Squared distance matrix of a weighted tree

Abstract

Let T be a tree with vertex set {1, …, n} such that each edge is assigned a nonzero weight. The squared distance matrix of T, denoted by Δ, is the n × 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 Δ. A formula for Δ − 1 is also obtained, under certain conditions. The results generalize known formulas for the unweighted case.

Keywords

• tree, distance matrix, squared distance matrix, determinant, inverse