Automorphisms of left Ideal relation graph over full matrix ring

TL;DR

This paper characterizes the automorphisms of the left-ideal relation graph over the full matrix ring over a finite field and explores various graph properties of the associated undirected graph.

Contribution

It provides a complete characterization of automorphisms of the left-ideal relation graph for matrix rings over finite fields and analyzes its fundamental graph properties.

Findings

Automorphisms of the left-ideal relation graph are fully characterized.

The undirected left relation graph's connectivity, girth, and clique number are determined.

Various structural properties of the graph are established.

Abstract

The left-ideal relation graph on a ring $R$, denoted by $\overrightarrow{\Gamma_{l-i}}(R)$, is a directed graph whose vertex set is all the elements of $R$ and there is a directed edge from $x$ to a distinct $y$ if and only if the left ideal generated by $x$, written as $[x]$, is properly contained in the left ideal generated by $y$. In this paper, the automorphisms of $\overrightarrow{\Gamma_{l-i}}(R)$ are characterized, where $R$ is the ring of all $n \times n$ matrices over a finite field $F_q$. The undirected left relation graph, denoted by $\Gamma_{l-i}(M_n(F_q))$, is the simple graph whose vertices are all the elements of $R$ and two distinct vertices $x, y$ are adjacent if and only if either $[x] \subset [y]$ or $[y] \subset [x]$ is considered. Various graph theoretic properties of $\Gamma_{l-i}(M_n(F_q))$ including connectedness, girth, clique number, etc. are studied.