Resistance distance in directed cactus graphs

Let G = (V, E) be a strongly connected and balanced digraph with vertex set V = {1, …, n}. The classical distance dij between any two vertices i and j in G is the minimum length of all the directed paths joining i and j. The resistance distance (or, simply the resistance) between any two vertices i and j in V is defined by rij:= lii† + l†jj − 2l†ij, where l†pq is the (p, q)th entry of the Moore-Penrose inverse of L which is the Laplacian matrix of G. In practice, the resistance rij is more significant than the classical distance. One reason for this is, numerical examples show that the resistance distance between i and j is always less than or equal to the classical distance, i.e., rij ≤ dij . However, no proof for this inequality is known. In this paper, it is shown that this inequality holds for all directed cactus graphs.