### 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 (*i**j*)-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.

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PDFDOI: http://dx.doi.org/10.5614/ejgta.2019.7.2.8

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