Showing papers on "Spectral graph theory published in 1998"

On the Quality of Spectral Separators

Abstract: Computing graph separators is an important step in many graph algorithms. A popular technique for finding separators involves spectral methods. However, there has not been much prior analysis of the quality of the separators produced by this technique; instead it is usually claimed that spectral methods "work well in practice." We present an initial attempt at such an analysis. In particular, we consider two popular spectral separator algorithms and provide counterexamples showing that these algorithms perform poorly on certain graphs. We also consider a generalized definition of spectral methods that allows the use of some specified number of the eigenvectors corresponding to the smallest eigenvalues of the Laplacian matrix of a graph, and we show that if such algorithms use a constant number of eigenvectors, then there are graphs for which they do no better than using only the second smallest eigenvector. Furthermore, using the second smallest eigenvector of these graphs produces partitions that are poor with respect to bounds on the gap between the isoperimetric number and the cut quotient of the spectral separator. Even if a generalized spectral algorithm uses $n^\epsilon$ for \mbox{$0 < \epsilon < \frac{1}{4}$} eigenvectors, there exist graphs for which the algorithm fails to find a separator with a cut quotient within \mbox{$n^{\frac{1}{4} - \epsilon} - 1$} of the isoperimetric number. We also introduce some facts about the structure of eigenvectors of certain types of Laplacian and symmetric matrices; these facts provide the basis for the analysis of the counterexamples. Finally, we discuss some developments in spectral partitioning that have occurred since these results first appeared.

Spectral Techniques in Graph Algorithms

Abstract: The existence of efficient algorithms to compute the eigenvectors and eigenvalues of graphs supplies a useful tool for the design of various graph algorithms. In this survey we describe several algorithms based on spectral techniques focusing on their performance for randomly generated input graphs.

Expanders are not hyperbolic

Abstract: We bound from above the number of vertices of a graph in terms of the Cheeger constant and the δ-hyperbolicity of the graph. As a corollary we get that expanders are not uniformly hyperbolic.

Graph, graph spectra and partitioning algorithms in a geodetic network structural analysis and adjustment

Abstract: Graph theory and its algorithms have been applied in geodesy, viz, structural analysis of networks, and sparse matrix technique in geodetic network adjustment. Fundamental in these applications is the representation of the geodetic network as a graph. Adjacency matrix and the spectra of the Laplacian matrix are shown to have useful properties for both structural analysis and partitioning of a geodetic network. Algorithms for the construction of adjacency matrix, structural analysis, and partitioning of a geodetic network are presented.