Publication1

Let G be a connected graph with (Formula presented.). Then the resistance distance between any two vertices i and j is given by (Formula presented.), where (Formula presented.) is the (Formula presented.) th entry of the Moore-Penrose inverse of the Laplacian matrix of G. For the resistance matrix (Formula presented.), there is an elegant formula to compute the inverse of R. This says that (Formula presented.) where (Formula presented.) A far reaching generalization of this result that gives an inverse formula for a generalized resistance matrix of a strongly connected and matrix weighted balanced directed graph is obtained in this paper. When the weights are scalars, it is shown that the generalized resistance is a non-negative real number. We also obtain a perturbation result involving resistance matrices of connected graphs and Laplacians of digraphs.