On resistance matrices of weighted balanced digraphs (2111.02051v1)

Abstract: Let $G$ be a connected graph with $V(G)={1,\dotsc,n}$. Then the resistance distance between any two vertices $i$ and $j$ is given by $r_{ij}:=l_{ii}^{\dag} + l_{jj}^{\dag}-2 l_{ij}^{\dag}$, where $l_{ij}^\dag$ is the $(i,j)^{\rm th}$ entry of the Moore-Penrose inverse of the Laplacian matrix of $G$. For the resistance matrix $R:=[r_{ij}]$, there is an elegant formula to compute the inverse of $R$. This says that [R^{-1}=-\frac{1}{2}L + \frac{1}{\tau' R \tau} \tau \tau', ] where [\tau:=(\tau_1,\dotsc,\tau_n)'\mbox{and} \tau_{i}:=2- \sum_{{j \in V(G):(i,j) \in E(G)}} r_{ij}~~~i=1,\dotsc,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.