On non-commuting graph of a finite ring

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 $ab e ba$. 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