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 |
| _version_ | 1797229415371898880 |
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| author | T. Tamizh Chelvam |
| author_facet | T. Tamizh Chelvam |
| author_sort | T. Tamizh Chelvam |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-24T15:12:14Z |
| format | Article |
| id | doaj.art-e76fc54f3cfa45cf921452be1f222569 |
| institution | Directory Open Access Journal |
| issn | 0972-8600 2543-3474 |
| language | English |
| last_indexed | 2024-04-24T15:12:14Z |
| publishDate | 2024-03-01 |
| publisher | Taylor & Francis Group |
| record_format | Article |
| series | AKCE International Journal of Graphs and Combinatorics |
| spelling | doaj.art-e76fc54f3cfa45cf921452be1f2225692024-04-02T10:40:56ZengTaylor & Francis GroupAKCE International Journal of Graphs and Combinatorics0972-86002543-34742024-03-0112510.1080/09728600.2024.2332780Graphs from matrices - a surveyT. Tamizh Chelvam0Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli, Tamil Nadu, IndiaAbstractLet 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.https://www.tandfonline.com/doi/10.1080/09728600.2024.2332780Commutative ringsgraphs from matricesmatrix ringszero-divisor graphtrace graph16S50 |
| spellingShingle | T. Tamizh Chelvam Graphs from matrices - a survey AKCE International Journal of Graphs and Combinatorics Commutative rings graphs from matrices matrix rings zero-divisor graph trace graph 16S50 |
| title | Graphs from matrices - a survey |
| title_full | Graphs from matrices - a survey |
| title_fullStr | Graphs from matrices - a survey |
| title_full_unstemmed | Graphs from matrices - a survey |
| title_short | Graphs from matrices - a survey |
| title_sort | graphs from matrices a survey |
| topic | Commutative rings graphs from matrices matrix rings zero-divisor graph trace graph 16S50 |
| url | https://www.tandfonline.com/doi/10.1080/09728600.2024.2332780 |
| work_keys_str_mv | AT ttamizhchelvam graphsfrommatricesasurvey |