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 $ab\ne ba$. We determine the diameter, girth and some dominating sets for ${\Gamma }_{S,K}$. Some connections between the ${\Gamma }_{S,K}$ and $\mathrm{Pr}(S,K)$ are also obtained. Further, $\mathbb{Z}$-isoclinism between two pairs of finite rings is defined and showed that the generalized non-commuting graphs of two $\mathbb{Z}$-isoclinic pairs are isomorphic under some condition.

13 pages