On balanced characteristic functions of canonical cliques in Paley graphs of square order

TL;DR

This paper proves that balanced characteristic functions of canonical cliques in Paley graphs of square order span a specific eigenspace, advancing the understanding of combinatorial structures in these graphs and contributing to a new proof of an Erdős-Ko-Rado type theorem.

Contribution

It establishes a novel spectral property of canonical cliques in Paley graphs of square order, aiding in a new proof of an Erdős-Ko-Rado type theorem.

Findings

Balanced characteristic functions span the -eigenspace of the graph.

First step towards a new proof of the Erdf6s-Ko-Rado theorem for Paley graphs.

Provides spectral characterization of canonical cliques.

Abstract

In this paper we prove that balanced characteristic functions of canonical cliques in a Paley graph of square order $P(q^2)$ span the $\frac{-1+q}{2}$-eigenspace of the graph. This is the first of two steps to a second proof of the analogue of Erd\"os-Ko-Rado theorem for Paley graphs of square order (the first proof was given by A. Blokhuis in 1984).