Also available on http://amc.imfm.si
ARS MATHEMATICA CONTEMPORANEA 1 (2008) 20–31
Coalesced and Embedded Nut Graphs in Singular
Graphs
Irene Sciriha †
Department of Mathematics, Faculty of Science
University of Malta, Msida MSD 06, Malta
Received 1 August 2007, accepted 9 May 2008, published online 19 June 2008
Abstract
A nut graph has a non-invertible (singular) 0-1 adjacency matrix with non-zero entries
in every kernel eigenvector. We investigate how the concept of nut graphs emerges as an
underlying theme in the theory of singular graphs. It is known that minimal configurations
(MCs) are necessarily found as subgraphs of singular graphs. We construct MCs having nut
graphs as subgraphs. Nut graphs can be coalesced with singular graphs at particular vertices
or grown into a family of core graphs of larger nullity by adding a vertex at a time. Moreover,
we propose a local enlargement of nut fullerenes and tetravalent nut graphs.
Keywords: Adjacency matrix, nut graph, kernel eigenvector, singular graphs, core, periphery, nut
fullerenes, line graphs of trees.
Math. Subj. Class.: 05C50, 05B20, 05C90, 92E10
1 Introduction
A graph G has vertex set VG = {1, 2, . . . , n} and edge set EG joining distinct vertices of VG.
The complement G of G has VG = VG and EG is the set of non-edges of G. The adjacency
matrix AG(= A or G), of a graph G, is the matrix (aij), where aij is one if ij ∈ EG
and zero otherwise. The graphs we consider are simple, i.e without loops or multiple edges;
thus each entry on the main diagonal is zero. Since a graph is uniquely determined by its
adjacency matrix, we use terminology for a graph and its adjacency matrix interchangeably.
For different labellings of the vertices of a graphG, the corresponding adjacency matrices are
similar. Thus the collection of eigenvalues λ1, λ2, . . . , λn, (with repetitions), which is said to
be the spectrum Sp(G) of G, is an invariant of G [1, 2].
†Research supported by the AWRF, University of MALTA.
E-mail address: irene.sciriha-aquilina@um.edu.mt (Irene Sciriha)
Copyright c© 2008 DMFA – založništvo
I. Sciriha: Coalesced and Embedded Nut Graphs in Singular Graphs 21
Singular graphs have the eigenvalue zero. In section 2, we survey properties of core
graphs and nut graphs as particular singular graphs. Minimal configurations (MCs)and sin-
gular configurations (SCs) are subgraphs determined by the vectors in ker(A). A graph of
nullity η has η independent MCs as subgraphs [17]. Catalogues of MCs of small order can
be found in [11] and [13]. The simplest MCs are trees including paths on an odd number of
vertices. A tree is a MC if and only if it is a subdivision of some tree [19]. In section 3, we
survey the properties of MCs and proceed, in section 4, to explore possible ways of convert-
ing singular graphs to larger ones. If a graph G has an eigenvalue λ0 of multiplicity k, then k
vertices can be deleted to produce a star complement for λ0, that is, a maximal induced sub-
graph of G that does not have λ0 as an eigenvalue. We show that a vertex deleted subgraph
Y − v of a nut graph Y is a star complement for λ0 = 0. Nut graphs can be coalesced with
singular graphs at particular vertices to obtain larger nut graphs or other particular types of
singular graphs. Moreover, if to nut graphs, vertices are added to increase the nullity at each
stage, a family of core graphs of increasing nullity, with a nut graph as an induced subgraph,
is produced. The last section presents a local enlargement for trivalent polyhedra, cylindrical
constructions for nut fullerenes and an extension of four-valent polyhedra to produce larger
core graphs.
2 Nut Graphs
A graph G is singular if A is singular, that is if A is not injective. Hence, there exist linearly
independent vectors x, s.t. Ax = 0, so that zero is an eigenvalue of A. The multiplicity η
of the eigenvalue zero is the dimension of the nullspace, ker(A) and is said to be the nullity
η(G) of G.
Definition 1. Let G be a singular graph. A non-zero vector x satisfying Ax = 0 is said to
be a kernel eigenvector of G.
There exist singular graphs, called core graphs, of nullity one or more, with a kernel
eigenvector having no zero entries. We say that a subgraph, F , of a singular graph G, is a
core of G, if there exists a kernel eigenvector x of G such that F is the subgraph induced by
the vertices corresponding to the non-zero entries (or support) of x. Hence a core F of G is
a core graph, with a kernel eigenvector xF , the non-zero restriction of x. The |F | vertices of
F are determined by the |F | labels corresponding to xF . We label a graph G, of nullity one,
so that a kernel eigenvector of the form x = (xF,0)
t, generates the nullspace ker(G), where
xF ∈ R|F | is the non-zero part of x with each of the |F | entries being non-zero. For nullity
greater than one, there are a number of cores, each corresponding to a non-trivial vector in
ker(G). Note that each core of G is a core graph in its own right.
The set of core vertices, CV , of G, consists of those vertices that lie in some core of
G. A vertex, not lying in any core, is said to be a core-forbidden vertex. Thus a core graph
has no core-forbidden vertices. Moreover, core graphs are distinguished from other singular
graphs, since the deletion of any vertex reduces the nullity (Corollaries 13 and 16).
Definition 2. A nut graph is a singular graph such that no kernel eigenvector has zero entries.
Proposition 3. A nut graph is a core graph of nullity one.
Proof. By definition, a nut graph has at least one kernel eigenvector with no zero entries.
Since linear combinations of linearly independent kernel eigenvectors can produce a kernel
eigenvector with a zero entry and this cannot be the case for nut graphs, it follows that all
22 Ars Mathematica Contemporanea 1 (2008) 20–31
kernel eigenvectors are linearly dependent. Hence, for a nut graph Y , the dimension of kerY
is one.
Theorem 4. [10] A nut graph
(i) is connected;
(ii) has no terminal vertex;
(iii) is not bipartite.
Remark 5. Nut graphs exist on seven or more vertices. The second and third graphs in
Figure 1 are nut graphs. From Proposition 4(iii), it follows that all nut graphs have odd
cycles. The girth of a graph is the size i of the smallest odd cycle Ci, on i vertices. It is given
by the index i of the first non-vanishing odd coefficient, ai, in the characteristic polynomial∑n
i=0 aiλ
n−i of the adjacency matrix of the graph [8]. There exist nut graphs with girth
three. Nut fullerenes are three–regular and have girth five.
Proposition 6. [10] If a triangle C3 = {v1, v2, v3} is a subgraph of the nut graph Y and the
vertex v1 is of degree two in Y , then v2 and v3 do not have the same neighbours. (Equiva-
lently, v2 and v3 are not co-duplicate vertices.)
3 Minimal Configurations
In [14], we show that a singular graph of nullity η has η subgraphs called minimal configura-
tions (MCs). In [17], a stronger result is obtained for singular configurations (SCs), namely
that a singular graph of nullity η has η SCs as induced subgraphs. MCs and SCs are singular
graphs of nullity one, determined by the non-zero part of their kernel eigenvector as follows
[11, 17]:
Definition 7. Let F be a core graph on at least two vertices, with nullity s ≥ 1 and a kernel
eigenvector xF having no zero entries. If a graph N , of nullity one, having xF as the non-
zero part of a kernel eigenvector, is obtained, by adding s − 1 independent vertices, whose
neighbours are vertices of F , then N is said to be a minimal configuration (MC).
The set of s−1 independent vertices added to F to produceN is said to be the periphery
P(N) of N . Figure 1 shows three MCs with s = 2, 1, 1 and |P(N)| = 1, 0, 0 respectively.
Figure 1: Three MCs
If the condition that the (s − 1) vertices added are independent is dropped, then the
singular graph S of nullity one obtained is said to be a SC. Various MCs may be constructed
for the same F and xF . The MCs, shown in Figure 2, have the same core N5 = K5 and
(1, 1,−2,−1, 2) or (1, 1,−1,−1, 1) as the non-zero part of the kernel eigenvector.
We present properties of MCs.
I. Sciriha: Coalesced and Embedded Nut Graphs in Singular Graphs 23
Figure 2: MCs with core K5 for two feasible 0-eigenvectors
Theorem 8. A MC is connected.
Proof. Suppose, for contradiction that a MC N is disconnected. Without loss of generality
N has a connected component N1 having a non-zero kernel eigenvector x1. If the vertices
of N1 are labelled first, then N
(
x1
x2
)
=
(
0
0
)
and N1x1 = 0. Since x1 6= 0, then x2 = 0,
otherwise
(
x1
0
)
and
(
0
x2
)
would be linearly independent kernel eigenvectors of N , con-
tradicting that the nullity of N is one. Thus x2 corresponds to vertices in the periphery and
hence there are no edges among them. But then N has isolated vertices which contribute to
the nullity, so that η(N) > 1, a contradiction. Thus N = N1.
Theorem 9. [16] Let N be a singular graph of order n ≥ 3. The graph N is a MC having
a core F with respect to the kernel eigenvector x and periphery P := V(N) − V(F ) if and
only if the following conditions are all satisfied:
(i) η(N) = 1,
(ii) P = ∅ or P induces a graph consisting of isolated vertices,
(iii) η(F ) = |P|+ 1.
Proof. Condition (i) follows from the definition of MC.
If the nullity of F is one, then F = N and P = ∅, so that both (ii) and (iii) are satisfied.
If the nullity s of F is more than one, then (ii) and (iii) follow from the construction of a
MC in Definition 7.
Conversely, if the three conditions hold, then
a. N has nullity one;
b. if P 6= ∅, then the neighbours of the vertices in P are vertices of F ;
c. by the Interlacing Theorem, condition (iii) requires that the nullity of the core F decreases
successively by one with each addition of the (s− 1) vertices in P , in turn.
The above properties of MCs do not depend on the edges or non-edges among the vertices
of the periphery, so that they hold also for SCs [17]. Note that a nut graph is a MC with
P = ∅. For η(G) = 1, a singular graph G has a unique core F and a unique periphery. For
η(G) > 1, it is convenient to use a minimal basis Bmin for the nullspace of G. The vectors
24 Ars Mathematica Contemporanea 1 (2008) 20–31
x1,x2, . . . ,xη in Bmin are chosen so that the total number of non-zero entries in the vectors
is a minimum. A somewhat surprising result is the following: if |xi| denotes the number of
non-zero entries in xi, then, every possible Bmin has a unique sequence |x1|, |x2|, . . . , |xη|
[15, 17].
4 Extensions
In [10], we explored the possibilities of growing nut graphs into larger nut graphs by adding
vertices and modifying the kernel eigenvector. This prompts us to explore other ways of
constructing singular graphs that have a nut graph embedded as an induced subgraph.
4.1 Coalescence with Nut Graphs
Definition 10. [9] If a vertex v1 of a graph G1 is identified with a vertex v2 of a graph G2,
then the graph G1.G2 obtained, of order |G1|+ |G2|− 1, is said to be the coalescence of G1
and G2.
We shall consider coalescence of singular graphs with nut graphs. The following theorem
[9], which we shall have occasion to use, gives an expression for the characteristic polynomial
φ(G,λ) of a graphG = K.H in terms of characteristic polynomials ofK andH , and of their
vertex-deleted subgraphs.
Theorem 11. The characteristic polynomial of the coalescence K.H of two rooted graphs
(K,u), (H,w) obtained by identifying the vertices u, and w so that this vertex v = u = w
becomes a cut-vertex of K.H is given by:
φ(K.H) = φ(K)φ(H − w) + φ(K − u)φ(H)− λφ(K − u)φ(H − w). (1)
Proposition 12. If G is a graph of nullity η, there exists a subset X of η vertices of V(G),
whose deletion yields a non-singular graph1.
Proof. Let G be labelled so that the vertices in CV are labelled first. If B = {z1, z2, . . . , zη}
is a basis for the nullspace of A(G), then the η × n matrix M whose rows are the η vectors
in B, has rank η. By row reduction, M can be reduced to Hermite Normal form M′, in
which, to the (column) position of the first non-zero entry of each of the η row vectors, there
corresponds a zero entry in all the other rows. Since row reduction is equivalent to taking
linear combinations of the kernel eigenvectors, the rows of M′ are also kernel eigenvectors
of G. Deleting any vertex, v, affects just one of the row vectors so that the remaining η − 1
rows of M′, restricted to G − v, are kernel eigenvectors of G − v. Moreover, there are no
more kernel eigenvectors linearly independent of these η − 1 row vectors as otherwise the
nullity of G would be more than η.
The following result follows from Proposition 12.
Corollary 13. If a core vertex v is deleted from a graph G, of nullity η, then the nullity of
G− v is η − 1.
The following result is also proved in [12], utilizing the adjugate of A.
1For the eigenvalue zero, the graph G − X is said to be a star complement of G and the set X a star set. See,
for instance, [7].
I. Sciriha: Coalesced and Embedded Nut Graphs in Singular Graphs 25
Corollary 14. If a core vertex v is deleted from a singular graph G of nullity one, then G−v
is non-singular.
Corollary 15. If any vertex v is deleted from a nut graph Y , then Y − v is non-singular.
Proof. This follows since Y has nullity one and every vertex of Y is a core vertex.
Corollary 16. If η(G− v) < η(G) for all v ∈ VG, then G is a core graph.
Proof. This follows since a basis for the nullspace of A(G) can be taken to be the row vectors
of M′ as constructed in the proof of Proposition 12.
Proposition 17. A star set for the eigenvalue zero is contained in CV.
Proof. For the nullity to decrease on deleting a vertex i, there needs to be a kernel eigenvector
in M ′ with its first non-zero entry at position i. These positions form a star set and are in CV
by definition.
Proposition 18. If the graph G′ is obtained by coalescing a nut graph Y with a singular
graph G, at a vertex vF of a core F of G, then the nullity remains unchanged.
Proof. Let η(G) = η. Now by Corollaries 13 and 15, η(G− vF ) = η− 1 and η(Y − v) = 0.
Thus η(G′) ≤ η by interlacing.
From Equation (i) of Theorem 11, the characteristic polynomial φ(G′) of the coalesence
G.Y is given by
φ(G′) = φ(G − vF , λ)φ(Y, λ) + φ(G,λ)φ(Y − vF , λ) − λφ(G − vF , λ)φ(Y − vF , λ).
Each term on the right hand side of the equation is divisible by λη , so that η(G′) ≥ η. This
completes the proof.
The construction of Proposition 18 may be done iteratively, running through all the core
vertices, to produce successively larger graphs of the same nullity.
Figure 3: An enlarged MC
Figure 3 shows the graph G′, an enlargement of the five vertex MC, H (made up of C4
and C3 with a common edge). The smallest nut graph is the seven vertex, eight edge graph
Y of Figure 1, formed by coalescing C5 with C3. If Y is coalesced, at each of the four core
vertices of H , a graph G′ is produced. By Proposition 18, η(H) = η(G′).
26 Ars Mathematica Contemporanea 1 (2008) 20–31
Proposition 19. Let N be a MC with core F . If nut graphs are coalesced with F at distinct
vertices of F , then a MC is produced.
Proof. Let G′ be constructed by coalescing nut graphs with F at distinct vertices of F . At
a vertex of F where no nut graph is coalesced, we say that K1 (the trivial nut graph) is
coalesced withN . If N =
(
F B
Bt 0
)
represents the adjacency matrix ofN , then, one way
to produce the adjacency matrix of G′, is to replace the diagonal entries of F in N by block
matrices Y1,Y2, . . . ,Y|F |, where Yi is the nut graph (or K1) coalesced at vertex i for 1 ≤
i ≤ |F |, leave the principal submatrix for the F−vertices unchanged and insert zero matrices
in the remaining blocks. Let u1,u2, . . . ,u|F | be the eigenvectors of Y1, Y2, . . . , Y|F |, scaled
so that the first entry of each ui is one. For Yi = K1, Yi = (0) and ui = (1). If Ny = 0
and y = (y1, y2, . . . , y|F |, 0, . . . , 0)t, then G′z = 0, where
z = (y1u1t, y2u2t, . . . , y|F |ut|F |, 0, . . . , 0)
t. Thus G′ is singular with a core F ′, which is the
coalescence of F with Y1, Y2, . . . , Y|F |.
To show that G is a MC, there remains to show that the three conditions of Theorem 9 are
satisfied. Indeed, by Proposition 18, iterative coalescence at core vertices of N preserves the
nullity at the value one, so that condition (i) is satisfied. The periphery P ′ = V(G′)\V(F ′)
is the same independent set as PF = V(G)\V(F ), satisfying condition (ii). Also, by Propo-
sition 18, η(F ′) = η(F ). Thus, deleting all the vertices of the periphery in turn from G′ has
the same effect on the nullity of G′ as deleting the corresponding vertices of P in G, so that
the nullity increases at each stage, as required for condition (iii). Hence G′ is also a MC.
Remark 20. Since the construction in Proposition 19 does not affect the vertices in the
periphery, the following result is immediate.
Corollary 21. Let Y1 and Y2 be nut graphs. The coalescence Y1.Y2 is a nut graph.
Figure 4: The coalescence of two nut graphs.
In Figure 4, the nut graph on 7 vertices is coalesced with the nut graph on 15 vertices, at
arbitrary vertices v1 and v2, to produce a larger nut graph.
4.2 Singular Graphs with a Nut Graph Embedding
Duplicate vertices are independent vertices having the same neighbours. They correspond
to identical rows (and columns) in A. A set of ` duplicate vertices contributes ` − 1 to the
nullity of a graph. A kernel eigenvector corresponding to a core graph consisting of a pair
of duplicate vertices has two non-zero entries only. A vertex can always be added to form a
duplicate pair with any vertex of a graph, thus increasing the nullity indefinitely. Following
I. Sciriha: Coalesced and Embedded Nut Graphs in Singular Graphs 27
Lepović, we call a graph canonical, if it has no duplicate vertices [5]. The kernel eigenvectors
of a singular canonical graph have three or more non-zero entries.
The Reconstruction Theorem [6, 7] applied to the eigenvalue zero states that a finite
number, k, of vertices are added to a zero-star complement to produce a canonical supergraph
of maximal nullity k. By Corollary 15, a vertex-deleted subgraph of a nut graph is a zero-star
complement.
Proposition 22. If a singular graph G of nullity η is obtained by adding η − 1 vertices to a
nut graph Y , then G is a core graph.
Proof. Let B1 be the basis containing only the nullspace generator of Y . Assume that j −
1 vertices have been added to Y to produce Gj with nullity j, j ≥ 1 and the basis Bj ,
containing j vectors, generates the nullspace of Gj . If a vertex v can be added to increase
the nullity by one, a kernel eigenvector xj+1, linearly independent of the ones in Bj , is
included to produce Bj+1. We can proceed inductively to produce a larger singular graph Gη
with a basis Bη , by adding to B1, vectors having a non-zero entry in positions |Y | + j − 1
corresponding to the additional vertices in turn, for 0 ≤ j ≤ η − 1. Note that an entry of
xj+1 in the position corresponding to v is necessarily non-zero. Thus the graph Gη is a core
graph.
Figure 5: A core graph of nullity three with a nut graph embedded.
In Figure 5, to the nut graph on 7 vertices, two vertices can be added to produce a singular
canonical graph of nullity three. The two graphs produced in the process are core graphs.
5 Nut Fullerenes
We conclude with a reference to the significance of nut graphs in chemistry, presenting ways
of transforming nut regular polyhedra to larger core graphs. The Hűckel Molecular Orbital
(HMO) model describes the distribution of the π-electrons in unsaturated organic molecules
and is an approximation to the solution of the Schrődinger Equation [4]. The molecular graph
G of a Carbon (C) structure has the C-atoms as vertices and the C-C σ bonds as edges. The
energy levels λi of the π-electrons, in the C-structure, are given by Axi = λixi, where xi
describes the molecular orbital for the energy λi and A is the adjacency matrix of G. For
singular molecular graphs, Ax = 0, A has zero eigenvalues, and the C-structure has non-
bonding molecular orbitals (NBMOs) described by x(6= 0) [4].
If G is a nut graph, then the NBMO x of the π-electron occupying this orbital has non-
zero entries. In chemistry, this is significant. In the radical molecule that would have the
NBO as partially occupied HOMO, all centres carry non-zero spin density. The charge,
spin and bond-order densities of a π-electron, occupying the NBMO, are distributed over
28 Ars Mathematica Contemporanea 1 (2008) 20–31
the whole structure and not on some substructure. Thus all C-centres are likely to be involved
in reactions. [18, 20, 21].
Figure 6: C36 : 14 and C12 : 2
There are 10, 190, 782 fullerene isomers on up to 120 vertices and only 41 are nut graphs.
In all hypothetical nut fullerenes found so far, occupation of the NBMO requires that the
fullerene carries at least five negative charges. All physically realizable fullerenes, known to
date, have isolated pentagons and none of these is a nut graph.
Figure 7: C60 and C70
The smallest nut fullerene is C36 : 14 [3] and the smallest nut trivalent polyhedron is
C12 : 2, both shown in Figure 6. Symmetry studies of polyhedra yield powerful mathemat-
ical results that can be immediately translated into chemical properties [18, 20]. For highly
I. Sciriha: Coalesced and Embedded Nut Graphs in Singular Graphs 29
symmetric vertex transitive or one–orbit graphs, the eigenvector entries for a simple eigen-
value, have the same non-zero absolute value. This implies equidistributivity of charge from
the single NBMO of a molecule. However Ax = 0, with each entry of x being±1, demands
an even number of bonds for each C-centre which is not the case in tri-valent polyhedra. Thus
vertex transitive nut fullerenes are ruled out. Several trivalent (+1,−1, 0) molecules with a
NBMO are possible. One is C70 which can be seen as the Buckminster fullerene C60 with
an additional ten C-atoms, contributing an equatorial ring of five extra hexagons, as shown
in Figure 7. Such a singular trivalent structure cannot be a MC, since every C-centre (even if
it corresponds to a zero entry of the kernel eigenvector) has an adjacent atom with no charge
from the NBMO. This forces two peripheral vertices to be adjacent, a configuration which is
not allowed in MCs.
The two nut trivalent polyhedra C36 : 14 and C12 : 2 mentioned above have a (−2,+1,
+1) NBMO, that is a kernel eigenvector with −2,+1 as its only entries. Such graphs are
said to be uniform nut graphs. Considering orbits, since every C-centre is adjacent to three
atoms corresponding to entries −2,+1 and +1, one can deduce that the number of C atoms
in uniform-nut fullerenes is a multiple of six [18, 20, 21].
Inspection of the pictures and spiral codes of the early nut fullerenes suggests some re-
curring patterns. We now investigate ways in which nut chemical graphs can be expanded,
while the original part of the nullspace generator remains in the new nullspace.
Figure 8: Local enlargement in regular chemical graphs
To preserve the regularity of a trivalent graph, coalescence cannot be employed. Instead
an expansion of cubic graphs that preserves the null eigenvector can be considered. The
following construction was suggested by P. W. Fowler. In a kernel eigenvector, the central
vertex carries entry a and by applying the zero-sum rule (or eigenvector equation for the zero
eigenvalue) the entries on its neighbours total b+c+d = 0. Expansion of the central vertex to
a half-cube, as shown in Figure 8, can be achieved without disruption of the null eigenvector,
if new eigenvector entries are assigned as shown. Clearly, if the parent graph is a uniform
nut graph and a = +1, the derived vector is uniform, but if a = (−2)r−1, r ≥ 3}, then the
entries of the derived vector are in {(−2)r−1, r ≥ 1}.
As the vertex count increases, cylindrical nut fullerenes occur, and can be grouped into
families according to their six-pentagon caps. In [21], it is shown that there exists a family
of tubular fullerenes, each with a cap of six pentagons and two hexagons, circumscribed by
successive rings of six hexagons and ending with the same cap, which have a kernel eigen-
vector with entries in {1,−2}. The pattern of coefficients is compatible with insertion of any
number of extra layers of six hexagons in the starting C48 structure, so that we have an infi-
30 Ars Mathematica Contemporanea 1 (2008) 20–31
Figure 9: Local enlargement in trivalent chemical graph.
nite series of singular fullerenes, with predictable spiral codes. All members of the series are
cores. Although we have no formal proof that the multiplicity of the zero eigenvector remains
at the value one after each stage of the expansion, the null eigenvector in each member of the
series is found to remain unique at least for all n ≤ 1000, and it appears plausible that we
have an infinite family of not only core fullerenes but also uniform-nut fullerenes.
We propose two similar expansions of tetravalent molecules where, in a kernel eigenvec-
tor, the central vertex carries entry a and the zero-sum rule necessitates that the addition of
the four neighbouring entries be zero. The bonds (edges) for the added atoms (vertices) are
shown in Figure 9. Note that in the first construction, two additional vertices, each carry-
ing charge −a, one above and one below the plane of the diagram, are required to produce
a four-valent molecule which is a core graph. In the second construction all the edges are
shown.
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