Graphs from matrices - a survey

Abstract

Let R be a commutative ring with identity. For a positive integer n ≥ 2 , let M n ( R ) be the set of all n × n matrices over R and M n ( R ) * be the set of all non-zero matrices of M n ( R ) . The zero-divisor graph of M n ( R ) is a simple directed graph with vertex set the non-zero zero-divisors in M n ( R ) and two distinct matrices A and B are adjacent if their product is zero. Given a matrix A ∈ M n ( R ) , Tr(A) is the trace of the matrix A. The trace graph of the matrix ring M n ( R ) , denoted by Γ t ( M n ( R ) ) , is the simple undirected graph with vertex set { A ∈ M n ( R ) * : there exists  B ∈ M n ( 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 Γ I t ( M n ( R ) ) , is a simple undirected graph with vertex set M n ( R ) ∖ M n ( 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.