Integer invariants of abelian Cayley graphs

TL;DR

This paper explores how the spectra of matrices associated with abelian Cayley graphs can determine their Smith normal forms, with applications to algebraic combinatorics and graph invariants.

Contribution

It introduces a spectral method to analyze Smith normal forms of matrices from abelian Cayley graphs, including applications to association schemes and critical groups.

Findings

Spectral techniques relate eigenvalues to Smith normal forms.

Applications to Hamming schemes and Cartesian products of complete graphs.

Recovery of known results on critical groups using new methods.

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.