Eigenvalues and the Smith normal form

On the Smith normal form of skew E-W matrices

“…• b 1 b 2 · · · b k divides the product of any k eigenvalues of A T + I (repetitions allowed), in the sense that the quotient is an algebraic integer, for 1 ≤ k ≤ 4t + 1 (see [13]). …”

Smith normal form and Laplacians

“…Conjecture 7 in [14] would imply that for threshold graphs G with an integer eigenvalue n > λ > 0 of multiplicity m(λ), then Φ(G) always contains a subgroup isomorphic to (Z/λZ) m(λ) , even when λ | n (see [14,Example 14]). Conjecture 7 in [14] also implies for threshold graphs G that each summand of Φ(G) has order a product of distinct eigenvalues of G, so that Φ(G) is killed by the product of the distinct eigenvalues of G. This latter fact follows for all graphs from our next proposition, which generalizes [35,Theorem 2].…”

“…Adding (or subtracting) all rows to the first row and all columns to the first column shows that the group Φ(M) is isomorphic to the group Φ(M (1,1) ). The minor M (1,1) is nonsingular, and thus we can use [35,Theorem 1], to show that Φ(M (1,1) ) contains an element of order L(λ) if M (1,1) has eigenvalue λ. This latter fact is true if m(λ) > 1, and is proved…”

“…In the more general context of any integer matrix M, this question is considered in [32] and [35]. In the special case of the adjacency matrix of a graph G, this question is considered in [34].…”

Integer invariants of abelian Cayley graphs

“…Generally speaking, the spectrum of an integer matrix has little connection to its elementary divisors. See [14,15] for a discussion in the context of the adjacency matrix, and [11] for what can be said about the Laplacian. For abelian Cayley graphs the connection to the Smith normal form of A E is made by an important observation of Sin [15, p. 1364], which we paraphrase in the following two theorems.…”