Generalised Paley graphs with a product structure

TL;DR

This paper characterizes when generalized Paley graphs can be decomposed into Cartesian prime factors, showing these factors are also generalized Paley graphs, thus advancing understanding of their automorphism groups.

Contribution

It provides a parameter-based characterization of Cartesian decomposability for generalized Paley graphs and proves their prime factors are also generalized Paley graphs, extending previous results.

Findings

Characterization of Cartesian decomposable generalized Paley graphs

Prime factors are themselves smaller generalized Paley graphs

Contributes to understanding automorphism groups of these graphs

Abstract

A graph is Cartesian decomposable if it is isomorphic to a Cartesian product of (more than one) strictly smaller graphs, each of which has more than one vertex and admits no such decomposition. These smaller graphs are called the Cartesian-prime factors of the Cartesian decomposition, and were shown, by Sabidussi and Vizing independently, to be uniquely determined up to isomorphism. We characterise by their parameters those generalised Paley graphs which are Cartesian decomposable, and we prove that for such graphs, the Cartesian-prime factors are themselves smaller generalised Paley graphs. This generalises a result of Lim and the second author which deals with the case where all the Cartesian-prime factors are complete graphs. These results contribute to the determination, by parameters, of generalised Paley graphs with automorphism groups larger than the 1-dimensional affine subgroups used to define them.