On generalized non-commuting graph of a finite ring

TL;DR

This paper introduces a new graph associated with subrings of finite rings, analyzing its properties like diameter and girth, and explores its relation to ring isoclinism, providing insights into the structure of finite rings.

Contribution

It defines the generalized non-commuting graph for subrings of finite rings and investigates its properties and connections to ring isoclinism, extending previous graph-theoretic approaches.

Findings

Determined the diameter and girth of the graph.

Established connections between the graph and the probability of non-commuting elements.

Proved isomorphism of graphs under Z-isoclinism of rings.

Abstract

Let $S, K$ be two subrings of a finite ring $R$. Then the generalized non-commuting graph of subrings $S, K$ of $R$, denoted by $\Gamma_{S, K}$, is a simple graph whose vertex set is $(S \cup K) \setminus (C_K(S) \cup C_S(K))$ and two distinct vertices $a, b$ are adjacent if and only if $a \in S$ or $b \in S$ and $ab \neq ba$. We determine the diameter, girth and some dominating sets for $\Gamma_{S, K}$. Some connections between the $\Gamma_{S, K}$ and $\Pr(S, K)$ are also obtained. Further, $\Z$-isoclinism between two pairs of finite rings is defined and showed that the generalized non-commuting graphs of two $\Z$-isoclinic pairs are isomorphic under some condition.