Subspace Sum Graph of a Vector Space

Abstract

In this paper we introduce a graph structure, called subspace sum graph $\mathcal{G}(\mathbb{V})$ on a finite dimensional vector space $\mathbb{V}$ where the vertex set is the collection of non-trivial proper subspaces of a vector space and two vertices $W_1,W_2$ are adjacent if $W_1 + W_2=\mathbb{V}$. The diameter, girth, connectivity, maximal independent sets, different variants of domination number, clique number and chromatic number of $\mathcal{G}(\mathbb{V})$ are studied. It is shown that two subspace sum graphs are isomorphic if and only if the base vector spaces are isomorphic. Finally some properties of subspace sum graph are studied when the base field is finite.