Squared distance matrix of a weighted tree

Abstract

Let $T$ be a tree with vertex set $\{1,\dots ,n\}$ 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.