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)


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.


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


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Albert, R., Barabási, A.L.: Statistical mechanics of complex networks. Reviews of modern physics 74, 47–97 (2002)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Alizadeh, F.: Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM J. Optimization 5, 13–51 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Alon, N., Duke, R.A., Rödl, V., Yuster, R.: The algorithmic aspects of the regularity lemma. J. of Algorithms 16, 80–109 (1994)zbMATHCrossRefGoogle Scholar
  4. 4.
    Alon, N., Naor, A.: Approximating the cut-norm via Grothendieck’s inequality. In: Proc. 36th STOC, pp. 72–80 (2004)Google Scholar
  5. 5.
    Bilu, Y., Linial, N.: Lifts, discrepancy and nearly optimal spectral gap. Combinatorica (to appear)Google Scholar
  6. 6.
    Butler, S.: On eigenvalues and the discrepancy of graphs. preprintGoogle Scholar
  7. 7.
    Chung, F., Graham, R.: Quasi-random graphs with given degree sequences. Preprint (2005)Google Scholar
  8. 8.
    Chung, F., Graham, R.: Sparse quasi-random graphs. Combinatorica 22, 217–244 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Chung, F., Graham, R., Wilson, R.M.: Quasi-random graphs. Combinatorica 9, 345–362 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Frieze, A., Kannan, R.: Quick approximation to matrices and applications. Combinatorica 19, 175–200 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Gerke, S., Steger, A.: A characterization for sparse ε-regular pairs. The Electronic J. Combinatorics 14, 4–12 (2007)MathSciNetGoogle Scholar
  12. 12.
    Grothendieck, A.: Résumé de la théorie métrique des produits tensoriels topologiques. Bol. Soc. Mat. Sao Paulo 8, 1–79 (1953)MathSciNetGoogle Scholar
  13. 13.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric algorithms and combinatorial optimization. Springer, Heidelberg (1988)zbMATHGoogle Scholar
  14. 14.
    Håstad, J.: Some optimal inapproximability results. J. of the ACM 48, 798–859 (2001)zbMATHCrossRefGoogle Scholar
  15. 15.
    Kohayakawa, Y.: Szemeredi’s regularity lemma for sparse graphs. In: Cucker, F., Shub, M. (eds.) Foundations of computational mathematics, pp. 216–230 (1997)Google Scholar
  16. 16.
    Kohayakawa, Y., Rödl, V., Thoma, L.: An optimal algorithm for checking regularity. SIAM J. Comput. 32, 1210–1235 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Krivine, J.L.: Sur la constante de Grothendieck. C. R. Acad. Sci. Paris Ser. A-B 284, 445–446 (1977)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Szemeredi, E.: Regular partitions of graphs. Problémes Combinatoires et Théorie des Graphes Colloques Internationaux CNRS 260, 399–401 (1978)MathSciNetGoogle Scholar
  19. 19.
    Trevisan, L., Sorkin, G., Sudan, M., Williamson, D.: Gadgets, approximation, and linear programming. SIAM J. Computing 29, 2074–2097 (2000)zbMATHCrossRefMathSciNetGoogle Scholar

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

Personalised recommendations