The genus of graphs associated with vector spaces

Let V be a k-dimensional vector space over a finite field F with a basis {alpha(1), ...,alpha(k)}. The nonzero component graph of V, denoted by Gamma(V), is a simple undirected graph with vertex set as nonzero vectors of V such that there is an edge between two distinct vertices x, y if and only if there exists at least one alpha(i) along which both x and y have nonzero scalars. In this paper, we find the vertex connectivity and girth of Gamma(V). We also characterize all vector spaces V for which Gamma(V) has genus either 0 or 1 or 2.