Bulletin, Classe des Sciences Mathématiques et Naturelles, Sciences mathématiques naturelles / sciences mathematiques EMIS ELibM Electronic Journals Bulletin, Classe des Sciences Mathématiques et Naturelles, Sciences mathématiques
Académie Serbe des Sciences et des Arts, Beograd

Vol. CXXIX, No. 29, pp. 15-23 (2004)

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Generalized inverse of the Laplacian matrix and some applications

I. Gutman and W. Xiao

Faculty of Science, University of Kragujevac, P. O. Box 60, 34000 Kragujevac, Serbia and Montenegro
Department of Computer Science, South China University of Technology, Guangzhou 510641, P. R. China, and Xiamen University, P. O. Box 1003, Xiamen 361005, P. R. China

Abstract: The generalized inverse $L^\dagger$ of the Laplacian matrix of a connected graph is examined and some of its properties are established. In some physical and chemical considerations the quantity $r_{ij} = (L^\dagger)_{ii} + (L^\dagger)_{jj} - (L^\dagger)_{ij} - (L^\dagger)_{ji}$ is encountered; it is called resistance distance. Based on the results obtained for $L^\dagger$ we prove some previously known and deduce some new properties of the resistance distance.

Keywords: Laplacian matrix; Laplacian eigenvector (of a graph); Laplacian eigenvalue (of a graph); resistance distance

Classification (MSC2000): 05C50

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Electronic version published on: 27 Jan 2005. This page was last modified: 27 Jan 2005.

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