Distance Matrices Perturbed by Laplacians

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 D_ij denote the sum of all the weights lying in the path connecting the vertices i and j of T. We now say that D_ij is the distance between i and j. Define D ≔ [D_ij], where D_ii is the s × s null matrix and for i ≠ j, D_ij 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 L_ij ≔ − W⁻¹_ij , where W_ij is the weight of the edge (i, j). Define ${L_{ii}}: = \sum\limits_{i \ne j,j = 1}^n {W_{ij}^{ - 1}}.$ The Laplacian of G is now the ns × ns block matrix L ≔ [L_ij]. 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.