Background: In Newman's PNAS 2006 paper Modularity and community structure in networks, the first eigenvector splits the graph in two clusters, and then each cluster can be further divided by eigenvector of a modified Laplacian of the nodes within this cluster. This leads to further splitting and a binary tree kind of structure of the nodes can be achieved. This makes sense in terms of maximizing the modularity of the clusters.
In Ng, Jordan and Weiss's NIPS 2001 paper On Spectral Clustering, they simply take the first $k$ eigenvectors and then use $k$-means to make clusters using the eigenvectors as features. I understand that this is not maximizing the modularity, but it makes sense for clustering. However, I am curious whether it maximizes any objective function.
Question: Are there other approaches to making more than two clusters from spectral clustering?
What if I take the first $k$ eigenvectors and make clusters based on the sign of the components? Will I get $2^k$ clusters? For $k=1$ will it reduce to Newman's algorithm? Does this make sense? If not, why not?
Are there any other approaches to make more than two clusters either by taking the first $k$ eigenvectors or by hierarchical partitioning?