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Combinatorica

, Volume 8, Issue 3, pp 261–277 | Cite as

Ramanujan graphs

  • A. Lubotzky
  • R. Phillips
  • P. Sarnak
Article

Abstract

A large family of explicitk-regular Cayley graphsX is presented. These graphs satisfy a number of extremal combinatorial properties.
  1. (i)

    For eigenvaluesλ ofX eitherλ=±k or ¦λ¦≦2 √k−1. This property is optimal and leads to the best known explicit expander graphs.

     
  2. (ii)

    The girth ofX is asymptotically ≧4/3 logk−1 ¦X¦ which gives larger girth than was previously known by explicit or non-explicit constructions.

     

AMS subject classification (1980)

05C35 

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References

  1. [1]
    N.Alon,Private communication 1986.Google Scholar
  2. [2]
    N. Alon, Eigenvalues, geometric expanders, sorting in rounds and Ramsey theory,Combinatorica,6 (1986), 207–219.Google Scholar
  3. [3]
    N. Alon, Eigenvalues and expanders,Combinatorica,6 (1986), 83–96.Google Scholar
  4. [4]
    B. Bollobás,Extremal graph theory, Academic Press, London 1978.Google Scholar
  5. [5]
    L. E. Dickson, Arithmetic of quaternions,Proc. London Math. Soc. (2)20 (1922), 225–232.Google Scholar
  6. [6]
    M. Eichler, Quaternäre quadratische Formen und die Riemannsche Vermutung für die kongruentz Zeta Funktion,Archiv. der Math. Vol. V, (1954), 355–366.Google Scholar
  7. [7]
    P. Erdős andH. Sachs, Reguläre Graphen gegenebener Teillenweite mit Minimaler Knotenzahl, Wiss. Z. Univ. Halle-Wittenberg,Math. Nat. R. 12 (1963), 251–258.Google Scholar
  8. [8]
    L.Gerritzen and N.Van der Put,Schottky groups and Mumford curves, Springer-Verlag, L. N. in Math. 817 (1980).Google Scholar
  9. [9]
    G.Hardy and E.Wright,An introduction to number theory, Oxford University Press 1978 (Fifth Edition).Google Scholar
  10. [10]
    E.Hecke, Analytische arithmetik der positiven quadratic formen,Collected works pp. 789–898, Göttingen, 1959.Google Scholar
  11. [11]
    A.Hofmann, On eigenvalues and colorings of graphs,in Graph theory and its applications (ed. B. Harris) Academic Press (1970), 79–91.Google Scholar
  12. [12]
    J. Igusa, Fibre systems of Jacobian varieties III,American Jnl. of Math. 81 (1959), 453–476.Google Scholar
  13. [13]
    Y.Ihara, Discrete subgroups of PL (2, k p),Proc. Symp. in Pure Math. IX, AMS (1968), 272–278.Google Scholar
  14. [14]
    W. Imrich, Explicit construction of regular graphs with no small cycles,Combinatorica 4 (1984), 53–59.Google Scholar
  15. [15]
    H. Kesten, Symmetric random walks on groups,Trans. AMS 92 (1959), 336–354.Google Scholar
  16. [16]
    M. Knesser, Strong approximation in:Algebraic Groups and Discontinuous Subgroups, Proc. Symp. Pure Math. Vol. IX, (1966), 187–196.Google Scholar
  17. [17]
    A. Lubotzky, R. Phillips, P. Sarnak, Ramanujan conjecture and explicit construction of expanders,Proc. Stoc. 86 (1986), 240–246.Google Scholar
  18. [18]
    A. Lubotzky, R. Phillips, P. Sarnak, Hecke operators and distributing points onS 2 I, IIComm. Pure and Applied Math. 39 (1986), 149–186,40 (1987), 401–420.Google Scholar
  19. [19]
    G. A. Margulis, Graphs without short cycles,Combinatorica 2 (1982), 71–78.Google Scholar
  20. [20]
    Mališev, On the representation of integers by positive definite forms,Mat. Steklov 65 (1962).Google Scholar
  21. [21]
    A. Ogg,Modular forms and Dirichlet series, W. A. Benjamin Inc., New York 1969.Google Scholar
  22. [22]
    S. Ramanujan, On certain arithmetical functions,Trans. Camb. Phil. Soc. 22 (1916), 159–184.Google Scholar
  23. [23]
    J. P. Serre,Trees, Springer Verlag, Berlin-Heidelberg-New York, (1980).Google Scholar
  24. [24]
    M. F.Vignéras,Arithmetique dè Algebras de Quaternions, Springer Lecture Notes; V. 800, (1980).Google Scholar
  25. [25]
    G. L. Watson, Quadratic diophantine equations,Royal Soc. of London, Phil. Trans., A 253, 227–2 (1960).Google Scholar
  26. [26]
    A.Weil, Sur les courbes algébriques et les varétés qui s'en déduisent,Actualites Sci. Et ind. No. 1041 (1948).Google Scholar
  27. [27]
    A. Weiss, Girths of bipartite sextet graphs,Combinatorica 4 (1984), 241–245.Google Scholar

Copyright information

© Akadémiai Kiadó 1988

Authors and Affiliations

  • A. Lubotzky
    • 1
  • R. Phillips
    • 2
  • P. Sarnak
    • 3
  1. 1.Institute of Mathematics and Computer ScienceHebrew UniversityJerusalemIsrael
  2. 2.Department of MathematicsStanford UniversityStanfordUSA
  3. 3.Department of MathematicsStanford UniversityStanfordUSA

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