The automorphism groups and identification of some Generalized Paley Graphs

TL;DR

This paper investigates the automorphism groups of generalized Paley graphs and establishes bounds on their Weisfeiler-Leman dimension, providing insights into their symmetry and graph isomorphism properties.

Contribution

It proves that for large prime power orders, the automorphism groups are subgroups of ${\mathrm{A\Gamma L}}(1,q)$ and bounds the Weisfeiler-Leman dimension to at most 5.

Findings

Automorphism groups are subgroups of ${\mathrm{A\Gamma L}}(1,q)$ for large q.

Weisfeiler-Leman dimension is at most 5 for these graphs.

The results also apply to Van Lint-Schrijver graphs.

Abstract

The family of generalized Paley graphs of prime power order $q$ and degree $(q-1)/k$ is studied. It is shown that the automorphism group of a graph in this family is a subgroup of ${\mathrm{A\Gamma L}}(1,q)$ whenever $q$ is sufficiently large relative to $k$. Furthermore, under the same conditions, the Weisfeiler-Leman dimension of these graphs is proved to be at most $5$. In particular, the same bound holds for the Van Lint-Schrijver graphs.