Symplectic inner product graphs and their automorphisms

Abstract

A new graph, called the symplectic inner product graph $Spi(2\nu ,q)$, over a finite field ${\mathbb{F}}_q$ is introduced. We show that $Spi(2\nu ,q)$ is connected with diameter $4$ if and only if $\nu \ge 2$ and the automorphism group of $Spi(2\nu ,q)$ is determined. Two necessary and sufficient conditions for two vertices of $Spi(2\nu ,q)$ and two edges of $Spi(2\nu ,q)$ respectively are in the same orbit under the action of the automorphism group of $Spi(2\nu ,q)$ are obtained.