Automorphisms of linear functional graphs over vector spaces

TL;DR

This paper characterizes the automorphisms of linear functional graphs over finite-dimensional vector spaces and determines the size of their automorphism groups.

Contribution

It provides a complete description of all automorphisms of these graphs and calculates the automorphism group cardinality, a novel contribution in the study of such graph structures.

Findings

Automorphisms are fully characterized.

The automorphism group size is explicitly determined.

The structure of the automorphism group is described.

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.