Second largest maximal cliques in small Paley graphs of square order

TL;DR

This paper investigates the structure of second largest maximal cliques in small Paley graphs of square order, focusing on deviations from the conjectured size and symmetry for specific small cases.

Contribution

It analyzes the algebraic and geometric structure of extra second largest maximal cliques in Paley graphs for small square orders, providing insights beyond the conjecture.

Findings

Extra second largest maximal cliques exist for q in {9,11,13,17,19,23}.

The structure of these extra cliques is analyzed algebraically and geometrically.

Deviations from the conjectured clique size and symmetry are characterized.

Abstract

There is a conjecture that the second largest maximal cliques in Paley graphs of square order $P(q^2)$ have size $\frac{q+\epsilon}{2}$, where $q \equiv \epsilon \pmod 4$, and split into two orbits under the full group of automorphisms whenever $q \ge 25$ (a symmetric description for these two orbits is known). However, some extra second largest maximal cliques (of this size) exist in $P(q^2)$ whenever $q \in \{9,11,13,17,19,23\}$. In this paper we analyse the algebraic and geometric structure of the extra cliques.