Strongly Regular Graphs with Generalized Denniston and Dual Generalized Denniston Parameters

TL;DR

This paper introduces two new families of strongly regular Cayley graphs with generalized parameters, unifying and extending previous constructions, and advancing the understanding of combinatorial structures in algebraic graph theory.

Contribution

It presents novel constructions of strongly regular graphs with generalized Denniston parameters, broadening the scope of known combinatorial designs and linking various existing structures.

Findings

Constructed two new families of strongly regular Cayley graphs

Unified and extended existing constructions in the literature

Provided new insights into partial difference sets and related combinatorial objects

Abstract

We construct two families of strongly regular Cayley graphs, or equivalently, partial difference sets, based on elementary abelian groups. The parameters of these two families are generalizations of the Denniston and the dual Denniston parameters, in contrast to the well known Latin square type and negative Latin square type parameters. The two families unify and subsume a number of existing constructions which have been presented in various contexts such as strongly regular graphs, partial difference sets, projective sets, and projective two-weight codes, notably including Denniston's seminal construction concerning maximal arcs in classical projective planes with even order. Our construction generates further momentum in this area, which recently saw exciting progress on the construction of the analogue of the famous Denniston partial difference sets in odd characteristic.