Generalized inverse of the Laplacian matrix and some applications

Abstract

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. .