Non-Zero Component Graph of a Finite Dimensional Vector Space

TL;DR

This paper introduces the non-zero component graph of finite dimensional vector spaces, explores its properties, and establishes a connection between vector space isomorphisms and graph isomorphisms, including degree calculations for finite fields.

Contribution

It defines the non-zero component graph, analyzes its properties, and proves that graph isomorphism corresponds exactly to vector space isomorphism.

Findings

The graph is connected.

Domination and independence numbers are determined.

Graph isomorphism iff vector space isomorphism.

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.