On distance and Laplacian matrices of trees with matrix weights

TL;DR

This paper characterizes matrix-weighted trees using Laplacian matrix rank, provides conditions for the invertibility of their distance matrices, and explores properties related to inverses and eigenvalues.

Contribution

It introduces a new characterization of matrix-weighted trees via Laplacian rank and derives conditions for the invertibility of their distance matrices.

Findings

Characterization of trees through Laplacian matrix rank

Necessary and sufficient conditions for distance matrix invertibility

Formulas for the inverse of the distance matrix

Abstract

The \emph{distance matrix} of a simple connected graph $G$ is $D(G)=(d_{ij})$, where $d_{ij}$ is the distance between the vertices $i$ and $j$ in $G$. We consider a weighted tree $T$ on $n$ vertices with edge weights are square matrix of same size. The distance $d_{ij}$ between the vertices $i$ and $j$ is the sum of the weight matrices of the edges in the unique path from $i$ to $j$. In this article we establish a characterization for the trees in terms of rank of (matrix) weighted Laplacian matrix associated with it. Then we establish a necessary and sufficient condition for the distance matrix $D$, with matrix weights, to be invertible and the formula for the inverse of $D$, if it exists. Also we study some of the properties of the distance matrices of matrix weighted trees in connection with the Laplacian matrices, g-inverses and eigenvalues.