Direct sum graph of the subspaces of a finite-dimensional vector space over finite fields

In this paper, we study a new graph structure, called the [Formula: see text] on a finite-dimensional vector space. We investigate the connectivity, diameter and the completeness of [Formula: see text]. We also determine the degree of each vertex in case the base field is finite. We establish some results about Eulerian and triangulation of the graph [Formula: see text]. Finally, we find the domination number and independence number, size, girth, edge-connectivity, clique number and the chromatic number of [Formula: see text].