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

Abstract

Let GP (q 2 , m) be the m-Paley graph defined on the finite field with order q 2 . We study eigenfunctions and maximal cliques in generalised Paley graphs GP (q 2 , m), where m | (q+1). In particular, we explicitly construct maximal cliques of size q+1 m or q+1 m + 1 in GP (q 2 , m), and show the weight-distribution bound on the cardinality of the support of an eigenfunction is tight for the smallest eigenvalue − q+1 m of GP (q 2 , 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 (q 2 , m) (first proved by Sziklai).