  The role of the normalized modularity matrix in finding homogeneous cuts will
be presented. We also discuss the testability of the structural eigenvalues and
that of the subspace spanned by the corresponding eigenvectors of this matrix.
In the presence of a spectral gap between the k-1 largest absolute value
eigenvalues and the remainder of the spectrum, this in turn implies the
testability of the sum of the inner variances of the k clusters that are
obtained by applying the k-means algorithm for the appropriately chosen vertex
representatives.
