The Smith and critical groups of Paley graphs

Abstract

There is a Paley graph for each prime power $$q$$q such that $$q\equiv 1\pmod 4$$q≡1(mod4). The vertex set is the field $${\mathbb {F}_q}$$Fq, and two vertices $$x$$x and $$y$$y are joined by an edge if and only if $$x-y$$x-y is a nonzero square of $${\mathbb {F}_q}$$Fq. We compute the Smith normal forms of the adjacency matrix and Laplacian matrix of a Paley graph.