A subspace based subspace inclusion graph on vector space

Let $\mathscr{W}$ be a fixed $k$-dimensional subspace of an $n$-dimensi\-onal vector space $\mathscr{V}$ such that $n-k\geq1.$ In this paper, we introduce a graph structure, called the subspace based subspace inclusion graph $\mathscr{I}_{n}^{\mathscr{W}}(\mathscr{V}),$ where the vertex set $\mathscr{V}(\mathscr{I}_{n}^{\mathscr{W}}(\mathscr{V}))$ is the collection of all subspaces $\mathscr{U}$ of $\mathscr{V}$ such that $\mathscr{U}+\mathscr{W}\neq\mathscr{V}$ and $\mathscr{U}\nsubseteq\mathscr{W},$ i.e., $\mathscr{V}(\mathscr{I}_{n}^{\mathscr{W}}(\mathscr{V}))= \{\mathscr{U}\subseteq\mathscr{V}~|~\mathscr{U}+\mathscr{W}\neq\mathscr{V}, \mathscr{U}\nsubseteq\mathscr{W}\}$ and any two distinct vertices $\mathscr{U}_{1}$ and $\mathscr{U}_{1}$ of $\mathscr{I}_{n}^{\mathscr{W}}(\mathscr{V})$ are adjacent if and only if either $\mathscr{U}_{1}+\mathscr{W}\subset\mathscr{U}_{2}+\mathscr{W}$ or $\mathscr{U}_{2}+\mathscr{W}\subset\mathscr{U}_{1}+\mathscr{W}.$ The diameter, girth, clique number, and chromatic number of $\mathscr{I}_{n}^{\mathscr{W}}(\mathscr{V})$ are studied. It is shown that two subspace based subspace inclusion graphs $\mathscr{I}_{n}^{\mathscr{W}_{1}}(\mathscr{V})$ and $\mathscr{I}_{n}^{\mathscr{W}_{2}}(\mathscr{V})$ are isomorphic if and only if $\mathscr{W}_{1}$ and $\mathscr{W}_{2}$ are isomorphic. Further, some properties of $\mathscr{I}_{n}^{\mathscr{W}}(\mathscr{V})$ are obtained when the base field is finite.