Distance Matrices Perturbed by Laplacians
Applications of Mathematics
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.
Ramamurthy, Balaji; Bapat, Ravindra Bhalchandra; and Goel, Shivani, "Distance Matrices Perturbed by Laplacians" (2020). Journal Articles. 118.