Domination in commuting graph and its complement

TL;DR

This paper investigates the domination and signed domination numbers of commuting graphs of non-commutative rings, providing specific results for rings of order four, prime cube, and products of rings, along with bounds for signed domination.

Contribution

It characterizes the domination sum for finite non-commutative rings and determines domination numbers for specific ring classes, introducing new bounds and exact values.

Findings

For rings of order four, the domination sum equals the ring size.

Domination number of product rings is explicitly determined.

Upper bounds for signed domination numbers are established.

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 $\Gamma(R)$ for non-commutative ring $R$ with $Z(R)=\{0\}$. For a finite ring $R$, it is shown that $\gamma(\Gamma(R)) + \gamma(\overline{\Gamma}(R))=|R|$ if and only if $R$ is non-commutative ring on 4 elements. Also we determine the domination number of $\Gamma(\prod_{i=1}^{t}R_i)$ 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 $\Gamma(\prod_{i=1}^{t}R_i)$.