The Smith and critical groups of Paley graphs

Abstract

There is a Paley graph for each prime power $q$ such that $q\equiv 1(\mathrm{mod}4)$. The vertex set is the field $\mathbb{F}q$ and two vertices $x$ and $y$ are joined by an edge if and only if $x-y$ is a nonzero square of $\mathbb{F}q$. We compute the Smith normal forms of the adjacency matrix and Laplacian matrix of a Paley graph.

10 pages. v2. Fixed typo 10 pages, v3. Minor corrections and clarifications