On non-commuting graph of a finite ring

TL;DR

This paper investigates the structure of non-commuting graphs of finite rings, establishing conditions for their isomorphism and exploring connections with commuting probabilities.

Contribution

It demonstrates that non-commuting graphs of non-commutative rings are not isomorphic to certain graphs, and shows isomorphism conditions for rings that are $ ext{Z}$-isoclinic.

Findings

Non-commuting graphs are not isomorphic to certain graphs of finite rings.

Isomorphism of non-commuting graphs depends on the order of the centers of $ ext{Z}$-isoclinic rings.

Connections between non-commuting graphs and commuting probabilities are established.

Abstract

The non-commuting graph $\Gamma_R$ of a finite ring $R$ with center $Z(R)$ is a simple undirected graph whose vertex set is $R \setminus Z(R)$ and two distinct vertices $a$ and $b$ are adjacent if and only if $ab \ne ba$. In this paper, we show that $\Gamma_R$ is not isomorphic to certain graphs of any finite non-commutative ring $R$. Some connections between $\Gamma_R$ and commuting probability of $R$ are also obtained. Further, it is shown that the non-commuting graphs of two $\mathbb{Z}$-isoclinic rings are isomorphic if the centers of the rings have same order