Orthogonal graphs over Galois rings of odd characteristic

TL;DR

This paper introduces orthogonal graphs over Galois rings of odd characteristic, proves their arc transitivity, and characterizes their parameters, revealing new classes of strongly regular and Deza graphs.

Contribution

It defines orthogonal graphs over Galois rings of odd characteristic and analyzes their symmetry and regularity properties, including their classification as strongly regular and Deza graphs.

Findings

Orthogonal graphs are arc transitive.

The graphs are quasi-strongly regular.

Special cases include strongly regular and Deza graphs.

Abstract

Assume that $\nu$ is a positive integer and $\delta=0, 1$ or $2$. In this paper we introduce the orthogonal graph $\Gamma^{2\nu+\delta}$ over a Galois ring of odd characteristic and prove that it is arc transitive. Moreover, we compute its parameters as a quasi-strongly regular graph. In particular, we show that $\Gamma^{2+\delta}$ is a strongly regular graph and $\Gamma^{2\nu+1}$ is a strictly Deza graph when $\nu\geq 2$.