On distance matrices and Laplacians

We consider distance matrices of certain graphs and of points chosen in a rectangular grid. Formulae for the inverse and the determinant of the distance matrix of a weighted tree are obtained. Results concerning the inertia and the determinant of the distance matrix of an unweighted unicyclic graph are proved. If D is the distance matrix of a tree, then we obtain certain results for a perturbation of D−1. As an example, it is shown that if L̃ is the Laplacian matrix of an arbitrary connected graph, then ( D−1 − L̃ )−1 is an entrywise positive matrix. We consider the distance matrix of a subset of a rectangular grid of points in the plane. If we choose m+ k − 1 points, not containing a closed path, in an m× k grid, then a formula for the determinant of the distance matrix of such points is obtained. © 2004 Elsevier Inc. All rights reserved.