Infinite classes of strongly regular graphs derived from $GL(n,F_2)$

TL;DR

This paper introduces eight infinite families of strongly regular graphs derived from orbit Cayley graphs over the elementary abelian 2-group, with six being new, and also provides their spectral properties.

Contribution

The paper constructs eight infinite families of strongly regular graphs from orbit Cayley graphs over Z_2^n, including six novel families, and characterizes their spectra.

Findings

Six new families of strongly regular graphs identified.

Spectral formulas for the orbit Cayley graphs derived.

Explicit construction methods provided.

Abstract

It is known that the automorphism group of the elementary abelian $2$-group $Z_2^n$ is isomorphic to the general linear group $GL(n,F_2)$ of degree $n$ over $F_2$. Let $W$ be the collection of permutation matrices of order $n$. It is clear that $W\le GL(n,F_2)$. In virtue of this, we consider the Cayley graph $Cay(Z_2^n,S)$, where $S$ is the union of some orbits under the action of $W$. We call such graphs the orbit Cayley graphs over $Z_2^n$. In this paper, we give eight infinite families of strongly regular graphs among orbit Cayley graphs over $Z_2^n$, in which six families are new as we know. By the way, we formulate the spectra of orbit Cayley graphs as well.