Graph of Linear Transformations over a Field

Abstract

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 ϵ 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] ϵ G([t] : Rᵐ → Rⁿ) are adjacent if and only if ker (x) ∩ ker (y) ≠ 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.