On Commuting graphs of triangular rings

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