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

TL;DR

This paper investigates eigenfunctions and maximal cliques in generalized Paley graphs of square order, providing explicit constructions and bounds that extend previous work and analyze stability of combinatorial theorems.

Contribution

It explicitly constructs maximal cliques in GP(q^2,m), proves tight bounds on eigenfunction support, and studies stability of Erdős-Ko-Rado theorem for these graphs.

Findings

Constructed maximal cliques of size (q+1)/m or (q+1)/m+1.

Proved the weight-distribution bound is tight for the smallest eigenvalue.

Analyzed stability of the Erdős-Ko-Rado theorem in this context.

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 $m \mid (q+1)$. In particular, we explicitly construct maximal cliques of size $\frac{q+1}{m}$ or $\frac{q+1}{m}+1$ 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 $-\frac{q+1}{m}$ 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 Erd\H{o}s-Ko-Rado theorem for GP$(q^2,m)$ (first proved by Sziklai).