The trace graph of the matrix ring over a finite commutative ring

Let R be a commutative ring and let ({n >1}) be an integer. We introduce a simple graph, denoted by ({Gamma_t(M_n(R))}), which we call the trace graph of the matrix ring ({M_n(R)}), such that its vertex set is ({M_n(R)^{ast}}) and such that two distinct vertices A and B are joined by an edge if and only if ({{rm Tr} (AB)=0}) where ({ {rm Tr} (AB)}) denotes the trace of the matrix AB. We prove that ({Gamma_t(M_n(R))}) is connected with ({{rm diam}(Gamma_{t}(M_{n}(R)))=2}) and ({{rm gr} (Gamma_t(M_n(R)))=3}). We investigate also the interplay between the ring-theoretic properties of R and the graph-theoretic properties of ({Gamma_t(M_n(R))}). Hence, we use the notion of the irregularity index of a graph to characterize rings with exactly one nontrivial ideal.