Orthogonal Inner Product Graphs over Finite Fields of Odd Characteristic

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