Squared distance matrices of trees with matrix weights

AbstractLet 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 [Formula: see text], 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 [Formula: see text] under some conditions.