The Laplace matrix of a graph as well as its eigenvalues and eigenvectors appear in several rather diverse areas such as graph partitioning, Euclidean embedding problems, rigidity and the analysis of mixing rates of Markov chains. Duality in semidefinite optimization allows to develop some intuition on the relation between these applications. Our main focus will be on an appealing geometric interpretation that arises when studying connections between the separator structure of the graph and eigenvectors to optimized extremal eigenvalues of the Laplacian.
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