ABSTRACT
In this chapter we focus on Laplacian matrices for weighted graphs, paying special attention to the interplay between the group inverse of the Laplacian matrix and the underlying structure of the weighted graph. In section 7.2, we consider the class of weighted trees, and derive formulas expressing the entries of the group inverse of the corresponding Laplacian matrices in terms of certain weighted distances between vertices. In particular, we characterise those weighted trees for which the group inverse of the Laplacian matrix is an M-matrix. Section 7.3 derives bounds on the algebraic connectivity of a weighted graph in terms of the group inverse of the corresponding Laplacian matrix, and discusses classes of graphs for which equality holds. In section 7.4, we deal with the resistance distance between vertices of a weighted graph, develop some of its main properties, and discuss the Kirchhoff index for weighted graphs. Finally, in section 7.5 we discuss the use of the group inverse of the Laplacian matrix in the analysis of electrical networks.
