Spectral Bisection with Two Eigenvectors

Abstrakt

We show a spectral bisection algorithm which makes use of the second and third eigenvector of the Laplacian matrix. This algorithm is guaranteed to return a cut that is smaller or equal to the one returned by the classic spectral bisection. To this end, we investigate combinatorial properties of certain configurations of a graph partition. These properties, that we call organized partitions, are shown to be related to the minimality and maximality of a cut. We show that organized partitions are related to the third eigenvector of the Laplacian matrix and give bounds on the minimum cut in terms of organized partitions and eigenvalues.