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

Let \( R \) be a noncommutative ring with identity and \( Z(R)^* \) be the non-zero zero-divisors of \( R \). The directed zero-divisor graph \(\Gamma(R)\) of \( R \) is a directed graph with vertex set \( Z(R)^* \) and for distinct vertices \( x \) and \( y \) of \( Z(R)^* \), 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 \(\Gamma(R)\) is not a star graph, then the domination number of the zero-divisor graph \(\Gamma(R)\) 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.