Integer invariants of abelian Cayley graphs

Abstract

Let G be a finite abelian group, let E be a subset of G, and form the Cayley (directed) graph of G with connecting set E. We explain how, for various matrices associated to this graph, the spectrum can be used to give information on the Smith normal form. This technique is applied to several interesting examples, including matrices in the Bose-Mesner algebra of the Hamming association scheme H(n, q). We also recover results of Bai and Jacobson-Niedermaier-Reiner on the critical group of a Cartesian product of complete graphs.2010 Mathematics Subject Classification. 05C50.

The distribution of sandpile groups of random graphs

“…The cokernel of the adjacency matrix is called the Smith group, and has been studied e.g. in [CSX14,DJ13].…”

The Smith group of the hypercube graph

“…Let p be given. We know from [2] that A is Z (p) -equivalent to a diagonal matrix whose nonzero are n − 2ℓ with multiplicity n ℓ , for 0 ≤ ℓ ≤ n. The latter is easily seen to be integrally equivalent to a diagonal matrix whose nonzero entries are 2k = 1,. . .…”

“…It is of interest to compute linear algebraic invariants of a graph, such as its eigenvalues and the invariant factors of an adjacency matrix or Laplacian matrix. In the case of Q n , previous work includes [1] and [2], where many of these invariants have been computed and some conjectures made about others. Here we shall consider the Smith group.…”

“…If X happens to have integer entries then its p-elementary divisors are the same whether it is considered as a matrix over Z or Z (p) Let A be an adjacency matrix for Q n . It was proved in [2] that for every odd prime p, A is Z (p) -equivalent to the diagonal matrix of the eigenvalues (all of which are integers).…”

Graphs with few trivial critical ideals

“…Smith groups were introduced in [31]. Recently, the computation of the Smith group for several families of graphs has attracted attention, see [10,13,19,20,36]. The critical group is especially interesting for connected graphs, since its order is equal to the number of spanning trees of the graph.…”