On r-noncommuting graph of finite rings

Let $R$ be a finite ring and $r\in R$. The $r$-noncommuting graph of $R$, denoted by ${\mathrm{\Gamma }}_R^r$, is a simple undirected graph whose vertex set is $R$ and two vertices $x$ and $y$ are adjacent if and only if $[x,y]\mathrm{\ne }r$ and $-r$. In this paper, we study several properties of ${\mathrm{\Gamma }}_R^r$. We show that ${\mathrm{\Gamma }}_R^r$ is not a regular graph, a lollipop graph and complete bipartite graph. Further, we consider an induced subgraph of ${\mathrm{\Gamma }}_R^r$ (induced by the non-central elements of $R$) and obtained some characterizations of $R$.