Graphs from matrices - a survey
AbstractLet 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 d...
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| Format: | Article |
| Language: | English |
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Taylor & Francis Group
2024-03-01
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| Series: | AKCE International Journal of Graphs and Combinatorics |
| Subjects: | |
| Online Access: | https://www.tandfonline.com/doi/10.1080/09728600.2024.2332780 |
| Summary: | AbstractLet 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. |
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| ISSN: | 0972-8600 2543-3474 |