Relative non-commuting graph of a finite ring

Let $S$ be a subring of a finite ring $R$ and $C_R(S) = {r \in R : rs = sr ;\forall; s \in S}$\mathrm{.}Therelativenon-commutinggraphofthesubring$S$in$R$,denotedby$\Gamma_{S, R}$, is a simple undirected graph whose vertex set is $R\setminus C_R(S)$ and two distinct vertices $a,b$ are adjacent if and only if $a$ or $b\in S$ and $ab\mathrm{\ne }ba$. In this paper, we discuss some properties of ${\mathrm{\Gamma }}_{S,R}$, determine diameter, girth, some dominating sets and chromatic index for ${\mathrm{\Gamma }}_{S,R}$. Also, we derive some connections between ${\mathrm{\Gamma }}_{S,R}$ and the relative commuting probability of $S$ in $R$. Finally, we show that the relative non-commuting graphs of two relative $\mathbb{Z}$-isoclinic pairs of rings are isomorphic under some conditions.