Resistance Distance in Directed Graphs

Let G = (V,E) be a strongly connected and balanced digraph with vertex set V = {1,...,n}. The Laplacian matrix of G is then the matrix (not necessarily symmetric) L:= D − A, where A is the adjacency matrix of G and D is the diagonal matrix such that the row sums and the column sums of L are equal to zero. Let L† = [lij† ] be the Moore-Penrose inverse of L.

Some interesting properties of the resistance and the corresponding resistance matrix [rij] will be discussed in the talk.

We define the resistance between any two vertices i and

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. 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 of this inequality is known. In the talk, we will show that this inequality holds for all directed cactus graphs. This is a joint work with Dr R. Balaji and Professor R. B. Bapat.