Orthogonal inner product graphs of odd characteristic and their automorphisms

Abstract

Let ${\mathbb{F}}_q$ be a finite field of odd characteristic and $2\nu +\delta \ge 2$ an integer number with $\delta =0,1$ or $2$. The orthogonal inner product graph $Oi(2\nu +\delta ,q)$ over ${\mathbb{F}}_q$ is defined and the automorphism groups of $Oi(2\nu +\delta ,q)$ are determined. We show that $Oi(2\nu +\delta ,q)$ 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 $Oi(2\nu +\delta ,q)$ and two edges of $Oi(2\nu +\delta ,q)$ respectively are in the same orbit under the action of the automorphism group of $Oi(2\nu +\delta ,q).$

arXiv admin note: text overlap with arXiv:2205.09426