Squared distance matrices of trees with matrix weights

Abstract

Let $T$ be a tree on $n$ vertices whose edge weights are positive definite matrices of order $s$. The squared distance matrix of $T$, denoted by $\Delta$, is the $ns\times ns$ block matrix with ${\Delta }_{ij}=d(i,j{)}^2$, where $d(i,j)$ is the sum of the weights of the edges in the unique $(i,j)$-path. In this article, we obtain a formula for the determinant of $\Delta$ and find ${\Delta }^{-1}$ under some conditions.

Preliminary version. Comments are welcome. 17 pages