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 L∼ is the Laplacian matrix of an arbitrary connected graph, then D-1-L∼-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.