The subspace structure of maximum cliques in pseudo-Paley graphs from unions of cyclotomic classes

TL;DR

This paper explores the structure of maximum cliques in pseudo-Paley graphs derived from unions of cyclotomic classes, revealing conditions for their existence and implications for clique bounds.

Contribution

It demonstrates that maximum cliques with subspace structure must evenly involve cyclotomic classes and shows most pseudo-Paley graphs lack such cliques, challenging existing bounds.

Findings

Most pseudo-Paley graphs do not admit maximum cliques with subspace structure.

Generalized Peisert graphs are not isomorphic to Paley or Peisert graphs.

The Delsarte bound on clique size can be improved for these graphs.

Abstract

Blokhuis showed that all maximum cliques in Paley graphs of square order have a subfield structure. Recently, it has been shown that in Peisert-type graphs, all maximum cliques are affine subspaces, and yet some maximum cliques do not arise from a subfield. In this paper, we investigate the existence of a clique of size $\sqrt{q}$ with a subspace structure in pseudo-Paley graphs of order $q$ from unions of semi-primitive cyclotomic classes. We show that such a clique must have an equal contribution from each cyclotomic class and that most such pseudo-Paley graphs do not admit such cliques, suggesting that the Delsarte bound $\sqrt{q}$ on the clique number can be improved in general. We also prove that generalized Peisert graphs are not isomorphic to Paley graphs or Peisert graphs, confirming a conjecture of Mullin.