Geodesic
Generalized inverse of the Laplacian matrix and some applications
Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 29 (2004) no. 1

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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 graph), Laplacian eigenvalue (of graph), resistance distance 
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     author = {I. Gutman and W. Xiao},
     title = {Generalized inverse of the {Laplacian} matrix and some applications},
     journal = {Bulletin de l'Acad\'emie serbe des sciences. Classe des sciences math\'ematiques et naturelles},
     pages = {15 - 23},
     publisher = {mathdoc},
     volume = {29},
     number = {1},
     year = {2004},
     zbl = {1079.05055},
     url = {http://geodesic.mathdoc.fr/item/BASS_2004_29_1_a1/}
}
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I. Gutman; W. Xiao. Generalized inverse of the Laplacian matrix and some applications. Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 29 (2004) no. 1. http://geodesic.mathdoc.fr/item/BASS_2004_29_1_a1/