Strongly Regular Graphs of Rank Four

TL;DR

This paper classifies strongly regular graphs of rank four by leveraging classification results on rank four permutation groups, revealing their automorphism groups and structural properties.

Contribution

It introduces a novel approach by using rank four permutation group classifications to analyze strongly regular graphs and determine their automorphism groups.

Findings

Classification of strongly regular graphs with automorphism groups of rank four

Identification of structural properties of these graphs

Connection between graph properties and permutation group classifications

Abstract

Strongly regular graphs are regular graphs with a constant number of common neighbours between adjacent vertices, and a constant number of common neighbours between non-adjacent vertices. These graphs have been of great interest over the last few decades and often give rise to interesting groups of automorphisms. In this paper we take a reverse approach, and leverage strong classification results on rank four permutation groups to classify the strongly regular graphs which yield such groups as a group of automorphisms.