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 Δ, is the ns × 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 Δ and find ${\Delta }^{-1}$ under some conditions.

KEYWORDS:

• Tree
• distance matrix
• squared distance matrix
• matrix weight
• determinant
• inverse

AMS CLASSIFICATION:

• 05C22
• 05C50

Acknowledgments

We are indebted to the referee for their valuable comments and suggestions.