Relative non-commuting graph of a finite ring

TL;DR

This paper introduces the relative non-commuting graph of a subring within a finite ring, exploring its properties, structural parameters, and connections to relative commuting probabilities, including isomorphism conditions under certain equivalences.

Contribution

It defines the relative non-commuting graph for subrings of finite rings and investigates its properties, including diameter, girth, dominating sets, chromatic index, and isomorphism conditions.

Findings

Determined the diameter and girth of the graph.

Identified bounds for the chromatic index.

Established conditions for graph isomorphism under relative Z-isoclinic pairs.

Abstract

Let $S$ be a subring of a finite ring $R$ and $C_R(S) = \{r \in R : rs = sr \;\forall\; s \in S\}$. The relative non-commuting graph of the subring $S$ in $R$, denoted by $\Gamma_{S, R}$, is a simple undirected graph whose vertex set is $R \setminus C_R(S)$ and two distinct vertices $a, b$ are adjacent if and only if $a$ or $b \in S$ and $ab \neq ba$. In this paper, we discuss some properties of $\Gamma_{S, R}$, determine diameter, girth, some dominating sets and chromatic index for $\Gamma_{S, R}$. Also, we derive some connections between $\Gamma_{S, R}$ and the relative commuting probability of $S$ in $R$. Finally, we show that the relative non-commuting graphs of two relative $\Z$-isoclinic pairs of rings are isomorphic under some conditions.