A note on Fiedler vectors interpreted as graph realizations

Authors:
Christoph Helmberg;Susanna Reiss
Affiliations:
Fakultät für Mathematik, Technische Universität Chemnitz, D-09107 Chemnitz, Germany;Fakultät für Mathematik, Technische Universität Chemnitz, D-09107 Chemnitz, Germany
Venue:
Operations Research Letters
Year:
2010

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Abstract

The second smallest eigenvalue of the Laplace matrix of a graph and its eigenvectors, also known as Fiedler vectors in spectral graph partitioning, carry significant structural information regarding the connectivity of the graph. Using semidefinite programming duality, we offer a geometric interpretation of this eigenspace as optimal solution to a graph realization problem. A corresponding interpretation is also given for the eigenspace of the maximum eigenvalue of the Laplacian.