Graphs from matrices - a survey

Abstract

Let R be a commutative ring with identity. For a positive integer [Formula: see text] let [Formula: see text] be the set of all n × n matrices over R and [Formula: see text] be the set of all non-zero matrices of [Formula: see text] The zero-divisor graph of [Formula: see text] is a simple directed graph with vertex set the non-zero zero-divisors in [Formula: see text] and two distinct matrices A and B are adjacent if their product is zero. Given a matrix [Formula: see text] Tr(A) is the trace of the matrix A. The trace graph of the matrix ring [Formula: see text] denoted by [Formula: see text] is the simple undirected graph with vertex set [Formula: see text] [Formula: see text] and two distinct vertices A and B are adjacent if and only if Tr [Formula: see text] For an ideal I of R, the notion of the ideal based trace graph, denoted by [Formula: see text] is a simple undirected graph with vertex set [Formula: see text] and two distinct vertices A and B are adjacent if and only if Tr [Formula: see text] In this survey, we present several results concerning the zero-divisor graph, trace graph and the ideal based trace graph of matrices over R.

Keywords

• Commutative rings
• graphs from matrices
• matrix rings
• zero-divisor graph
• trace graph
• 16S50