We provide a graph theoretic background for the analysis of financial networks and review some technique recently proposed for the extraction of financial networks. We develop new measures of network connectivity, that are Von Neumann entropies and disagreement persistence index, using the spectrum of normalized Laplacian and Diplacian. We show that the new measures account for global connectivity patterns given by paths and walks of the network. We apply the new measures to a sequence of inferred pairwise-Granger networks. In the application, we employ the proposed measures for the system immunization and early warning for banking crises.