Cliques and colorings in generalized Paley graphs and an approach to synchronization

TL;DR

This paper explores the properties of generalized Paley graphs, focusing on their clique and chromatic numbers, and connects these findings to the synchronization in certain permutation groups.

Contribution

It introduces a new analysis of generalized Paley graphs where clique and chromatic numbers match and links this to synchronization in affine permutation groups.

Findings

Identified conditions for clique and chromatic numbers to coincide in generalized Paley graphs.

Established a connection between graph properties and synchronization in permutation groups.

Provided new insights into the structure of primitive affine permutation groups.

Abstract

Given a finite field, one can form a directed graph using the field elements as vertices and connecting two vertices if their difference lies in a fixed subgroup of the multiplicative group. If -1 is contained in this fixed subgroup, then we obtain an undirected graph that is referred to as a generalized Paley graph. In this paper we study generalized Paley graphs whose clique and chromatic numbers coincide and link this theory to the study of the synchronization property in 1-dimensional primitive affine permutation groups.