Graphs from matrices - a survey

Let R be a commutative ring with identity. For a positive integer n≥2, let Mn(R) be the set of all n × n matrices over R and Mn(R)* be the set of all non-zero matrices of Mn(R). The zero-divisor graph of Mn(R) is a simple directed graph with vertex set the non-zero zero-divisors in Mn(R) and two distinct matrices A and B are adjacent if their product is zero. Given a matrix A∈Mn(R), Tr(A) is the trace of the matrix A. The trace graph of the matrix ring Mn(R), denoted by Γt(Mn(R)), is the simple undirected graph with vertex set {A∈Mn(R)*:there exists B∈Mn(R)* such that Tr(AB)=0} and two distinct vertices A and B are adjacent if and only if Tr (AB)=0. For an ideal I of R, the notion of the ideal based trace graph, denoted by ΓIt(Mn(R)), is a simple undirected graph with vertex set Mn(R)∖Mn(I) and two distinct vertices A and B are adjacent if and only if Tr (AB)∈I. In this survey, we present several results concerning the zero-divisor graph, trace graph and the ideal based trace graph of matrices over R.