  We investigate combinatorial properties of certain configurations of a graph
partition which are related to the minimality of a cut. We show that such
configurations are related to the third eigenvector of the Laplacian matrix. It
is well known that the second eigenvector encodes structural information, and
that can be used to approximate a minimum bisection. In this paper, we show
that the third eigenvector carries structural information as well. We then
provide a new spectral bisection algorithm using both eigenvectors. The new
algorithm is guaranteed to return a cut that is smaller or equal to the one
returned by the classic spectral bisection. Also, we provide a spectral
algorithm that can refine a given partition and produce a smaller cut.
