Relative non-commuting graph of a finite ring

Let $S$ be a subring of a finite ring $R$ and $C_R(S) = \{r \in R : rs = sr \;\forall\; s \in S\}$. The relative non-commuting graph of the subring $S$ in $R$, denoted by $\Gamma_{S, R}$, 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 $ab \neq ba$. In this paper, we discuss some properties of $\Gamma_{S, R}$, determine diameter, girth, some dominating sets and chromatic index for $\Gamma_{S, R}$. Also, we derive some connections between $\Gamma_{S, R}$ and the relative commuting probability of $S$ in $R$. Finally, we show that the relative non-commuting graphs of two relative $\Z$-isoclinic pairs of rings are isomorphic under some conditions.