We present a new method for upper bounding the second eigenvalue of the Laplacian of graphs. Our approach uses multi-commodity flows to deform the geometry of the graph; we embed the resulting metric into Euclidean space to recover a bound on the Rayleigh quotient. Using this, we show that every @math -vertex graph of genus @math and maximum degree @math satisfies @math . This recovers the @math bound of Spielman and Teng for planar graphs, and compares to Kelner's bound of @math , but our proof does not make use of conformal mappings or circle packings. We are thus able to extend this to resolve positively a conjecture of Spielman and Teng, by proving that @math whenever @math is @math -minor free. This shows, in particular, that spectral partitioning can be used to recover @math -sized separators in bounded degree graphs that exclude a fixed minor. We extend this further by obtaining nearly optimal bounds on @math for graphs which exclude small-depth minors in the sense of Plotkin, Rao, and Smith. Consequently, we show that spectral algorithms find small separators in a general class of geometric graphs. Moreover, while the standard "sweep" algorithm applied to the second eigenvector may fail to find good quotient cuts in graphs of unbounded degree, our approach produces a vector that works for arbitrary graphs. This yields an alternate proof of the result of Alon, Seymour, and Thomas that every excluded-minor family of graphs has @math -node balanced separators.
Connections with discrete conformal mappings. One can view the minimizer of (or, more appropriately, the maximizer of the vertex version ) as a sort of global uniformizing'' metric for general graphs. In the setting of discrete conformal mappings, a number of variationally defined objects appear, and duality is often an important component in their analysis. We mention, for instance, the extremal length @cite_23 as a prominent example. It also often happens that one chooses a weight function @math as the minimizer of some convex functional, and this weight function plays the role of a discrete Riemannian metric (much as is the case in Section ); see, e.g. the work of Schramm @cite_27 and He and Schramm @cite_20 .
@cite_27 @cite_20 @cite_23
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We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . Part I. An Overview of Circle Packing: 1. A circle packing menagerie 2. Circle packings in the wild Part II. Rigidity: Maximal Packings: 3. Preliminaries: topology, combinatorics, and geometry 4. Statement of the fundamental result 5. Bookkeeping and monodromy 6. Proof for combinatorial closed discs 7. Proof for combinatorial spheres 8. Proof for combinatorial open discs 9. Proof for combinatorial surfaces Part III. Flexibility: Analytic Functions: 10. The intuitive landscape 11. Discrete analytic functions 12. Construction tools 13. Discrete analytic functions on the disc 14. Discrete entire functions 15. Discrete rational functions 16. Discrete analytic functions on Riemann surfaces 17. Discrete conformal structure 18. Random walks on circle packings Part IV: 19. Thurston's Conjecture 20. Extending the Rodin Sullivan theorem 21. Approximation of analytic functions 22. Approximation of conformal structures 23. Applications Appendix A. Primer on classical complex analysis Appendix B. The ring lemma Appendix C. Doyle spirals Appendix D. The brooks parameter Appendix E. Schwarz and buckyballs Appendix F. Inversive distance packings Appendix G. Graph embedding Appendix H. Square grid packings Appendix I. Experimenting with circle packings. Let @math denote a set of polynomials with complex coefficients. Let @math denote any set of sample points . For any @math , the discrete polynomial transform of @math (with respect to @math and @math ) is defined as the collection of sums, @math , where @math for some associated weight function @math . These sorts of transforms find important applications in areas such as medical imaging and signal processing. In this paper, we present fast algorithms for computing discrete orthogonal polynomial transforms. For a system of @math orthogonal polynomials of degree at most @math , we give an @math algorithm for computing a discrete polynomial transform at an arbitrary set of points instead of the @math operations required by direct evaluation. Our algorithm depends only on the fact that orthogonal polynomial sets satisfy a three-term recurrence and thus it may be applied to any such set of discretely sampled functions. In particular, sampled orthogonal polynomials generate the vector space of functions on a distance transitive graph. As a direct application of our work, we are able to give a fast algorithm for computing subspace decompositions of this vector space which respect the action of the symmetry group of such a graph. This has direct applications to treating computational bottlenecks in the spectral analysis of data on distance transitive graphs, and we discuss this in some detail.