Distance Matrices Perturbed by Laplacians

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 ij −1 , where W ij is the weight of the edge ( i, j ). Define L i i = ∑ i ≠ j j = 1 n W i j − 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.