Domination in commuting graph and its complement

Abstract

For each non-commutative ring R, the commuting graph of R is a graph with vertex set $R\setminus Z(R)$ and two vertices $x$ and $y$ are adjacent if and only if $x\neq y$ and $xy=yx$. In this paper, we consider the domination and signed domination numbers on commuting graph $Γ(R)$ for non-commutative ring $R$ with $Z(R)=\{0\}$. For a finite ring $R$, it is shown that $γ(Γ(R)) + γ(\overlineΓ(R))=|R|$ if and only if $R$ is non-commutative ring on 4 elements. Also we determine the domination number of $Γ(\prod_{i=1}^{t}R_i)$ and commuting graph of non-commutative ring $R$ of order $p^3$, where $p$ is prime. Moreover we present an upper bound for signed domination number of $Γ(\prod_{i=1}^{t}R_i)$.