Subspace Inclusion Graph of a Vector Space

Abstract

In this paper, the authors introduce a graph structure, called subspace inclusion graph ℐn(𝕍) on a finite dimensional vector space 𝕍 where the vertex set is the collection of nontrivial proper subspaces of a vector space and two vertices are adjacent if one is contained in other. The diameter, girth, clique number, and chromatic number of ℐn(𝕍) are studied. It is shown that two subspace inclusion graphs are isomorphic if and only if the base vector spaces are isomorphic. Finally, some properties of subspace inclusion graph are studied when the base field is finite.