  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.
