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.
On distance matrices of wheel graphs with an odd number of vertices
Let (Formula presented.) denote the wheel graph having n-vertices. If i and j are any two vertices of (Formula presented.), define (Formula presented.) Let D be the (Formula presented.) matrix with (Formula presented.) entry equal to (Formula presented.). The matrix D is called the distance matrix of (Formula presented.). Suppose (Formula presented.) is an odd integer. In this paper, we deduce a formula to compute the Moore-Penrose inverse of D. More precisely, we obtain an (Formula presented.) matrix (Formula presented.) and a rank one matrix (Formula presented.) such that (Formula presented.) Here, (Formula presented.) is positive semidefinite, (Formula presented.) and all row sums are equal to zero.