (z) and Q̃(z) are monic polynomials of degree deg(P̃ ) and deg(Q̃) respectively. We set Q̃(z) = zn − 1 and note that deg(P̃ ) = 2sk is even. Then we obtain n−1∏ j=0 P̃ (εjn) = Res (P̃ (z), z n − 1) = Res (zn − 1, P̃ (z)) = ∏ z:P̃ (z)=0 (zn − 1) = ∏ z:P (z)=0 (zn − 1) = sk∏ p=1 (znp − 1)(z−np − 1) = (−1)sk sk∏ p=1 (2Tn(wp)− 2). We used the identity Tn( 12 (z + z −1)) = 12 (z n + z−n). Here wp = 12 (zp + 1 zp ), p = 1, . . . , sk. These numbers are the roots of algebraic equation ∑k j=1(2Tsj (w)−2) = 1. To finish the proof of the theorem we use Lemma 3.2 and take absolute value of the righthand side of the Equation 3.1. 4 The number of rooted spanning forests in circulant graphs of odd valency This section is devoted to investigation of the numbers of rooted spanning forests in circu- lant graph C2n(s1, s2, . . . , sk, n) in terms of Chebyshev polynomials. 680 Ars Math. Contemp. 22 (2022) #P4.10 / 675–686 Theorem 4.1. Let C2n(s1, s2, . . . , sk, n), 1 ≤ s1 < s2 < . . . < sk < n, be a circulant graph of odd degree. Then the number fG(n) of rooted spanning forests in the graph G = C2n(s1, s2, . . . , sk, n) is given by the formula fG(n) = sk∏ p=1 (2Tn(up)− 2)(2Tn(vp) + 2), where the numbers up and vp, p = 1, 2, . . . , sk are respectively the roots of the algebraic equations Q(u)− 1 = 0 and Q(v) + 1 = 0, Q(w) = 2k + 2− 2 ∑k i=1 Tsi(w) and Ts(w) is the Chebyshev polynomial of the first kind. Proof. In order to find the number of rooted spanning forests fG(n) in the graph G we need to find the determinant det(I2n + L(G)). The matrix I2n + L(G) can be represented in the form I2n + L(G) = (2k + 2)I2n − k∑ j=1 (T sj + T−sj )− Tn, where T is 2n × 2n circulant matrix circ(0, 1, 0, . . . , 0). The eigenvalues of circulant ma- trix T are εj2n, j = 0, 1, . . . , 2n − 1, where ε2n = e 2πi 2n . Since all of them are distinct, the matrix T is similar to the matrix T = diag(1, ε2n, . . . , ε2n−12n ) with diagonal entries 1, ε2n, . . . , ε 2n−1 2n . To find the determinant det(I2n + L(G)) we use the product of all eigenvalues of matrix I2n+L(G). The matrix I2n+L(G) is similar to the diagonal matrix with eigenvalues λj = 2k + 2− k∑ l=1 (εj sl2n + ε −j sl 2n )− ε jn 2n, j = 0, 1, . . . , 2n− 1. All of them are non-zero. Consider the following Laurent polynomial P (z) = 2k+2− ∑k i=1(z si +z−si). Since εn2n = −1, we can write λj = P (ε j 2n)− 1 if j is even and λj = P (ε j 2n) + 1 if j is odd. By Formula 2.1 we have fG(n) = 2n−1∏ j=0 λj = n−1∏ s=0 (P (ε2s2n)− 1) n−1∏ s=0 (P (ε2s+12n ) + 1) = n−1∏ s=0 (P (ε2s2n)− 1) ∏2n−1 p=0 (P (ε p 2n) + 1)∏n−1 s=0 (P (ε 2s 2n) + 1) = n−1∏ s=0 (P (εsn)− 1) ∏2n−1 p=0 (P (ε p 2n) + 1)∏n−1 s=0 (P (ε s n) + 1) . By making use of Lemma 3.2 and arguments from the proof of Theorem 3.1 we obtain (i) ∏n−1 s=0 (P (ε s n)− 1) = (−1)n (sk+1) ∏sk p=1(2Tn(up)− 2), (ii) ∏n−1 s=0 (P (ε s n) + 1) = (−1)n (sk+1) ∏sk p=1(2Tn(vp)− 2), and (iii) ∏2n−1 p=0 (P (ε p 2n) + 1) = ∏sk p=1(2T2n(vp)− 2), L. A. Grunwaldr and I. Mednykh: The number of rooted forests in circulant graphs 681 where up and vp are the same as in the statement of the theorem. Hence, fG(n) = sk∏ p=1 (2Tn(up)− 2) sk∏ p=1 T2n(vp)− 1 Tn(vp)− 1 . Finally, taking into account the identity T2n(w)−1 = 2(Tn(w)−1)(Tn(w)+1) we obtain fG(n) = sk∏ p=1 (2Tn(up)− 2)(2Tn(vp) + 2). 5 Arithmetic properties of the number of rooted spanning forests for circulant graphs It has been proved in the paper [23] that the number of spanning trees τ(n) in circulant graph Cn(s1, s2, . . . , sk) is given by the formula τ(n) = p n a(n)2, where a(n) is an integer sequence and p is a natural number depending only on the parity of n. The aim of the next theorem is to find a similar property for the number of rooted spanning forests. Recall that any positive integer p can be uniquely represented in the form p = q r2, where p and q are positive integers and q is square-free. We will call q the square-free part of p. Theorem 5.1. Let fG(n) be the number of rooted spanning forests in the circulant graph Cn(s1, s2, . . . , sk), 1 ≤ s1 < s2 < . . . < sk < n 2 . Denote by p the number of odd elements in the sequence s1, s2, . . . , sk and let q be the square-free part of 4p+ 1. Then there exists an integer sequence a(n) such that (1) fG(n) = a(n)2, if n is odd; (2) fG(n) = q a(n)2, if n is even. Proof. The number of odd elements in the sequence s1, s2, . . . , sk is counted by the for- mula p = ∑k i=1 1−(−1)si 2 . We already know that all eigenvalues of the In + L(G) are given by the formulas λj = P (ε j n), j = 0, . . . , n−1, where P (z) = 2k+1− ∑k i=1(z si +z−si) and εn = e 2πi n . We note that λn−j = P (εn−jn ) = P (ε j n) = λj . Since λ0 = P (ε0n) = P (1) = 1 by Formula 2.1 we have fG(n) = ∏n−1 j=1 λj . Since λn−j = λj , we obtain fG(n) = ( ∏n−1 2 j=1 λj) 2 if n is odd and fG(n) = λn2 ( ∏n 2 −1 j=1 λj) 2 if n is even. We note that each algebraic number λj comes with all its Galois conjugates [20]. So, the numbers b(n) = ∏n−1 2 j=1 λj and c(n) = ∏n 2 −1 j=1 λj are integers. Also, for even n we have λn 2 = 2k + 1 − ∑k i=1((−1)si + (−1)−si) = 1 + 2 ∑k i=1(1 − (−1)si) = 4p + 1. Hence, fG(n) = b(n)2 if n is odd and fG(n) = (4p + 1) c(n)2 if n is even. Let q be the square-free part of 4p + 1 and 4p + 1 = q r2. Setting a(n) = b(n) in the first case and a(n) = r c(n) in the second, we conclude that number a(n) is always integer which completes the proof. 682 Ars Math. Contemp. 22 (2022) #P4.10 / 675–686 The following theorem clarifies some number-theoretical properties of the number of rooted spanning forests fG(n) for circulant graphs of odd valency. Theorem 5.2. Let fG(n) be the number of rooted spanning forests in the circulant graph G = C2n(s1, s2, . . . , sk, n), 1 ≤ s1 < s2 < . . . < sk < n. Denote by p the number of odd elements in the sequence s1, s2, . . . , sk. Let q be the square- free part of 4p + 1 and r be the square-free part of 4p + 3. Then there exists an integer sequence a(n) such that (1) fG(n) = q a(n)2, if n is even; (2) fG(n) = r a(n)2, if n is odd. Proof. The number p of odd elements in the sequence s1, s2, . . . , sk is equal to∑k i=1 1−(−1)si 2 . The eigenvalues of the matrix I2n + L(G) are given by the formulas λj = P (ε j 2n)− (−1)j , j = 0, 1, . . . , 2n− 1, where P (z) = 2k + 2− ∑k l=1(z sl + z−sl) and ε2n = e πi n . Since λ0 = P (1) − 1 = 1 by the Formula 2.1 we have fG(n) = ∏2n−1 j=1 λj . Since λ2n−j = λj , we obtain fG(n) = λn( ∏n−1 j=1 λj) 2, where λn = P (−1) − (−1)n. Now we have λn = 2k+2− (−1)n−2 k∑ l=1 (−1)sl = 2− (−1)n+4 k∑ l=1 1− (−1)sl 2 = 4 p+2− (−1)n. So, λn = 4 p + 1, if n is even and λn = 4 p + 3, if n is odd. We note that each algebraic number λj comes into the product ∏n−1 j=1 λj together with all its Galois conjugates, so the number c(n) = ∏n−1 j=1 λj is an integer [20]. Hence, fG(n) = (4 p+ 1)c(n)2, if n is even and fG(n) = (4 p+ 3) c(n)2, if n is odd. Let q and r be the square-free parts of 4 p + 1 and of 4p + 3 respectively. Then for some integers x and y we have 4 p+ 1 = q x2 and 4 p+ 3 = r y2. Now, the integer number fG(n) can be represented in the form (1) fG(n) = q (x c(n))2 if n is even and (2) fG(n) = r (y c(n))2 if n is odd. Setting a(n) = x c(n) in the first case and a(n) = y c(n) in the second, we conclude that number a(n) is always integer. The theorem is proved. 6 Asymptotics for the number of spanning forests In this section we give asymptotic formulas for the number of rooted spanning forests in circulant graphs. Theorem 6.1. The number of rooted spanning forests in the circulant graph G = Cn(s1, s2, . . . , sk), 1 ≤ s1 < s2 < . . . < sk < n 2 L. A. Grunwaldr and I. Mednykh: The number of rooted forests in circulant graphs 683 has the following asymptotics fG(n) ∼ An, as n → ∞ where A = exp( ∫ 1 0 log(P (e2πit))dt) is the Mahler measure of Laurent polynomial P (z) = 2k + 1− ∑k i=1(z si + z−si). Proof. By Theorem 3.1, the number of rooted spanning forests fG(n) is given by fG(n) = sk∏ s=1 |2Tn(ws)− 2|, where ws = (zs + z−1s )/2. We have Tn(ws) = 1 2 (z n s + z −n s ), where zs and 1/zs, s = 1, . . . , sk are all the roots of the polynomial P (z). If φ ∈ R then P (eiφ) = 2k + 1 − ∑k j=1(e sjiφ + e−sjiφ) = 2k + 1 − 2 ∑k j=1 cos(sjφ) ≥ 1, so |zs| ≠ 1 for all s. Replacing zs by 1/zs, if it is necessary, we can assume that |zs| > 1 for all s. Then Tn(ws) ∼ 12z n s , as n tends to ∞. So, |2Tn(ws)− 2| ∼ |zs|n, n → ∞. Hence sk∏ s=1 |2Tn(ws)− 2| ∼ sk∏ s=1 |zs|n = ∏ P (z)=0, |z|>1 |z|n = An, where A = ∏ P (z)=0, |z|>1 |z| is the Mahler measure of P (z). By the results mentioned in the preliminary part, it can be found by the formula A = exp( ∫ 1 0 log(P (e2πit))dt). Finally, fG(n) = sk∏ s=1 |2Tn(ws)− 2| ∼ An, n → ∞. The next theorem is a direct consequence of Theorem 4.1 and can be proved by the same arguments as Theorem 6.1. Theorem 6.2. The number of rooted spanning forests fG(n) in the circulant graph G = C2n(s1, s2, . . . , sk, n), 1 ≤ s1 < s2 < . . . < sk < n has the following asymp- totics fG(n) ∼ Kn, as n → ∞. Here K = exp( ∫ 1 0 log |P (e2πit)2−1|dt) is the Mahler measure of the Laurent polynomial P (z)2 − 1, where P (z) = 2k + 2− ∑k i=1(z si + z−si). 7 Examples (1) Cycle graph G = Cn(1) = Cn. We need to solve the equation 1+2− 2T1(w) = 0. We have w = 3/2. So, fG(n) = 2Tn(3/2) − 2. Then fG(n) ∼ n→∞ ( 3+ √ 5 2 ) n. Also, we have fG(n) = 5F 2n , if n is even, and fG(n) = L2n, if n is odd, where Fn and Ln are the Fibonacci and Lucas numbers respectively. The latter result was independently obtained in [6]. 684 Ars Math. Contemp. 22 (2022) #P4.10 / 675–686 (2) Graph G = Cn(1, 2). We need to solve the equation 1 + 4 − 2T1(w) − 2T2(w) = 0. Its roots are w1 = 1 4 (−1 + √ 29) and w2 = 14 (−1− √ 29). By Theorem 5.1, there exists an integer sequence a(n) such that fG(n) = 5a(n)2, if n is even, and fG(n) = a(n)2, if n is odd. (3) Graph G = Cn(1, 3). Let w1, w2 and w3 be the roots of the cubic equation 1+4−2T1(w)−2T3(w) = 0. Then fG(n) = (2Tn(w1)− 2)(2Tn(w2)− 2)(2Tn(w3)− 2). In this case, fG(n) ∼ An1,3, n → ∞, where A1,3 ≈ 4.48461 . . . is the Mahler measure of the Laurent polynomial 5−z−z−1−z3−z−3. One can check that A1,3 is a root of the equation 1−x− 2x2− 4x3+x4 = 0. By Theorem 5.1, we have fG(n) = a(n)2, where a(n) is an integer sequence. (4) Graph Möbius ladder G = C2n(1, n). We have to solve the equations 3−2T1(w) = 0 and 5−2T1(w) = 0. Their roots are u1 = 3/2 and v1 = 5/2 respectively. Then fG(n) = (2Tn(3/2) − 2)(2Tn(5/2) + 2) ∼ Kn, where K = 14 (3 + √ 5)(5 + √ 21) ≈ 12.5438 . . . . By Theorem 5.2, fG(n) = 5a(n) 2, if n is even, and fG(n) = 7a(n)2, if n is odd for some integer sequence a(n). ORCID iDs Lilya A. Grunwald https://orcid.org/0000-0003-4622-5259 Ilya Mednykh https://orcid.org/0000-0001-7682-3917 References [1] T. L. Austin, The enumeration of point labelled chromatic graphs and trees, Can. J. Math. 12 (1960), 535–545, doi:10.4153/cjm-1960-047-1. [2] F. T. Boesch and Z. R. Bogdanowicz, The number of spanning trees in a prism, Int. J. Comput. Math. 21 (1987), 229–243, doi:10.1080/00207168708803568. [3] F. T. Boesch and H. Prodinger, Spanning tree formulas and Chebyshev polynomials, Graphs Comb. 2 (1986), 191–200, doi:10.1007/bf01788093. [4] D. Callan, A combinatorial derivation of the number of labeled forests, J. Integer Seq. 6 (2003), Article 03.4.7, 9, https://cs.uwaterloo.ca/journals/JIS/vol6.html. [5] A. Cayley, A theorem on trees, Quart. J. Pure Appl. Math. 23 (1889), 376–378. [6] P. Chebotarev, Spanning forests and the golden ratio, Discrete Appl. Math. 156 (2008), 813– 821, doi:10.1016/j.dam.2007.08.030. [7] P. Chebotarev and E. Shamis, Matrix forest theorems, 2006, arXiv:math/0602575 [math.CO]. [8] X. Chen, Q. Lin and F. Zhang, The number of spanning trees in odd valent circulant graphs, Discrete Math. 282 (2004), 69–79, doi:10.1016/j.disc.2003.12.006. [9] P. J. Davis, Circulant Matrices, New York, NY: AMS Chelsea Publishing, 1994. [10] G. Everest and T. Ward, Heights of polynomials and entropy in algebraic dynamics, Universi- text, Springer-Verlag London, Ltd., London, 1999, doi:10.1007/978-1-4471-3898-3. L. A. Grunwaldr and I. Mednykh: The number of rooted forests in circulant graphs 685 [11] A. J. Guttmann and M. D. Rogers, Spanning tree generating functions and Mahler measures, J. Phys. A 45 (2012), 494001, 24, doi:10.1088/1751-8113/45/49/494001. [12] A. J. W. Hilton, Spanning trees and Fibonacci and Lucas numbers, Fibonacci Quart. 12 (1974), 259–262, https://www.fq.math.ca/12-3.html. [13] Y. Jin and C. Liu, Enumeration for spanning forests of complete bipartite graphs, Ars Comb. 70 (2004), 135–138, https://www.researchgate.net/publication/220620306_ Enumeration_for_spanning_forests_of_complete_bipartite_graphs. [14] A. K. Kelmans and V. M. Chelnokov, A certain polynomial of a graph and graphs with an extremal number of trees, J. Comb. Theory Ser. B 16 (1974), 197–214, doi:10.1016/ 0095-8956(74)90065-3. [15] G. Kirchhoff, Über die auflösung der gleichungen, auf welche man bei der untersuchung der linearen verteilung galvanischer ströme geführt wird,, Ann. Phys. Chem. 72 (1847), 497–508, doi:10.1002/andp.18471481202. [16] O. Knill, Cauchy-Binet for pseudo-determinants, Linear Algebra Appl. 459 (2014), 522–547, doi:10.1016/j.laa.2014.07.013. [17] Y. S. Kwon, A. D. Mednykh and I. A. Mednykh, On Jacobian group and complexity of the generalized Petersen graph GP(n, k) through Chebyshev polynomials, Linear Algebra Appl. 529 (2017), 355–373, doi:10.1016/j.laa.2017.04.032. [18] D. H. Lehmer, Factorization of certain cyclotomic functions., Ann. Math. (2) 34 (1934), 461– 479, doi:10.2307/1968172. [19] C. J. Liu and Y. Chow, Enumeration of forests in a graph, Proc. Am. Math. Soc. 83 (1981), 659–662, doi:10.2307/2044142. [20] D. Lorenzini, Smith normal form and Laplacians, J. Comb. Theory, Ser. B 98 (2008), 1271– 1300, doi:10.1016/j.jctb.2008.02.002. [21] K. Mahler, On some inequalities for polynomials in several variables, J. Lond. Math. Soc. 37 (1962), 341–344, doi:10.1112/jlms/s1-37.1.341. [22] J. C. Mason and D. C. Handscomb, Chebyshev polynomials, Boca Raton, FL: Chapman & Hall/CRC, 2003, http://inis.jinr.ru/sl/. [23] A. D. Mednykh and I. A. Mednykh, The number of spanning trees in circulant graphs, its arithmetic properties and asymptotic, Discrete Math. 342 (2019), 1772–1781, doi:10.1016/j. disc.2018.08.030. [24] I. A. Mednykh, On Jacobian group and complexity of I-graph I(n, k, l) through Chebyshev polynomials, Ars Math. Contemp. 15 (2018), 467–485, doi:10.26493/1855-3974.1355.576. [25] A. J. Schwenk, Computing the characteristic polynomial of a graph, Graphs Comb. 406 (1974), 153–172, doi:10.1007/bfb0066438. [26] J. Sedlacek, On the spanning trees of finite graphs, Čas. Pěstovánı́ Mat. 91 (1966), 221–226. [27] J. Sedlacek, On the skeletons of a graph or digraph, Combinat. Struct. Appl., 1970, https://www.researchgate.net/publication/265917830_On_the_ skeletons_of_a_graph_or_digraph. [28] R. Shrock and F. Y. Wu, Spanning trees on graphs and lattices in d-dimensions, J. Phys. A, Math. Gen. 33 (2000), 3881–3902, doi:10.1088/0305-4470/33/21/303. [29] D. S. Silver and S. G. Williams, Graph complexity and mahler measure, 2017, arXiv:1701.06097v1 [math.CO]. [30] C. Smyth, The Mahler measure of algebraic numbers: a survey, in: Number Theory and Polynomials, Cambridge: Cambridge University Press, pp. 322–349, 2010, doi:10.1017/ cbo9780511721274.021. 686 Ars Math. Contemp. 22 (2022) #P4.10 / 675–686 [31] W. Sun, S. Wang and J. Zhang, Counting spanning trees in prism and anti-prism graphs, J. Appl. Anal. Comput. 6 (2016), 65–75, doi:10.11948/2016006. [32] L. Takács, On the number of distinct forests, SIAM J. Discrete Math. 3 (1990), 574–581, doi: 10.1137/0403050. [33] L. Weinberg, Number of trees in graph, Proc. IRE 46 (1958), 1954–1955. [34] Y. Zhang, X. Yong and M. J. Golin, The number of spanning trees in circulant graphs, Discrete Math. 223 (2000), 337–350, doi:10.1016/s0012-365x(99)00414-8. [35] Y. Zhang, X. Yong and M. J. Golin, Chebyshev polynomials and spanning tree formulas for circulant and related graphs, Discrete Math. 298 (2005), 334–364, doi:10.1016/j.disc.2004.10. 025. Author Guidelines Before submission Papers should be written in English, prepared in LATEX, and must be submitted as a PDF file. The title page of the submissions must contain: • Title. The title must be concise and informative. • Author names and affiliations. 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