Domination in the Directed Zero-Divisor Graph of Ring of Matrices

Abstract

Let $R$ be a noncommutative ring with identity and $Z(R{)}^{\ast }$ be the non-zero zero-divisors of $R$. The directed zero-divisor graph $\mathrm{\Gamma }(R)$ of $R$ is a directed graph with vertex set $Z(R{)}^{\ast }$ and for distinct vertices $x$ and $y$ of $Z(R{)}^{\ast }$, there is a directed edge from $x$ to $y$ if and only if $xy=0$ in $R$. S.P. Redmond has proved that for a finite commutative ring $R$, if $\mathrm{\Gamma }(R)$ is not a star graph, then the domination number of the zero-divisor graph $\mathrm{\Gamma }(R)$ equals the number of distinct maximal ideals of $R$. In this paper, we prove that such a result is true for the noncommutative ring $M_2(\mathbb{F})$, where $\mathbb{F}$ is a finite field. Using this, we obtain a class of graphs for which all six fundamental domination parameters are equal.