On the trace graph of matrices

Abstract

Let R be a commutative ring with identity, ${M_n(R)}$ be the set of all ${n \times n}$ matrices over R and ${M_n(R) ^{*} }$ be the set of all non-zero matrices of ${M_n(R)}$ where ${n \geq 2}$. For a matrix ${A \in M_n(R)}$, ${{\rm Tr} (A)}$ is the trace of A. The trace graph of the matrix ring ${M_n(R)}$, denoted by ${\Gamma_t(M_n(R))}$, is the simple undirected graph denoted by ${\Gamma_t(M_n(R))}$ with vertex set $\{{A \in M_n(R) ^{*} : }$ there exists ${B \in M_n(R) ^{*} }$ such that ${{\rm Tr}(AB)=0}\}$ and two distinct vertices A and B are adjacent if and only if ${{\rm Tr} (AB) = 0}$. First, we prove that ${\Gamma_t(M_n(R))}$ is 2-connected and hence obtain Eulerian properties of ${\Gamma_t(M_n(R))}$. Also we obtain the domination number of ${\Gamma_t(M_n(R))}$ of a commutative semisimple ring R and obtain the domination number for ${\Gamma_t(M_n(\mathbb Z_2^m))}$. Finally, it is proved that for a commutative ring R with identity, ${\Gamma_t(M_n(R))}$ is non-planar and classified all finite commutative rings R with identity for which the trace graph has thickness 2.