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Generalized inverse of the Laplacian matrix and some applications

Gutman Ivan (University of Kragujevac, Faculty of Science)
Xiao W. (South China University of Technology, Department of Computer Science, Guangzhou, P.R. China + Xiamen University, Xiamen, P.R. China)

The generalized inverse L† 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 rij = {L†)ii + (L†)jj — (L†)ij - (L†)ji is encountered; it is called resistance distance. Based on the results obtained for L† we prove some previously known and deduce some new properties of the resistance distance. .

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