Strongly regular edge-transitive graphs

TL;DR

This paper investigates the structure of strongly regular, edge-transitive graphs, revealing their automorphism groups are quasiprimitive and establishing constraints on their parameters, with implications for classification.

Contribution

It demonstrates that irreducible graphs in this family have quasiprimitive automorphism groups and rules out holomorphic simple automorphism groups using the Classification of Finite Simple Groups.

Findings

Irreducible graphs have quasiprimitive automorphism groups

No graph has a holomorphic simple automorphism group

Constraints on graph parameters reduce to complete graphs

Abstract

In this paper, we examine the structure of vertex- and edge-transitive strongly regular graphs, using normal quotient reduction. We show that the irreducible graphs in this family have quasiprimitive automorphism groups, and prove (using the Classification of Finite Simple Groups) that no graph in this family has a holomorphic simple automorphism group. We also find some constraints on the parameters of the graphs in this family that reduce to complete graphs.