dc.contributor.author Badawi, Ayman dc.contributor.author El-Ashi, Yasmine Ahmed dc.date.accessioned 2022-12-07T07:48:45Z dc.date.available 2022-12-07T07:48:45Z dc.date.issued 2022-12-01 dc.identifier.citation 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 en_US dc.identifier.isbn 9789811938986 dc.identifier.uri http://hdl.handle.net/11073/25087 dc.description.abstract 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. en_US dc.language.iso en_US en_US dc.publisher Springer en_US dc.relation.uri https://doi.org/10.1007/978-981-19-3898-6_31 en_US dc.subject Zero-divisor graph en_US dc.subject Total graph en_US dc.subject Unitary graph en_US dc.subject Dot product graph en_US dc.subject Annihilator graph en_US dc.subject Linear transformations graph en_US dc.title Graph of Linear Transformations Over R en_US dc.type Book chapter en_US dc.type Postprint en_US dc.identifier.doi 10.1007/978-981-19-3898-6_31
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