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

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