Automorphisms of left Ideal relation graph over full matrix ring

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.