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 |
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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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