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Cliques in Paley graphs of square order and in Peisert graphs

Andries E. Brouwer, Sergey Goryainov, Leonid Shalaginov, Chi Hoi Yip

TL;DR

The paper analyzes maximal and near-maximal cliques in the collinearity graphs arising from Desarguesian nets, with a focus on Paley graphs $P(q)$ and Peisert graphs $P^*(q)$. It develops structural results for maximal cliques $C_{x,L}$ containing a fixed small clique $igl"{x}igr floor ext{and }(x^otigcap L)$, including transitivity properties and size formulas $|C_{x,L}|=m-1+h p^f$ subject to divisibility constraints, and complements these with extensive numerical data on clique sizes for small square-order graphs and their Taylor extensions. The work confirms known results for square-order Paley graphs and extends understanding in Peisert graphs, highlighting how subfields yield maximum cliques and how second-largest cliques arise through conic- and orbit-based constructions; it also provides evidence about orbit structures of 2nd-largest cliques and offers conjectural guidance for larger nets. Overall, the paper contributes both analytic and computational insights into the clique structure of incidence-based SRGs, with implications for design theory and algebraic graph theory.

Abstract

We study maximal cliques in the collinearity graphs of Desarguesian nets, give some structural results and some numerical information.

Cliques in Paley graphs of square order and in Peisert graphs

TL;DR

The paper analyzes maximal and near-maximal cliques in the collinearity graphs arising from Desarguesian nets, with a focus on Paley graphs and Peisert graphs . It develops structural results for maximal cliques containing a fixed small clique , including transitivity properties and size formulas subject to divisibility constraints, and complements these with extensive numerical data on clique sizes for small square-order graphs and their Taylor extensions. The work confirms known results for square-order Paley graphs and extends understanding in Peisert graphs, highlighting how subfields yield maximum cliques and how second-largest cliques arise through conic- and orbit-based constructions; it also provides evidence about orbit structures of 2nd-largest cliques and offers conjectural guidance for larger nets. Overall, the paper contributes both analytic and computational insights into the clique structure of incidence-based SRGs, with implications for design theory and algebraic graph theory.

Abstract

We study maximal cliques in the collinearity graphs of Desarguesian nets, give some structural results and some numerical information.

Paper Structure

This paper contains 12 sections, 12 theorems.

Table of Contents

  1. Introduction
  2. Paley graphs
  3. Peisert graphs
  4. Nets
  5. Automorphism groups
  6. Maximum Cliques
  7. 2nd Largest Cliques
  8. Maximal cliques in Desarguesian nets
  9. Numerical data
  10. Maximal cliques in small Paley graphs
  11. Maximal cliques in the Taylor extension of small Paley graphs
  12. Maximal cliques in small Peisert graphs

Key Result

Proposition 1.1

(Goryainov et al. Goryainov-et-al18) Let $\beta$ be primitive in ${\mathbb F}_{r^2}$, and put $\omega := \beta^{q-1}$. Let $Q := \langle \omega^2 \rangle$. If $q \equiv 1~({\rm mod}~4)$ the set $Q$ is a maximal coclique of size $(q+1)/2$ in $P(r^2)$. If $q \equiv 3~({\rm mod}~4)$ the set $Q \cup \{0

Theorems & Definitions (12)

  • Proposition 1.1
  • Proposition 1.2
  • Theorem 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Theorem 2.7
  • Corollary 2.8
  • ...and 2 more