On non-commuting graph of a finite ring

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