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 , 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 under some conditions.