Pseudoinverse of the Laplacian and best spreader node in a network

Determining a set of "important" nodes in a network constitutes a basic endeavor in network science. Inspired by electrical flows in a resistor network, we propose the best conducting node j in a graph G as the minimizer of the diagonal element Q_{jj}^{†} of the pseudoinverse matrix Q^{†} of the weighted Laplacian matrix of the graph G. We propose a new graph metric that complements the effective graph resistance R_{G} and that specifies the heterogeneity of the nodal spreading capacity in a graph. Various formulas and bounds for the diagonal element Q_{jj}^{†} are presented. Finally, we compute the pseudoinverse matrix of the Laplacian of star, path, and cycle graphs and derive an expansion and lower bound of the effective graph resistance R_{G} based on the complement of the graph G.