On distance matrices and Laplacians

Bapat, R. ; Kirkland, S. J. ; Neumann, M. (2005) On distance matrices and Laplacians Linear Algebra and its Applications, 401 . pp. 193-209. ISSN 0024-3795

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Official URL: http://www.sciencedirect.com/science/article/pii/S...

Related URL: http://dx.doi.org/10.1016/j.laa.2004.05.011

Abstract

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.

Item Type:Article
Source:Copyright of this article belongs to Elsevier Science.
Keywords:Trees; Distance Matrices; Laplacians; Determinants; Nonnegative Matrices
ID Code:77904
Deposited On:14 Jan 2012 15:26
Last Modified:14 Jan 2012 15:26

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