On Commuting graphs of triangular rings

Let $R$ be a noncommutative ring with identity. The commuting graph of $R$, denoted by $\Gamma(R)$, is a graph with vertex set $R \setminus Z(R)$, and two vertices $a$, $b$ are adjacent if $a\neq b$ and $ab=ba$. Let $T=Tr(R)$ be the ring of all $2\times 2$ upper triangular matrices over $R$ and $\Gamma(T)$ be the commuting graph of $T$. In this article, we find the number of edges, cliques, clique number, and independence number of $\Gamma (T)$ when $R$ is a finite field. Moreover, we show that for the case when $R= \mathbb{Z}_{n}$ is not a field, $\Gamma (T)$ is connected with diameter 3. Some useful related results are also obtained, some examples are presented and a question is posed.