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

Abstract

In this paper, we introduce a new graph structure, called the $directsumgraph$ 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 ${\Gamma }_{U\oplus W}(\mathbb{V})$.

19 pages