On some properties of vector space based graphs.

In this paper, we study some problems related to subspace inclusion graph $ \mathop {\mathcal {I}n}(\mathbb {V}) $ I n ⁡ (V) and subspace sum graph $ \mathcal {G}(\mathbb {V}) $ G (V) of a finite-dimensional vector space $ \mathbb {V} $ V . Namely, we prove that $ \mathop {\mathcal {I}n}(\mathbb {V}) $ I n ⁡ (V) is a Cayley graph as well as Hamiltonian when the dimension of $ \mathbb {V} $ V is 3. We also find the exact value of independence number of $ \mathcal {G}(\mathbb {V}) $ G (V) when the dimension of $ \mathbb {V} $ V is odd. The above two problems were left open in previous works in the literature. Moreover, we prove that the determining numbers of $ \mathop {\mathcal {I}n}(\mathbb {V}) $ I n ⁡ (V) and $ \mathcal {G}(\mathbb {V}) $ G (V) are bounded above by 6. Finally, we study some forbidden subgraphs of these two graphs. [ABSTRACT FROM AUTHOR]