On Non-Zero Component Graph of Vector Spaces over Finite Fields

TL;DR

This paper explores the properties of the non-zero component graph of finite-dimensional vector spaces over finite fields, revealing its Hamiltonian nature, clique structure, and graph invariants.

Contribution

It characterizes maximal cliques, determines the clique number in specific cases, and analyzes key graph properties like size, edge-connectivity, and chromatic number.

Findings

The graph is Hamiltonian but not Eulerian.

Two classes of maximal cliques are identified.

Exact clique number computed for certain cases.

Abstract

In this paper, we study non-zero component graph $\Gamma(\mathbb{V})$ on a finite dimensional vector space $\mathbb{V}$ over a finite field $\mathbb{F}$. We show that the graph is Hamiltonian and not Eulerian. We also characterize the maximal cliques in $\Gamma(\mathbb{V})$ and show that there exists two classes of maximal cliques in $\Gamma(\mathbb{V})$. We also find the exact clique number of $\Gamma(\mathbb{V})$ for some particular cases. Moreover, we provide some results on size, edge-connectivity and chromatic number of $\Gamma(\mathbb{V})$.