  We consider the minimum-cut partitioning of a graph into more than two parts
using spectral methods. While there exist well-established spectral algorithms
for this problem that give good results, they have traditionally not been well
motivated. Rather than being derived from first principles by minimizing graph
cuts, they are typically presented without direct derivation and then proved
after the fact to work. In this paper, we take a contrasting approach in which
we start with a matrix formulation of the minimum cut problem and then show,
via a relaxed optimization, how it can be mapped onto a spectral embedding
defined by the leading eigenvectors of the graph Laplacian. The end result is
an algorithm that is similar in spirit to, but different in detail from,
previous spectral partitioning approaches. In tests of the algorithm we find
that it outperforms previous approaches on certain particularly difficult
partitioning problems.
