Symplectic inner product graphs and their automorphisms

TL;DR

This paper introduces the symplectic inner product graph over finite fields, analyzes its connectivity and diameter, and determines its automorphism group, providing conditions for vertex and edge orbits under automorphisms.

Contribution

It defines the new symplectic inner product graph and characterizes its automorphism group and orbit conditions, advancing understanding of its symmetry properties.

Findings

The graph is connected with diameter 4 for ν ≥ 2.

Automorphism group of the graph is explicitly determined.

Necessary and sufficient conditions for orbit equivalences are established.

Abstract

A new graph, called the symplectic inner product graph $Spi\big(2\nu,q\big)$, over a finite field $\mathbb{F}_q$ is introduced. We show that $Spi\big(2\nu,q\big)$ is connected with diameter $4$ if and only if $\nu\geq2$ and the automorphism group of $Spi\big(2\nu,q\big)$ is determined. Two necessary and sufficient conditions for two vertices of $Spi\big(2\nu,q\big)$ and two edges of $Spi\big(2\nu,q\big)$ respectively are in the same orbit under the action of the automorphism group of $Spi\big(2\nu,q\big)$ are obtained.