Joshua E. Ducey, Deelan M. Jalil
Published 2013-08-10, updated 2013-12-12Version 2
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.
Comments: 8 pages. Many small changes, mostly to make references more precise and to explain notation and terminology. The biggest changes are to Section 3, where we are now more careful with the statements of the theorems and their proofs
More on the skew-spectra of bipartite graphs and Cartesian products of graphs
On Linkedness of Cartesian Product of Graphs
Collinear triples and quadruples for Cartesian products in $\mathbb{F}_p^2$
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