Squared distance matrix of a tree: Inverse and inertia

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...

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Bibliographic Details
Published in:Linear algebra and its applications 2016-02, Vol.491, p.328-342
Main Authors: Bapat, R.B., Sivasubramanian, Sivaramakrishnan
Format: Article
Language:English
Subjects:
Distance matrix
Inertia
Squared distance
Tree
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Summary: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.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2015.09.008