The EKR-module property of pseudo-Paley graphs of square order

TL;DR

This paper demonstrates that certain pseudo-Paley graphs of square order possess the EKR-module property, with a purely combinatorial approach that extends known results for Paley graphs and addresses a previously open problem.

Contribution

It proves the EKR-module property for a new family of pseudo-Paley graphs using combinatorial methods, expanding the understanding of their clique structure.

Findings

Pseudo-Paley graphs of square order satisfy the EKR-module property.

Characteristic vectors of maximum cliques are linear combinations of canonical clique vectors.

The approach models these graphs as block graphs of orthogonal arrays.

Abstract

We prove that a family of pseudo-Paley graphs of square order obtained from unions of cyclotomic classes satisfies the Erd\H{o}s-Ko-Rado (EKR) module property, in a sense that the characteristic vector of each maximum clique is a linear combination of characteristic vectors of canonical cliques. This extends the EKR-module property of Paley graphs of square order and solves a problem proposed by Godsil and Meagher. Different from previous works, which heavily rely on tools from number theory, our approach is purely combinatorial in nature. The main strategy is to view these graphs as block graphs of orthogonal arrays, which is of independent interest.