Embedded in the Shadow of the Separator

Frank Göring (frank.goeringmathematik.tu-chemnitz.de)
Christoph Helmberg (helmbergmathematik.tu-chemnitz.de)
Markus Wappler (markus.wapplermathematik.tu-chemnitz.de)

Abstract: We study the problem of maximizing the second smallest eigenvalue of the Laplace matrix of a graph over all nonnegative edge weightings with bounded total weight. The optimal value is the \emph{absolute algebraic connectivity} introduced by Fiedler, who proved tight connections of this value to the connectivity of the graph. Using semidefinite programming techniques and exploiting optimality conditions we show that the problem is equivalent to finding an embedding of the $n$ nodes in $n-$space so that their barycenter is at 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 that for any separator in the graph, at least one of the two separated node sets is embedded in the shadow (with the origin being the light source) of the convex hull of the separator. In particular, the barycenters of partitions induced by separators are separated by the affine subspace spanned by the nodes of the separator. Furthermore, we show that there always exists an optimal embedding whose dimension is bounded by the tree width of the graph plus one.

Keywords: spectral graph theory, semidefinite programming, eigenvalue optimization, embedding, graph partitioning, tree-width

Category 1: Combinatorial Optimization (Graphs and Matroids )

Category 2: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Citation: Preprint 2005-12, Technische Universität Chemnitz, Fakultät für Mathematik, September 2005

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Entry Submitted: 09/02/2005
Entry Accepted: 09/02/2005
Entry Last Modified: 09/02/2005

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