Resistance matrices of graphs with matrix weights

TL;DR

This paper extends the concept of resistance matrices to weighted graphs with matrix weights, deriving formulas for their determinants and inverses, and analyzing their eigenvalue properties.

Contribution

It introduces formulas for resistance matrix determinants and inverses for matrix-weighted graphs, and studies their eigenvalue interlacing and inertia.

Findings

Derived the determinant formula for resistance matrices with matrix weights.

Established an interlacing inequality for eigenvalues of resistance and Laplacian matrices.

Determined the inertia of the resistance matrix for weighted graphs.

Abstract

The \emph{resistance matrix} of a simple connected graph $G$ is denoted by $R$, and is defined by $R =(r_{ij})$, where $r_{ij}$ is the resistance distance between the vertices $i$ and $j$ of $G$. In this paper, we consider the resistance matrix of weighted graph with edge weights being positive definite matrices of same size. We derive a formula for the determinant and the inverse of the resistance matrix. Then, we establish an interlacing inequality for the eigenvalues of resistance and Laplacian matrices. Using this interlacing inequality, we obtain the inertia of the resistance matrix.