Distance Matrices Perturbed by Laplacians

Article Type

Research Article

Publication Title

Applications of Mathematics

Abstract

Let T be a tree with n vertices. To each edge of T we assign a weight which is a positive definite matrix of some fixed order, say, s. Let Dij denote the sum of all the weights lying in the path connecting the vertices i and j of T. We now say that Dij is the distance between i and j. Define D ≔ [Dij], where Dii is the s × s null matrix and for i ≠ j, Dij is the distance between i and j. Let G be an arbitrary connected weighted graph with n vertices, where each weight is a positive definite matrix of order s. If i and j are adjacent, then define Lij ≔ − Wij−1, where Wij is the weight of the edge (i, j). Define Lii=∑i≠jj=1nWij−1. The Laplacian of G is now the ns × ns block matrix L ≔ [Lij]. In this paper, we first note that D−1 − L is always nonsingular and then we prove that D and its perturbation (D−1 − L)−1 have many interesting properties in common.

First Page

599

Last Page

607

DOI

10.21136/AM.2020.0362-19

Publication Date

10-1-2020

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