Is there any easy way to detect using the SVD that a matrix $A$ is an adjacency matrix of a graph with disjoint cliques (i.e. that $A$ is a clustering matrix?). Up to a reordering of columns a cluster matrix is a block diagonal matrix with blocks filled with $1$'s.
The SVD of a clustering matrix will be low rank with integer values for the singular values. But given the SVD of a matrix $A$, can we conclude that $A$ is or is not a clustering matrix?