Graph of Linear Transformations Over R
In this paper, we study a connection between graph theory and linear transformations of finite dimensional vector spaces over R (the set of all real numbers). Let Rm, Rn be finite vector spaces over R, and let L be the set of all non-trivial linear transformations from Rm into Rn. An equivalence rel...
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2022
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| Online Access: | http://hdl.handle.net/11073/25087 |
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| _version_ | 1864513433361186816 |
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| author | Badawi, Ayman |
| author2 | El-Ashi, Yasmine Ahmed |
| author2_role | author |
| author_facet | Badawi, Ayman El-Ashi, Yasmine Ahmed |
| author_role | author |
| dc.creator.none.fl_str_mv | Badawi, Ayman El-Ashi, Yasmine Ahmed |
| dc.date.none.fl_str_mv | 2022-12-07T07:48:45Z 2022-12-07T07:48:45Z 2022-12-01 |
| dc.format.none.fl_str_mv | application/pdf |
| dc.identifier.none.fl_str_mv | Badawi, A., El-Ashi, Y. (2022). Graph of Linear Transformations Over R. In: Ashraf, M., Ali, A., De Filippis, V. (eds) Algebra and Related Topics with Applications. ICARTA 2019. Springer Proceedings in Mathematics & Statistics, vol 392. Springer, Singapore. https://doi.org/10.1007/978-981-19-3898-6_31 9789811938986 http://hdl.handle.net/11073/25087 10.1007/978-981-19-3898-6_31 |
| dc.language.none.fl_str_mv | en_US |
| dc.publisher.none.fl_str_mv | Springer |
| dc.relation.none.fl_str_mv | https://doi.org/10.1007/978-981-19-3898-6_31 |
| dc.subject.none.fl_str_mv | Zero-divisor graph Total graph Unitary graph Dot product graph Annihilator graph Linear transformations graph |
| dc.title.none.fl_str_mv | Graph of Linear Transformations Over R |
| dc.type.none.fl_str_mv | Postprint info:eu-repo/semantics/publishedVersion info:eu-repo/semantics/bookPart |
| description | In this paper, we study a connection between graph theory and linear transformations of finite dimensional vector spaces over R (the set of all real numbers). Let Rm, Rn be finite vector spaces over R, and let L be the set of all non-trivial linear transformations from Rm into Rn. An equivalence relation ∼ is defined on L such that two elements f, k ∈ L are equivalent, f ∼ k, if and only if ker (f ) = ker (k). Let m, n ≥ 1 be positive integers and Vm,n be the set of all equivalence classes of ∼. We define a new graph, Gm,n, to be the undirected graph with vertex set equals to Vm,n, such that two vertices, [x] , [y] ∈ Vm,n are adjacent if and only if ker (x) ∩ ker (y) 6 = 0. The relationship between the connectivity of the graph Gm,n and the values of m and n has been investigated. We determine the values of m and n so that Gm,n is a complete graph. Also, we determine the diameter and the girth of Gm,n. |
| format | bookPart |
| id | aus_8e8876dcce0e41d5d3e1f8d5aa98e957 |
| identifier_str_mv | Badawi, A., El-Ashi, Y. (2022). Graph of Linear Transformations Over R. In: Ashraf, M., Ali, A., De Filippis, V. (eds) Algebra and Related Topics with Applications. ICARTA 2019. Springer Proceedings in Mathematics & Statistics, vol 392. Springer, Singapore. https://doi.org/10.1007/978-981-19-3898-6_31 9789811938986 10.1007/978-981-19-3898-6_31 |
| language_invalid_str_mv | en_US |
| network_acronym_str | aus |
| network_name_str | aus |
| oai_identifier_str | oai:repository.aus.edu:11073/25087 |
| publishDate | 2022 |
| publisher.none.fl_str_mv | Springer |
| repository.mail.fl_str_mv | |
| repository.name.fl_str_mv | |
| repository_id_str | |
| spelling | Graph of Linear Transformations Over RBadawi, AymanEl-Ashi, Yasmine AhmedZero-divisor graphTotal graphUnitary graphDot product graphAnnihilator graphLinear transformations graphIn this paper, we study a connection between graph theory and linear transformations of finite dimensional vector spaces over R (the set of all real numbers). Let Rm, Rn be finite vector spaces over R, and let L be the set of all non-trivial linear transformations from Rm into Rn. An equivalence relation ∼ is defined on L such that two elements f, k ∈ L are equivalent, f ∼ k, if and only if ker (f ) = ker (k). Let m, n ≥ 1 be positive integers and Vm,n be the set of all equivalence classes of ∼. We define a new graph, Gm,n, to be the undirected graph with vertex set equals to Vm,n, such that two vertices, [x] , [y] ∈ Vm,n are adjacent if and only if ker (x) ∩ ker (y) 6 = 0. The relationship between the connectivity of the graph Gm,n and the values of m and n has been investigated. We determine the values of m and n so that Gm,n is a complete graph. Also, we determine the diameter and the girth of Gm,n.Springer2022-12-07T07:48:45Z2022-12-07T07:48:45Z2022-12-01Postprintinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/bookPartapplication/pdfBadawi, A., El-Ashi, Y. (2022). Graph of Linear Transformations Over R. In: Ashraf, M., Ali, A., De Filippis, V. (eds) Algebra and Related Topics with Applications. ICARTA 2019. Springer Proceedings in Mathematics & Statistics, vol 392. Springer, Singapore. https://doi.org/10.1007/978-981-19-3898-6_319789811938986http://hdl.handle.net/11073/2508710.1007/978-981-19-3898-6_31en_UShttps://doi.org/10.1007/978-981-19-3898-6_31oai:repository.aus.edu:11073/250872024-08-22T12:01:50Z |
| spellingShingle | Graph of Linear Transformations Over R Badawi, Ayman Zero-divisor graph Total graph Unitary graph Dot product graph Annihilator graph Linear transformations graph |
| status_str | publishedVersion |
| title | Graph of Linear Transformations Over R |
| title_full | Graph of Linear Transformations Over R |
| title_fullStr | Graph of Linear Transformations Over R |
| title_full_unstemmed | Graph of Linear Transformations Over R |
| title_short | Graph of Linear Transformations Over R |
| title_sort | Graph of Linear Transformations Over R |
| topic | Zero-divisor graph Total graph Unitary graph Dot product graph Annihilator graph Linear transformations graph |
| url | http://hdl.handle.net/11073/25087 |