On resistance matrices of weighted balanced digraphs

TL;DR

This paper generalizes the inverse formula for resistance matrices from undirected graphs to weighted balanced directed graphs, providing new theoretical insights and a perturbation result.

Contribution

It extends the inverse resistance matrix formula to matrix-weighted balanced digraphs and explores properties of generalized resistance in this context.

Findings

Derived an inverse formula for generalized resistance matrices of weighted balanced digraphs.

Showed that scalar weights lead to non-negative resistance values.

Established a perturbation result relating resistance matrices and Laplacians.

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.