Embedded in the Shadow of the Separator

Eigenvectors to the second smallest eigenvalue of the Laplace matrix of a graph, also known as Fiedler vectors, are the basic ingredient in spectral graph partitioning heuristics. Maximizing this second smallest eigenvalue over all nonnegative edge weightings with bounded total weight yields the absolute algebraic connectivity introduced by Fiedler, who proved tight connections of this value to the connectivity of the graph. Our objective is to gain a better understanding of the connections between separators and the eigenspace of this eigenvalue by studying the dual semidefinite optimization problem to the absolute algebraic connectivity. Exploiting optimality conditions we show that this problem is equivalent to finding an embedding of the n nodes of the graph in n-space so that their barycenter is the origin, the distance between adjacent nodes is bounded by one and the nodes are spread as much as possible (the sum of the squared norms is maximized). For connected graphs we prove t...

• Type: Journal
• Year: 2008
• Where: SIAMJO
• Authors: Frank Göring, Christoph Helmberg, Markus Wappler
• Keywords: Absolute Algebraic Connectivity, Nonnegative Edge Weightings, SIAMJO 2008, Smallest Eigenvalue