On Vector Spaces Associated with a Graph

For a given graph G, let V( Q), V( B), and $V(G)$ be the vector spaces associated with the sets of cutsets, circuits, and subgraphs of G, respectively. It is shown that V( Q) and V( B) are orthogonal complements of $V(G)$ if and only if G has an even number of forests. Also, it is shown that the intersection of V( Q) and V( B) is the null space of the matrix whose rows correspond to the bases of V( Q) and V( B).