Laplacian matrix of a weighted graph with new pendant vertices

The Laplacian matrix of a simple graph has been widely studied, as a consequence of its applications. However the Laplacian matrix of a weighted graph is still a challenge. In this work we provide the Moore-Penrose inverse of the Laplacian matrix of the graph obtained adding new pendant vertices to an initial graph, in terms of the Moore-Penrose inverse of the Laplacian matrix of the original graph. As an application we can compute the effective resistances and the Kirchhoff index of the new network.