Squared distance matrix of a weighted tree

TL;DR

This paper derives formulas for the determinant and inverse of the squared distance matrix of a weighted tree, generalizing known results from the unweighted case.

Contribution

It provides explicit formulas for the determinant and inverse of the squared distance matrix of weighted trees, extending previous unweighted results.

Findings

Derived a formula for the determinant of the squared distance matrix.

Obtained an explicit formula for the inverse of the squared distance matrix under certain conditions.

Generalized known formulas from unweighted to weighted trees.

Abstract

Let $T$ be a tree with vertex set $\{1, \ldots, n\}$ such that each edge is assigned a nonzero weight. The squared distance matrix of $T,$ denoted by $\Delta,$ is the $n \times n$ matrix with $(i,j)$-element $d(i,j)^2,$ where $d(i,j)$ is the sum of the weights of the edges on the $(ij)$-path. We obtain a formula for the determinant of $\Delta.$ A formula for $\Delta^{-1}$ is also obtained, under certain conditions. The results generalize known formulas for the unweighted case.