Automorphisms of linear functional graphs over vector spaces

Abstract

Let $\mathbb{F}_q$ be a finite field with $q$ elements, $n\geq2$ a positive integer, $\mathbb{V}_0$ a $n$-dimensional vector space over $\mathbb{F}_q$ and $\mathbb{T}_0$ the set of all linear functionals from $\mathbb{V}_0$ to $\mathbb{F}_q$. Let $\mathbb{V}=\mathbb{V}_0\setminus\{0\}$ and $\mathbb{T}=\mathbb{T}_0\setminus\{0\}$. The \emph{linear functional graph} of $\mathbb{V}_0$ dented by $\digamma(\mathbb{V})$, is an undirected bipartite graph, whose vertex set $V$ is partitioned into two sets as $V=\mathbb{V}\cup \mathbb{T}$ and two vertices $v\in \mathbb{V}$ and $f\in \mathbb{T}$ are adjacent if and only if $f$ sends $v$ to the zero element of $\mathbb{F}_q$ (i.e. $f(v)=0$). In this paper, the structure of all automorphisms of this graph is characterized and formolized. Also the cardinal number of automorphisms group for this graph is determined.