Relative $r$-noncommuting graph of finite rings

Let $S$ be a subring of a finite ring $R$ and $r\in R$. The relative $r$-noncommuting graph of $R$ relative to $S$, denoted by $\Gamma_{S, R}^r$, is a simple undirected graph whose vertex set is $R$ and two vertices $x$ and $y$ are adjacent if and only if $x \in S$ or $y \in S$ and $[x,y] \neq r$, $[x,y]\neq -r$. In this paper, we determine degree of any vertex in $\Gamma_{S, R}^r$ and characterize all finite rings such that $\Gamma_{S, R}^r$ is a star, lollipop or a regular graph. We derive connections between relative $r$-noncommuting graphs of two isoclinic pairs of rings. We also derive certain relations between the number of edges in $\Gamma_{S, R}^r$ and various generalized commuting probabilities of $R$. Finally, we conclude the paper by studying an induced subgraph of $\Gamma_{S, R}^r$.