On resistance matrices of weighted balanced digraphs.

Let G be a connected graph with V (G) = { 1 , ... , n }. Then the resistance distance between any two vertices i and j is given by r i j := l i i † + l j j † − 2 l i j † , where l i j † is the (i , j) th entry of the Moore-Penrose inverse of the Laplacian matrix of G. For the resistance matrix R := [ r i j ] , there is an elegant formula to compute the inverse of R. This says that R − 1 = − 1 2 L + 1 τ ′ R τ τ τ ′ , where τ := (τ 1 , ... , τ n) ′ and τ i := 2 − ∑ j ∼ i r i j i = 1 , ... , n. 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. [ABSTRACT FROM AUTHOR]