On eigenfunctions and maximal cliques of generalised Paley graphs of square order

Abstract

Let GP (q2,m) be the m-Paley graph defined on the finite field with order q2. We study eigenfunctions and maximal cliques in generalised Paley graphs GP (q2,m), where m|(q+1). In particular, we explicitly construct maximal cliques of size q+1m or q+1m+1 in GP (q2,m), and show the weight-distribution bound on the cardinality of the support of an eigenfunction is tight for the smallest eigenvalue −q+1m of GP (q2,m). These new results extend the work of Baker et al. and Goryainov et al. on Paley graphs of square order. We also study the stability of the Erdős-Ko-Rado theorem for GP (q2,m) (first proved by Sziklai).