On two conjectures on the subspace inclusion 6 graph of a vector space

Abstract

The subspace inclusion graph on a vector space [Formula: see text], denoted by [Formula: see text], is a graph whose vertex set consists of nontrivial proper subspaces of [Formula: see text] and two vertices are adjacent if one is properly contained in another. In a recent paper, Das posed the following four conjectures on the subspace inclusion graph [Formula: see text]: If [Formula: see text] is a [Formula: see text]-dimensional vector space over a finite field [Formula: see text] with [Formula: see text] elements, then: (1) The domination number of [Formula: see text] is [Formula: see text]. (2) [Formula: see text] is distance regular. (3) [Formula: see text] is Hamiltonian. (4) [Formula: see text] is a Cayley graph. In the present paper, we prove the first two conjectures: If [Formula: see text] is a [Formula: see text]-dimensional vector space over a finite field [Formula: see text] with [Formula: see text] elements, then: (1) The domination number of [Formula: see text] is [Formula: see text]. (2) [Formula: see text] is distance regular.