The Smith and critical groups of Paley graphs

David Chandler, Peter Sin, Qing Xiang

Published 2014-01-31, updated 2014-09-30Version 3

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