Direct sum graph of the subspaces of a finite dimensional vector space over finite fields

TL;DR

This paper introduces a new graph called the direct sum graph on finite-dimensional vector spaces over finite fields, analyzing its structural properties and parameters such as connectivity, diameter, and chromatic number.

Contribution

It defines the direct sum graph for subspaces, investigates its properties, and computes various graph invariants, providing new insights into the interplay between vector space substructures and graph theory.

Findings

The graph's connectivity, diameter, and completeness are characterized.

The degree of vertices is determined for finite fields.

The graph is shown to be non-Eulerian and conditions for triangulation are provided.

Abstract

In this paper, we introduce a new graph structure, called the $direct~ sum ~graph$ on a finite dimensional vector space. We investigate the connectivity, diameter and the completeness of $\Gamma_{U\oplus W}(\mathbb{V})$. Further, we find its domination number and independence number. We also determine the degree of each vertex in case the base field is finite and show that the graph $\Gamma_{U\oplus W}(\mathbb{V})$ is not Eulerian. We also show that under some mild conditions the graph $\Gamma_{U\oplus W}(\mathbb{V})$ is triangulated. We determine the clique number of $\Gamma_{U\oplus W}(\mathbb{V})$ for some particular cases. Finally, we find the size, girth, edge-connectivity and the chromatic number of $\Gamma_{U\oplus W}(\mathbb{V})$.