On the trace graph of matrices

摘要

Let R be a commutative ring with identity, Mn(R) be the set of all nxn matrices over R and Mn(R) be the set of all non-zero matrices of Mn(R) where n2. For a matrix AMn(R), Tr(A) is the trace of A. The trace graph of the matrix ring Mn(R), denoted by t(Mn(R)), is the simple undirected graph denoted by t(Mn(R)) with vertex set {AMn(R): there exists BMn(R) such that Tr(AB)=0} and two distinct vertices A and B are adjacent if and only if Tr(AB)=0. First, we prove that t(Mn(R)) is 2-connected and hence obtain Eulerian properties of t(Mn(R)). Also we obtain the domination number of t(Mn(R)) of a commutative semisimple ring R and obtain the domination number for t(Mn(Z2m))Finally, it is proved that for a commutative ring R with identity, t(Mn(R)) is non-planar and classified all finite commutative rings R with identity for which the trace graph has thickness 2.