THE DOMINATION NUMBER OF COMMUTING GRAPHS OVER MATRIX DIRECT SUMS

Abstract

Let $L$ be a finite commutative ring with unity and $M(m\oplus m, L)$ be the collection of all direct sum matrices over $L$. The commuting graph $\Gamma(M(m \oplus m, L))$ has vertex set $M(m \oplus m, L) \setminus Z(M(m \oplus m, L))$, where two distinct vertices are adjacent if they commute. In this paper, we investigate the domination number of $\Gamma(M(m \oplus m, L))$ and establish bounds for various ring structures. We prove that for rings, the domination number satisfies $\gamma(\Gamma(M(m \oplus m, L))) \geq 2$.