A classification of one dimensional affine rank three graphs

TL;DR

This paper classifies one-dimensional affine rank three graphs, revealing they are either Paley graphs or constructed by Van Lint and Schrijver or Peisert, using elementary group theory without relying on the full classification of affine groups.

Contribution

It provides a simplified classification of these graphs, connecting them to well-known graph families without complex group classification methods.

Findings

Graphs are either Paley, Van Lint-Schrijver, or Peisert graphs.

The classification is achieved through elementary group theory.

The approach simplifies previous complex classifications.

Abstract

The rank three subgroups of a one-dimensional affine group over a finite field were classified in 1978 by Foulser and Kallaher. Although one can use their results for a classification of corresponding rank three graphs, the author did not find such a classification in a literature. The goal of this note is to present such a classification. It turned out that graph classification is much simpler than the group one. More precisely, it is shown that the graphs in the title are either the Paley graphs or one of the graphs constructed by Van Lint and Schrijver or by Peisert. Our approach is based on elementary group theory and does not use the classification of rank three affine groups.