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Graph of Linear Transformations Over a Field

Yasmine Ahmed El-Ashi

出版	American University of Sharjah, 2019
URL	http://books.google.com.hk/books?id=zyoj0QEACAAJ&hl=&source=gbs_api

註釋

"This research is an attempt to introduce a connection between graph theory and linear transformations of finite dimensional vector spaces over a field F (in our case we will be considering R). Let Rm, Rn be finite vector spaces over R, and let L be the set of all non-trivial linear transformations from Rm to 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 V be the set of all equivalence classes of∼. We define a new graph, G([t] :Rm→Rn), to be the undirected graph with vertex set equal to V, such that two vertices,[x],[y]∈G([t] :Rm→Rn)are adjacent if and only if ker (x) ∩ ker (y)6=0. The relationship between the connectivity of the graph G ([t]: Rm→Rn) and the values of m and n has been investigated. In addition, we determine the values of m and n for a complete and totally disconnected graph, as well as the diameter and girth of the graph if connected."--Abstract.