Integral quadratic forms and graphs

Linear Algebra and its Applications 585 (2020) 60–70

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Linear Algebra and its Applications www.elsevier.com/locate/laa

Integral quadratic forms and graphs Larry J. Gerstein Department of Mathematics, University of California, Santa Barbara, United States of America

a r t i c l e

i n f o

Article history: Received 5 January 2019 Accepted 25 September 2019 Available online 30 September 2019 Submitted by R. Brualdi MSC: 05C50 05C60 11E08 11E12 11E41

a b s t r a c t The structure of an undirected graph is completely determined by a symmetric matrix: its adjacency matrix with respect to an ordering of its vertices; and that matrix can be used to define an integral quadratic form. The main purpose of this paper is to raise this question: “What can quadratic forms tell us about graphs?” As an initial answer, the theory of quadratic forms will be applied to the graph isomorphism problem. The essential definitions and facts from the theory of quadratic forms will be sketched without proof. © 2019 Published by Elsevier Inc.

Keywords: Matrix Graph Isomorphism Quadratic form Lattice Genus

1. Introduction Let G1 and G2 be undirected graphs of order n, with A1 and A2 their associated adjacency matrices with respect to some vertex orderings. Then G1 and G2 are isomorphic if and only if there is a permutation matrix P such that A2 = t P A1 P . If n is large it is impractical to check for isomorphism by plugging in all n! candidates for such a maE-mail addresses: [email protected], [email protected]. https://doi.org/10.1016/j.laa.2019.09.032 0024-3795/© 2019 Published by Elsevier Inc.

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trix P . On the other hand, when n is large there are a great many isomorphism classes of graphs of order n; so, in the absence of a lot more information it is very likely that the given graphs are not isomorphic. This raises the question of how to demonstrate non-isomorphism. A permutation matrix P satisfies t P = P −1 . That is, P is an orthogonal matrix. So for there to be a graph isomorphism G1 ≈ G2 , their adjacency matrices must be similar. This leads to the notion of the spectrum of a graph, and much of algebraic graph theory is concerned with relating the spectrum of a graph to graph properties. Of course if one has checked that the adjacency matrices have the same eigenvalues and even that they are similar, the matrix that carries out the similarity need not be a permutation matrix. So similarity of the adjacency matrices is no guarantee of graph isomorphism. How else can we proceed on the graph isomorphism question? One approach is based on the notion of matrix equivalence. Recall that two n × n matrices A1 and A2 over a principal ideal domain R are equivalent, denoted A1 ∼ A2 , if there are unimodular matrices U, V ∈ GLn (R) such that A2 = U A1 V . In fact every matrix A ∈ Mn (R) is equivalent to its Smith normal form: A ∼ S(A) = diag(s1 , . . . , sn ), where the si satisfy si |si+1 for 1 ≤ i ≤ n − 1. The si are unique up to unit factors. (If R = Z we will usually assume the si are positive.) They are the invariant factors of A. Thus A1 ∼ A2 if and only if S(A1 ) = S(A2 ). See Brouwer-Haemers ([3], Chapter 13), for some relations between the invariant factors and the eigenvalues of adjacency matrices. Since permutation matrices are unimodular, if the adjacency matrices of two graphs are not equivalent then the graphs are not isomorphic. Examples illustrating this approach to demonstrating graph non-isomorphism can be found in [3]. If R is an integral domain, matrices A1 , A2 ∈ Mn (R) are congruent over R if there is a unimodular matrix T ∈ GLn (R) such that t T A1 T = A2 . Matrix congruence is fundamental to the theory of quadratic forms, a subject for which we will sketch the necessary details. The idea of linking quadratic forms to the graph isomorphism problem was first explored by Shmuel Friedland in two papers ([4], [5]) in the 1990s. Since an n × n permutation matrix is in Mn (Q), for two graphs to be isomorphic their adjacency matrices must be congruent over Q, a matter that can be resolved with the Hasse-Minkowski theorem. But a permutation matrix is also orthogonal, and so Friedland pushes beyond Hasse-Minkowski to determine conditions when two matrices are congruent via a rational orthogonal matrix. Friedland went on to conjecture in [5] that matrices that are orthogonally congruent over Q and orthogonally congruent over the rings of p-adic integers for all primes p must be orthogonally congruent over Z. It turns out that in 2009 Wang [10] provided counterexamples to this conjecture; nevertheless, the work of Friedland and Wang raises the question of what in addition to rational and local integral orthogonal congruence would force orthogonal congruence over Z. In the present paper we follow Friedland in pursuing a local-global approach to matrix classification, but now focusing on the unimodularity of permutation matrices instead of on their orthogonality. First we give some background on the theory of quadratic forms, including the notion of the genus of a lattice. Then we present an example illustrating that

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the genus concept may demonstrate non-isomorphism of associated graphs even when the adjacency matrices of those graphs are known to be both similar and equivalent. Another example will show that this approach has limitations. Finally, we offer some suggestions for further research. 2. Lattices on quadratic spaces Let V be an n-dimensional vector space over the field Q of rational numbers. A Z-lattice L on V is a free Z-module spanning V . Given a symmetric matrix A = (aij ) ∈ Mn (Z) and a basis v1 , . . . , vn for L (and hence for V ), define a symmetric bilinear form B : V × V → Q by setting B(vi , vj ) = aij and extending bilinearly to all of V × V . The map q : V → Q defined by q(v) = B(v, v) is the associated quadratic form; and V , together with B and q, is a quadratic space over Q, or quadratic Q-space. The matrix A is the Gram matrix of V and L with respect to the given basis, and we sometimes write V ∼ = A, respectively, to indicate that this is the case with respect to some = A and L ∼ basis. A lattice L is unimodular if it has a unimodular Gram matrix; L is modular if its Gram matrix is a scalar multiple of a unimodular matrix, and k-modular if that scalar is k. Now suppose V1 and V2 are n-dimensional quadratic Q-spaces, and L1 and L2 are Z-lattices on V1 , V2 , respectively. The spaces are isometric, denoted V1 ∼ = V2 , if there is an isomorphism ϕ : V1 → V2 such that B(ϕx, ϕy) = B(x, y) for all x, y ∈ V1 . If also ϕ(L1 ) = L2 then the lattices are isometric, denoted L1 ∼ = L2 . Isometric lattices on the same space are in the same isometry class. If A1 and A2 are Gram matrices for L1 and L2 , then L1 ∼ = L2 if and only if their Gram matrices are congruent over Z; that is, there is a unimodular Z-matrix T such that t T A1 T = A2 . The common determinant of all the Gram matrices of a given Z-lattice L is the discriminant of L, denoted dL. Recall that a signed permutation matrix is a matrix obtained by negating a subset (possibly empty) of the entries of a permutation matrix. The following proposition illustrates the preceding terminology. Proposition. A unimodular Z-matrix U is orthogonal if and only if U is a signed permutation matrix. Proof. It is trivial to check that a signed permutation matrix is orthogonal. For the converse, first note that the standard integer lattice Zn satisfies Zn ∼ = In with respect to the standard basis {e1 , . . . , en }. If the unimodular matrix U satisfies t U = U −1 , then t U In U = In ; so U is the matrix of an isometry of Zn onto itself with respect to the standard basis. But a vector v ∈ Zn satisfies q(v) = 1 if and only if v = ±ej for some j. The result follows. 2 Remark. If the adjacency matrices A1 and A2 for graphs G1 and G2 were orthogonally congruent over Z, it would follow from the preceding proposition that there is a signed

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permutation matrix S such that t SA1 S = A2 . If S is actually a permutation matrix—or can be replaced by a permutation matrix—this would show that G! and G2 are isomorphic graphs. If G1 , G2 are graphs with adjacency matrices A1 , A2 , for there to be an isomorphism G1 ≈ G2 it is necessary that lattices L1 and L2 with A1 and A2 as Gram matrices be isometric. A necessary weaker condition is that the underlying spaces V1 , V2 be isometric, which is equivalent to their Gram matrices being congruent over Q. Whether or not two Q-spaces are isometric can be determined effectively by means of the Hasse–Minkowski theorem, which reduces the problem to a finite computation. See [6] for details. So, in checking for isometry of two given Z-lattices, we can assume without loss of generality that their underlying quadratic Q-spaces are isometric. The general problem of determining whether two given Z-lattices are isometric has not been solved. The genus of a lattice—which we will define—contains its isometry class. Fortunately, determining whether two lattices with the same rank and discriminant are in the same genus can be carried out via computer algebra systems, provided that the prime factorization of the discriminant is known, and this will suffice for our application to the graph isomorphism problem. I thank Mark Watkins, of the Computational Algebra Group at the University of Sydney, for pointing out that the genus determinations can be carried out in polynomial time. 3. The local theory, and the genus of a lattice For each prime number p, let Qp denote the field of p-adic numbers and Zp the ring of p-adic integers. (For an introduction to p-adic numbers, see [6], Chapter 3, or [2].) Let V be a quadratic Q-space, and let p be a prime number. Roughly speaking, we now define Vp to be the quadratic Qp -space generated by V . More formally, define Vp = V ⊗Q Qp and then do a suitable identification, after which we can view V as a subset of Vp with the property that every basis for V over Q is also a basis for Vp over Qp . If A is a Gram matrix for V with respect to a basis B, then Vp becomes a quadratic Qp -space by taking A to be its Gram matrix, again with respect to B. If L is a Z-lattice on V and p is a prime, let Lp denote the Zp -lattice (that is, Zp -module) spanned by L. We call Lp the localization of L at p. The discriminant dL of a Zp -lattice L is well-defined only up to squares of Zp -units. In what follows, we use n for the dimension of the underlying quadratic spaces. Two Zp -lattices with Gram matrices A1 , A2 on a quadratic Qp -space are isometric if and only if there is a matrix T ∈ GLn (Zp ) such that t T A1 T = A2 . Now suppose L1 and L2 are lattices on a quadratic Q-space. Since for every prime p there is the inclusion GLn (Z) ⊂ GLn (Zp ), it follows that a necessary condition for an isometry L1 ∼ = L2 is that there be an isometry L1p ∼ = L2p for every prime p. In the vernacular: for global isometry it is necessary that there be local isometry at all primes. Two lattices on the same quadratic Q-space are said to be in the same genus if they are locally isometric at

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all primes. Thus the isometry class of a lattice L is contained in its genus. The number of isometry classes in the genus is the class number of L, denoted h(L), and it is known to be finite. Now let’s discuss the local structure of a Z-lattice L. Suppose L’s Gram matrix A has invariant factors s1 , . . . , sn , and let p be a prime. Form the list of n p-powers fully dividing the n successive invariant factors. These are the p-power elementary divisors of A; and among these, assume there are exactly t distinct values; say pi1 , . . . , pit , with 0 ≤ i1 < · · · < it . For 1 ≤ k ≤ t, let nk be the frequency of pik on the list of elementary divisors. Then there is an orthogonal splitting of Lp into t modular components. Explicitly, (t) Lp = L(1) p ⊥ · · · ⊥ Lp , (k)

(k)

where for 1 ≤ k ≤ t the component Lp is pik -modular and rank Lp = nk . Thus the (k) Gram matrix of Lp is of the form pik Uk , with Uk ∈ GLnk (Zp ) for k = 1, . . . , t. Such a decomposition of Lp into the orthogonal sum of modular components is a Jordan splitting (k) of Lp , and the Lp are the Jordan components of the splitting. In particular, if p is relatively prime to the discriminant dL, and hence relatively prime to its invariant factors, then Lp is a unimodular Zp -lattice, and therefore has only one Jordan component: Lp itself. Incidentally, the rank (possibly 0) of the unimodular Jordan component of Lp is what in [3] is called the p-rank of L. For the purposes of this paper, it is important to note that while the invariant factors of the Gram matrix A of L determine the ranks and modularities of the Jordan components of the localized lattices Lp uniquely, more invariants are needed to determine the structure of the Lp up to isometry. That is, suppose M is another Z-lattice whose Gram matrix has the same invariant factors as L’s. Then for every prime p there is a Jordan splitting Mp = Mp(1) ⊥ · · · ⊥ Mp(t) with rank Mk = rank L(k) p = nk

and Mk ∼ = pk W k

for some Wk ∈ GLnk (Zp ).

Such a Zp -lattice Mp need not be isometric to Lp . For example, if p is odd, there is the following additional necessary requirement for an isometry Lp ∼ = Mp : for all k, the unimodular matrices Uk and Wk appearing in the expressions for the Gram matrices of Lp and Mp must satisfy det Uk = ε2k det Wk for some Zp -unit εk . (The conditions are more complicated over Z2 , the 2-adic integers.) In general there can be many different genera of Z-lattices whose Gram matrices have the same invariant factors. If p is odd, two unimodular Zp -lattices with the same rank and discriminant are isometric. This observation underlies the fact that determining whether two Z-lattices with the same discriminant and whose Gram matrices have the same invariant factors

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are in the same genus requires only a finite number of computations: only the local invariants when p = 2 and when p is an odd prime divisor of the common discriminant need to be considered. 4. Examples In the examples and discussion that follows, the computer algebra system Magma will be used for the computations. Example 1. Let K4 denote the complete graph on four vertices, and let G1 be the cartesian product graph K4 K4 . Thus G1 has vertex set {1, 2, 3, 4} × {5, 6, 7, 8}, and two vertices are adjacent if and only if they agree in exactly one coordinate. Let G2 be the Shrikhande graph, which was first considered in [8] in connection with the study of block designs. Also see Biggs [1], Chapter 3, for a diagram. Both G1 and G2 have order 16, and both are regular of degree 6. We will show that these two graphs are not isomorphic. This is not a new result; it is our method of proof that is new. Let A1 and A2 be their respective adjacency matrices. Thus ⎛0 1 1 1 1 0 0 0 1 0 0 0 1 0 0 0⎞

A1 =

1 ⎜1 ⎜1 ⎜ ⎜1 ⎜0 ⎜0 ⎜ ⎜0 ⎜1 ⎜ ⎜0 ⎜0 ⎜0 ⎜ ⎜1 ⎝0 0 0

0 1 1 0 1 0 0 0 1 0 0 0 1 0 0

1 0 1 0 0 1 0 0 0 1 0 0 0 1 0

1 1 0 0 0 0 1 0 0 0 1 0 0 0 1

0 0 0 0 1 1 1 1 0 0 0 1 0 0 0

1 0 0 1 0 1 1 0 1 0 0 0 1 0 0

0 1 0 1 1 0 1 0 0 1 0 0 0 1 0

0 0 1 1 1 1 0 0 0 0 1 0 0 0 1

0 0 0 1 0 0 0 0 1 1 1 1 0 0 0

1 0 0 0 1 0 0 1 0 1 1 0 1 0 0

0 1 0 0 0 1 0 1 1 0 1 0 0 1 0

0 0 1 0 0 0 1 1 1 1 0 0 0 0 1

0 0 0 1 0 0 0 1 0 0 0 0 1 1 1

1 0 0 0 1 0 0 0 1 0 0 1 0 1 1

0 1 0 0 0 1 0 0 0 1 0 1 1 0 1

0 0⎟ ⎟ 1⎟ 0⎟ 0⎟ ⎟ 0⎟ 1⎟ 0⎟ ⎟ 0⎟ 0⎟ ⎟ 1⎟ 1⎟ 1⎠ 1 0

⎛0 1 0 1 1 1 0 0 0 0 0 0 1 0 0 1⎞ 1010011000001100

⎜0 1 0 1 0 0 1 1 0 0 0 0 0 1 1 0⎟ ⎜1 0 1 0 1 0 0 1 0 0 0 0 0 0 1 1⎟ ⎜ ⎟ ⎜1 0 0 1 0 1 0 1 1 1 0 0 0 0 0 0⎟ ⎜1 1 0 0 1 0 1 0 0 1 1 0 0 0 0 0⎟ ⎜0 1 1 0 0 1 0 1 0 0 1 1 0 0 0 0⎟ ⎜ ⎟ ⎜ ⎟ and A2 = ⎜ 00 00 10 10 11 00 10 01 10 01 00 11 01 01 00 00 ⎟ ⎜ ⎟ ⎜0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0⎟ ⎜0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1⎟ ⎜0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1⎟ ⎜ ⎟ ⎜1 1 0 0 0 0 0 0 1 0 0 1 0 1 0 1⎟ ⎝0 1 1 0 0 0 0 0 1 1 0 0 1 0 1 0⎠ 0011000001100101 1001000000111010

The Magma command IsSimilar(A1 , A2 ) tells us that these matrices are similar; so for this example similarity is no help in settling the graph isomorphism question. For that matter, the Magma command SmithForm(A1 , A2 ) tells us that the matrices are equivalent, again leaving the graph isomorphism problem unresolved. Explicitly, as pointed out in [3] (§13.8), they have a common Smith normal form: A1 ∼ A2 ∼ diag(1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 4, 4, 4, 4, 4, 12). Now we would like to show that Z-lattices with A1 and A2 as Gram matrices are not in the same genus. But Magma can perform this sort of check only if the given matrices are positive definite. That is, each of the associated quadratic forms q must have the property that q(v) > 0 for every nonzero vector v. For each of the given lattices, the zeros on the diagonal of its Gram matrix mean that q(v) = 0 for each vector in the

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lattice basis associated with that matrix, defying “definiteness.” We need to get around this obstruction. Notice that there is a graph isomorphism G1 ≈ G2 if and only if the graphs remain isomorphic when r loops are added at each vertex of both graphs. In matrix language: A2 = t P A1 P

⇐⇒

A2 + 2rIn = t P (A1 + 2rIn )P

for every permutation matrix P and every nonnegative integer r. In fact this matrix equation also follows without the observation on graph isomorphism just from the fact that t P = P −1 and the fact that 2rIn is in the center of GLn (C). Also from this latter fact, we see that if A1 and A2 are similar, then so are A1 + 2rIn and A2 + 2rIn . Given any symmetric n × n matrix with real entries, from the formula for the determinant it follows that adding a diagonal matrix xIn to the given matrix will yield a positive definite matrix if the real number x is a sufficiently large positive number. Returning now to our example, we observe that adding two loops to each vertex of G1 and G2 yields graphs with positive definite adjacency matrices C1 = A1 + 4I16 and C2 = A2 + 4I16 . (Positive definiteness was checked with Magma.) Call the graphs obtained from G1 and G2 by this loop supplementation H1 and H2 . Thus G1 ≈ G2

⇐⇒

H1 ≈ H2 .

From our previous remarks, without needing Magma we know that the Gram matrices of H1 and H2 are similar. Let M1 and M2 be Z-lattices with C1 and C2 as Gram matrices. From Magma we have SmithForm(C1 ) = SmithForm(C2 ) = diag(1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 12, 12, 12, 12, 12, 60), so both matrices have the following nontrivial elementary divisors: four 2’s,

six 4’s,

six 3’s,

and one 5.

From this listing we learn that both M1 and M2 , when localized at a prime p, have Jordan splittings of the following sort: At p = 2: a 2-modular component of rank 4, a 4-modular component of rank 6, and a unimodular component of rank 6. At p = 3: a 3-modular component of rank 6 and a unimodular component of rank 10. At p = 5: a 5-modular component of rank 1 and a unimodular component of rank 15. At all other primes the localized lattice is unimodular. The product of the invariant factors is the common discriminant: 238878720. On the other hand, if we don’t first compute the invariant factors but instead use the Magma command M1 := LatticeWithGram(C1 )

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the program responds “Determinant 238878720, Factored Determinant 216 ∗ 36 ∗ 5.” (And similarly for M2 .) This tells us that only the local behavior when p = 2, 3, 5 is of interest when comparing the genera of M1 and M2 , because the localizations of both lattices at all other primes are isometric. (Recall from the earlier discussion: when p is an odd prime, two unimodular Zp -lattices of the same rank and discriminant are isometric.) But the factorization of the discriminant by itself is not enough to give the ranks of the Jordan components when the exponent on a particular prime in the discriminant’s factorization is ≥ 2; for this we need multiplicities of the elementary divisors. Finally, we come to the genus comparison via Magma. Set GMi = Genus(Mi ) for i = 1, 2, and then give the command GM1 eq GM2 , to which the machine replies “false.” At this point we’re done, except for our natural curiosity as to which localizations of M1 and M2 are not isometric. We need to consider only the primes p = 2, 3, 5. This is done with the Magma commands LocalGenera(GM1 ) and LocalGenera(GM2 ), whereupon Magma displays the Gram matrices of the three relevant Jordan splittings for each lattice, from which we can see immediately that M1p ∼ = M2p when p = 3, 5. So, given that the genera are different, it must be the case that M1p  M2p when p = 2. In fact, differences in the 2-adic Jordan components of M1p and M2p make this evident, from Theorem 8.7 in [6]. (To check this, note that the norm n(L) of a lattice X is the ideal generated by the range of the quadratic form q when restricted to X.) Let’s summarize Example 1. The goal was to determine whether two graphs, G1 and G2 were isomorphic. We saw that there is an isomorphism G1 ≈ G2 if and only if there is an isomorphism H1 ≈ H2 between two graphs obtained by adjoining two loops to each vertex of G1 and G2 . The adjacency matrices C1 and C2 of H1 and H2 are similar and equivalent. But when viewed as the Gram matrices of Z-lattices M1 and M2 , it turns out that M1 and M2 are not in the same genus. It follows that the graphs H1 and H2 are not isomorphic, so G1 and G2 are not isomorphic. Example 1 shows that the genus concept for quadratic forms may demonstrate nonisomorphism of two graphs when similarity and equivalence criteria applied to their associated adjacency matrices fail to do so. But the following example shows that the genus approach has limitations. The computations use Magma as in Example 1, and only a brief comment will be given. Example 2. The Chang graphs are three graphs obtained from the line graph L(K8 ) by applying the “switching” procedure to suitable subsets of vertices in L(K8 ). These graphs each have 28 vertices (the lines of K8 ). Their adjacency matrices (see [9]) are isospectral and equivalent. Adding two loops to each vertex yields graphs with positive definite adjacency matrices, and Magma shows that all three Z-lattices with those as

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their Gram matrices are in the same genus. But the three Chang graphs are known not to be isometric. Hence the genus concept is not sufficiently strong to demonstrate this. 5. Remarks and questions (1) Use of the genus failed to show that the Chang graphs are not isomorphic. But actually the Chang graphs have much in common: they are all strongly regular graphs with parameters (28, 12, 6, 4) That is, they all have 28 vertices, the vertices all have degree 12, every pair of adjacent vertices have 6 common neighbors, and every pair of nonadjacent vertices have 4 common neighbors. In summary, they share many structural features. This raises the following question. Suppose the adjacency matrices of two graphs are shown to be Gram matrices of Z-lattices in the same genus. Does this tell us any new information about common properties of the two graphs? (2) While the isometry problem for Z-lattices of large rank n has not been solved in the positive definite case, it is nevertheless interesting to ask this: suppose Z-lattices with Gram matrices A1 , A2 are actually isometric. What would this tell us about common properties of graphs with A1 and A2 as adjacency matrices? The class numbers h(∗) of positive definite Z-lattices of rank n—that is, the number of isometry classes in their genera—are known to grow rapidly with n. For example, from Pfeuffer [7] we have the inequality  h(L) ≥

n Γ( i ) 2 i

i=1

π2

× 22−n

if L is a positive definite lattice of rank n. For instance, if n = 50 then a Magma computation gives h(L) ≥ 8 × 10142 And if one chooses a representative set of lattices for the massive collection of isometry classes partitioning such a genus, no two of their Gram matrices are congruent, even though they all have the same invariant factors. Thus the condition of being isometric is much stronger than being in the same genus, just as being in the same genus is stronger than having the same associated invariant factors. It is therefore reasonable to expect that graphs with congruent adjacency matrices will share some interesting properties. (3) The minimum of a Z-lattice L is the value min L = min{|q(x)| | 0 = x ∈ L}

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Suppose A is a symmetric n × n matrix in Mn (Z). Does the minimum of a Z-lattice having A as a Gram matrix tell us anything about a graph having A as an adjacency matrix? (4) Some interesting matrix-theoretic questions arise from proceeding as we have in the examples. For instance, the matrices A1 , A2 in Example 1 are equivalent, as are the matrices obtained by adjoining r loops to each graph vertex when r = 2 (the case used in the example) and r = 5, but not when r = 1, 3 or 4. (These facts are thanks to Magma.) Can one predict values of r for which the resulting matrices are again equivalent? Of course any r that yields different invariant factors or different lattice genera would demonstrate non-isomorphism of the graphs. (In fact, in [3] the authors deduce the non-isomorphism of the Shrikhande graph and K4 K4 by noting that using r = 1 yields adjacency matrices with different invariant factors.) With respect to the graph isomorphism problem, each time one encounters an r for which the resulting matrices have the same matrix invariants (or the associated lattices are in the same genus) heightens our expectation that the graphs are isomorphic. How should one quantify the probabilities that this is the case? That is, for given ε > 0, how many values of r must one test—finding that the resulting matrices continue to have the same invariant factors or actually represent lattices in the same genus—so that the probability that the graphs are not isomorphic is less than ε? And what if Pfeuffer’s lower bound on class numbers is included in the computation as well? (5) The procedure used in this paper has been to add r loops to each of the n graph vertices in order to obtain associated matrices for consideration. One could alternatively insert c edges between each pair of vertices of the given graphs and consider the associated matrices. So if an initial adjacency matrix is A then the new matrix would be A + c(Jn − In ), where Jn is the n × n matrix of all 1’s. More generally, one can consider matrices of the form A + 2rIn + c(Jn − In ), with r and c nonnegative integers. Again, probabilistic questions as in remark (4) can be asked. In general, how does the number of elementary divisors change when matrix sums of this kind are carried out? If it increases significantly, this will cause a big increase in the number of genera of lattices whose Gram matrices have those invariant factors. (6) The work of Friedland mentioned earlier in this paper leads to this question: determine necessary and conditions for two Zp -matrices to be orthogonally congruent over Zp . (7) Examples 1 and 2 showed the utility and limitations of the genus concept for lattices in connection with the graph isomorphism problem. But as noted in Remark 2, knowing that two lattices are actually isometric is much stronger than knowing that they are in the same genus. As the computational power of software develops, it will be interesting to explore the strength of lattice isometry when applied to the graph isomorphism problem, and we leave this for further investigation. For instance, the lattices associated with the Chang graphs are in the same genus. But are they isometric?

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Declaration of competing interest There is no competing interest. Acknowledgements Markus Kirschmer provided me with much-needed assistance with Magma and alerted me to Pfeuffer’s result. Mark Watkins advised me on Magma’s capacity and limitations. Finally, Edward Spence [9] was kind enough to supply me with adjacency matrices for the Chang graphs. References [1] N. Biggs, Algebraic Graph Theory, second edition, Cambridge University Press, 1993. [2] G. Bachman, Introduction to p-Adic Numbers and Valuation Theory, Academic Press, New YorkLondon, 1964. [3] A.E. Brouwer, W.H. Haemers, Spectra of Graphs, Springer-Verlag, New York, 2012. [4] S. Friedland, Quadratic forms and the graph isomorphism problem, Linear Algebra Appl. 150 (1991) 423–442. [5] S. Friedland, Rational orthogonal similarity of rational symmetric matrices, Linear Algebra Appl. 192 (1993) 109–114. [6] L.J. Gerstein, Basic Quadratic Forms, Graduate Studies in Mathematics, vol. 90, American Mathematical Society, Providence, 2008. [7] H. Pfeuffer, Einklassige Geschlechter totalpositive quadratischer Formen in totalreelen algebraischen Zahlkörpern, J. Number Theory 3 (1971) 371–411. [8] S.S. Shrikhande, The uniqueness of the L2 association scheme, Ann. Math. Stat. 30 (1959) 781–798. [9] E. Spence, Matrices 1, 3, 4, http://www.maths.gla.ac.uk/~es/SRGs/28-12-6-4. [10] W. Wang, A counterexample to a conjecture of Friedland, Linear Algebra Appl. 430 (2009) 2026–2029.