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<SUP>-1</SUP>. As an example, it is shown that if L˜ is the Laplacian matrix of an arbitrary connected graph, then (D<SUP>-1</SUP>-L˜)<SUP>-1</SUP> 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