Nonzero Component Graph of a Finite Dimensional Vector Space

Abstract

In this paper, we introduce a graph structure, called non-zero component graph on finite dimensional vector spaces. We show that the graph is connected and find its domination number and independence number. We also study the inter-relationship between vector space isomorphisms and graph isomorphisms and it is shown that two graphs are isomorphic if and only if the corresponding vector spaces are so. Finally, we determine the degree of each vertex in case the base field is finite.

Automorphism Group and Other Properties of Zero Component Graph over a Vector Space

“…A binary relation on a vector space, group, or ring can be studied with the help of considering the associating graph defined by this relation. Symplectic graphs, orthogonal graphs, subspace inclusion graphs, and nonzero component graphs are such examples which have been considered recently (see [1][2][3][4]).…”

“…All graphs considered in this paper are undirected simple graphs (without loops and multiedges). e nonzero component graph Γ c (V) of V, introduced by Das [1], is a graph with vertex set V * � V\ 0 { }, in which for distinct α, β ∈ V * , there exists an edge joining α and β if and only if S A (α) ∩ S A (β) ≠ ∅. In [1], the author studied the diameter, dominating number, and dependent number of Γ c (V).…”

“…e nonzero component graph Γ c (V) of V, introduced by Das [1], is a graph with vertex set V * � V\ 0 { }, in which for distinct α, β ∈ V * , there exists an edge joining α and β if and only if S A (α) ∩ S A (β) ≠ ∅. In [1], the author studied the diameter, dominating number, and dependent number of Γ c (V). When the base field F is finite, Das [5] considered the clique number, edge connectivity, and chromatic number of Γ c (V) and showed that Γ c (V) is Hamiltonian, but not Eulerian.…”

“…Generally, it is an important but difficult work to describe the full automorphism group both in graph theory and in algebra. In [1], the author characterizes the automorphisms of Γ c (V). For the automorphisms of symplectic graphs, orthogonal graphs, and subspace inclusion graphs, the reader is referred to [3,4,8].…”

Detecting Similarities in Posts Using Vector Space and Matrix

“…A vector is a geometric object that has both a magnitude and a direction (Sentosa, 2016). Vector space are finite dimensional over a field and dim ( ) nv  (Das, 2016a). In addition to vector space, this study uses cosine similarity.…”