Orthogonal inner product graphs of odd characteristic and their automorphisms

TL;DR

This paper studies the orthogonal inner product graphs over finite fields of odd characteristic, determines their automorphism groups, and characterizes the conditions for vertices and edges to be in the same orbit under these automorphisms.

Contribution

It introduces the orthogonal inner product graph over finite fields of odd characteristic and fully characterizes its automorphism group and orbit structure.

Findings

The graph is disconnected if and only if $2 u+ ext{delta}=2$.

Automorphism groups are explicitly determined.

Necessary and sufficient conditions for vertices and edges to be in the same orbit.

Abstract

Let $\mathbb{F}_q$ be a finite field of odd characteristic and $2\nu+\delta\geq2$ an integer number with $\delta=0,1$ or $2$. The orthogonal inner product graph $Oi\big(2\nu+\delta,q\big)$ over $\mathbb{F}_q$ is defined and the automorphism groups of $Oi\big(2\nu+\delta,q\big)$ are determined. We show that $Oi\big(2\nu+\delta,q\big)$ is a disconnected graph if $2\nu+\delta=2$; otherwise it is not. Moreover, we have two necessary and sufficient conditions for two vertices of $Oi\big(2\nu+\delta,q\big)$ and two edges of $Oi\big(2\nu+\delta,q\big)$ respectively are in the same orbit under the action of the automorphism group of $Oi\big(2\nu+\delta,q\big).$