On generalized non-commuting graph of a finite ring

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.