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