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Nodal theorems for generalized modularity matrices ensure that the clusterlocated by the positive entries of the leading eigenvector of variousmodularity matrices induces a connected subgraph. In this paper we obtain lowerbounds for the modularity of that set of nodes showing that, under certainconditions, the nodal domains induced by eigenvectors corresponding to highlypositive eigenvalues of the normalized modularity matrix have indeed positivemodularity, that is they can be recognized as modules inside the network.Moreover we establish Cheeger-type inequalities for the cut-modularity of thegraph, providing a theoretical support to the common understanding that highlypositive eigenvalues of modularity matrices are related with the possibility ofsubdividing a network into communities.

Modularity bounds for clusters located by leading eigenvectors of the normalized modularity matrix