Automorphisms of strongly regular graphs

TL;DR

This paper extends Benson's theorem to strongly regular graphs and directed variants, providing numerical constraints on automorphisms, with applications in algebraic combinatorics and finite geometry.

Contribution

It generalizes a key theorem to broader classes of graphs, offering new tools for analyzing automorphisms in strongly regular and directed strongly regular graphs.

Findings

Provides bounds on fixed vertices under automorphisms

Establishes restrictions on vertices mapped to adjacent vertices

Links results to partial difference sets and projective two-weight sets

Abstract

In this article we generalize a theorem of Benson for generalized quadrangles to strongly regular graphs and directed strongly regular graphs. The main result provides numerical restrictions on the number of fixed vertices and the number of vertices mapped to adjacent vertices under an automorphism. It is explained how these results can be used when studying partial difference sets in Abelian groups and projective two-weight sets. The underlying ideas are linear algebraic in nature.