Squared distance matrix of a tree: Inverse and inertia

Article Type

Research Article

Publication Title

Linear Algebra and Its Applications

Abstract

Let T be a tree with vertices V(T)={1,...,n}. The distance between vertices i,j∈V(T), denoted dij, is defined to be the length (the number of edges) of the path from i to j. We set dii=0,i=1,...,n. The squared distance matrix Δ of T is the n×n matrix with (i,j)-element equal to 0 if i=j, and dij2 if i≠j. It is known that Δ is nonsingular if and only if the tree has at most one vertex of degree 2. We obtain a formula for Δ-1, if it exists. When the tree has no vertex of degree 2, the formula is particularly simple and depends on a certain "two-step" Laplacian of the tree. We determine the inertia of Δ. The inverse and the inertia of the edge orientation matrix are also described.

First Page

328

Last Page

342

DOI

10.1016/j.laa.2015.09.008

Publication Date

2-15-2016

Comments

Open Access; Bronze Open Access

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