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Quasi-randomness and Algorithmic Regularity for Graphs with General Degree Distributions

  • Noga Alon
  • Amin Coja-Oghlan
  • Hiệp Hàn
  • Mihyun Kang
  • Vojtěch Rödl
  • Mathias Schacht
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4596)

Abstract

We deal with two very related subjects: quasi-randomness and regular partitions. The purpose of the concept of quasi-randomness is to measure how much a given graph “resembles” a random one. Moreover, a regular partition approximates a given graph by a bounded number of quasi-random graphs. Regarding quasi-randomness, we present a new spectral characterization of low discrepancy, which extends to sparse graphs. Concerning regular partitions, we present a novel concept of regularity that takes into account the graph’s degree distribution, and show that if G = (V,E) satisfies a certain boundedness condition, then G admits a regular partition. In addition, building on the work of Alon and Naor [4], we provide an algorithm that computes a regular partition of a given (possibly sparse) graph G in polynomial time.

Keywords

quasi-random graphs Laplacian eigenvalues sparse graphs regularity lemma Grothendieck’s inequality 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Noga Alon
    • 1
  • Amin Coja-Oghlan
    • 2
    • 3
  • Hiệp Hàn
    • 3
  • Mihyun Kang
    • 3
  • Vojtěch Rödl
    • 4
  • Mathias Schacht
    • 3
  1. 1.School of Mathematics and Computer Science, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978Israel
  2. 2.Carnegie Mellon Univsersity, Department of Mathematical Sciences, Pittsburgh, PA 15213USA
  3. 3.Humboldt-Universität zu Berlin, Institut für Informatik, Unter den Linden 6, 10099 BerlinGermany
  4. 4.Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322USA

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