Integer invariants of abelian Cayley graphs

Understand

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.