On generalised Paley graphs and their automorphism groups

TL;DR

This paper investigates the structure and automorphism groups of generalized Paley graphs, identifying conditions for connectivity, isomorphism to Hamming graphs, and describing their automorphism groups as affine primitive groups.

Contribution

It characterizes when generalized Paley graphs are connected, identifies when they are isomorphic to Hamming graphs, and describes their automorphism groups as affine primitive groups.

Findings

Generalized Paley graphs are connected under specific parameters.

They are isomorphic to Hamming graphs for certain parameters.

Automorphism groups are primitive affine groups in the non-Hamming case.

Abstract

The generalised Paley graphs are, as their name suggests, a generalisation of the Paley graphs, first defined by Paley in 1933 (see \cite{Paley}). They arise as the relation graphs of symmetric cyclotomic association schemes. However, their automorphism groups may be much larger than the groups of the corresponding schemes. We determine the parameters for which the graphs are connected, or equivalently, the schemes are primitive. Also we prove that generalised Paley graphs are sometimes isomorphic to Hamming graphs and consequently have large automorphism groups, and we determine precisely the parameters for this to occur. We prove that in the connected, non-Hamming case, the automorphism group of a generalised Paley graph is a primitive group of affine type, and we find sufficient conditions under which the group is equal to the one-dimensional affine group of the associated cyclotomic association scheme. The results have been applied in \cite{LLP} to distinguish between cyclotomic schemes and similar twisted versions of these schemes, in the context of homogeneous factorisations of complete graphs.