  Nodal theorems for generalized modularity matrices ensure that the cluster
located by the positive entries of the leading eigenvector of various
modularity matrices induces a connected subgraph. In this paper we obtain lower
bounds for the modularity of that set of nodes showing that, under certain
conditions, the nodal domains induced by eigenvectors corresponding to highly
positive eigenvalues of the normalized modularity matrix have indeed positive
modularity, that is they can be recognized as modules inside the network.
Moreover we establish Cheeger-type inequalities for the cut-modularity of the
graph, providing a theoretical support to the common understanding that highly
positive eigenvalues of modularity matrices are related with the possibility of
subdividing a network into communities.
