On eigenfunctions and maximal cliques of Paley graphs of square order

TL;DR

This paper identifies new maximal cliques in Paley graphs of square order and constructs eigenfunctions with minimal support, advancing understanding of their combinatorial and spectral properties.

Contribution

It introduces new maximal cliques in Paley graphs of order q^2 and constructs eigenfunctions with minimal support, linking clique structure to spectral properties.

Findings

New maximal cliques of size (q+1)/2 or (q+3)/2 identified.

Eigenfunctions with support size q+1 constructed.

Results depend on the congruence class of q modulo 4.

Abstract

In this paper we find new maximal cliques of size $\frac{q+1}{2}$ or $\frac{q+3}{2}$, accordingly as $q\equiv 1(4)$ or $q\equiv 3(4)$, in Paley graphs of order $q^2$, where $q$ is an odd prime power. After that we use new cliques to define a family of eigenfunctions corresponding to both non-principal eigenvalues and having the cardinality of support $q+1$, which is the minimum by the weight-distribution bound.