, Volume 12, Issue 3, pp 275–285 | Cite as

Cubic Ramanujan graphs

  • Patrick Chiu


A fimily of cubic Ramanujan graph is explicitly constructed. They are realized as Cayley graphs of a certain free group acting on the 3-regular tree; this group is obtained from a definite quaternion algebra that splits at the prime 2 and has a maximal order of class number 1.

AMS Subject Classification code (1991)

05 C 35 


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  1. [1]
    T. Apostol:Introduction to analytic number theory, Springer-Verlag, 1976.Google Scholar
  2. [2]
    B. Bollobás:Graph theory — an introductory course, Springer-Verlag GTM 63, 1979.Google Scholar
  3. [3]
    M. Eichler:Lectures on modular correspondences, Tata Institute of Fundamental Research, Bombay, 1955–56.Google Scholar
  4. [4]
    M. Eichler: The basis problem for modular forms and the traces of the Hecke operators,Modular Functions of One Variable, Springer-Verlag Lecture Notes in Math. 320, 1973.Google Scholar
  5. [5]
    M. Eichler: Quaternäre quadratische Formen und die Riemannsche Vermutung für die kongruentz Zeta Funktion,Archiv. der Math. V (1954), 355–366.Google Scholar
  6. [6]
    E. Hecke: Analytische arithmetik der positiven quadratic formen,Collected Works, pp. 789–898, Gottingen, 1959.Google Scholar
  7. [7]
    J. Igusa: Fibre systems of Jacobian varieties III,American Jnl. of Math. 81 (1959), 453–476.Google Scholar
  8. [8]
    A. Lubotzky:Discrete groups, expanding graphs and invariant measures, NSF-CBMS Regional Conference Lecture Notes, U. of Oklahoma, 1989.Google Scholar
  9. [9]
    A. Lubotzky, R. Phillips andP. Sarnak: Ramanujan graphs,Combinatorica 8 (1988), 261–277.Google Scholar
  10. [10]
    A. Lubotzky, R. Phillips andP. Sarnak: Hecke operators and distributing points onS 2, parts I and II,Comm. Pure and Applied Math. 39 (1986), 149–186,40 (1987), 401–420.Google Scholar
  11. [11]
    Malisev: On the representation of integers by positive definite forms,Math. Steklov 65 (1962).Google Scholar
  12. [12]
    G. A. Margulis: Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and concentrators,Prob. of Info. Trans. (1988) 39–46.Google Scholar
  13. [13]
    A. Ogg:Modular forms and Dirichlet series, W. A. Benjamin Enc., New York, 1977.Google Scholar
  14. [14]
    R. Rankin:Modular forms and functions, Cambridge University Press, 1977.Google Scholar
  15. [15]
    P. Sarnak:Some applications of modular forms, Cambridge University Press, Cambridge, 1990.Google Scholar
  16. [16]
    B. Schoeneberg:Elliptic modular functions, Springer Verlag, 1974.Google Scholar
  17. [17]
    J. P. Serre:Trees, Springer Verlag, 1980.Google Scholar
  18. [18]
    M. Vigneras:Arithmetique de algebras de quaternions, Springer-Verlag Lecture Notes in Math. 800 1980.Google Scholar

Copyright information

© Akademiai Kiado 1992

Authors and Affiliations

  • Patrick Chiu
    • 1
  1. 1.Stanford UniversityStanfordU.S.A.

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