Subspace-based nonzero component graph of a finite dimensional vector space

Abstract

Let [Formula: see text] be a vector space over a field [Formula: see text] with basis [Formula: see text]. Let [Formula: see text] be an [Formula: see text]-dimensional subspace of [Formula: see text] with basis [Formula: see text], where [Formula: see text] is some [Formula: see text]. We introduce the subspace-based nonzero component graph, denoted by [Formula: see text], of the finite dimensional vector space [Formula: see text] with respect to [Formula: see text] and [Formula: see text] as follows: [Formula: see text] and for [Formula: see text], there is an edge between [Formula: see text] and [Formula: see text] if and only if either [Formula: see text] or [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text]. We investigate the connectivity, diameter and completeness of [Formula: see text]. Further, we find its domination number and independence number. If [Formula: see text] is a subspace of a vector space, we show that the graph is complete if and only if [Formula: see text] is a hyperspace. Finally, we determine the degree of each vertex in case the base field is finite.