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