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Cubic Ramanujan graphs

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Abstract

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.

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References

  1. T. Apostol:Introduction to analytic number theory, Springer-Verlag, 1976.

  2. B. Bollobás:Graph theory — an introductory course, Springer-Verlag GTM 63, 1979.

  3. M. Eichler:Lectures on modular correspondences, Tata Institute of Fundamental Research, Bombay, 1955–56.

    Google Scholar 

  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.

  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. E. Hecke: Analytische arithmetik der positiven quadratic formen,Collected Works, pp. 789–898, Gottingen, 1959.

  7. J. Igusa: Fibre systems of Jacobian varieties III,American Jnl. of Math. 81 (1959), 453–476.

    Google Scholar 

  8. A. Lubotzky:Discrete groups, expanding graphs and invariant measures, NSF-CBMS Regional Conference Lecture Notes, U. of Oklahoma, 1989.

  9. A. Lubotzky, R. Phillips andP. Sarnak: Ramanujan graphs,Combinatorica 8 (1988), 261–277.

    Google Scholar 

  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. Malisev: On the representation of integers by positive definite forms,Math. Steklov 65 (1962).

  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.

  13. A. Ogg:Modular forms and Dirichlet series, W. A. Benjamin Enc., New York, 1977.

    Google Scholar 

  14. R. Rankin:Modular forms and functions, Cambridge University Press, 1977.

  15. P. Sarnak:Some applications of modular forms, Cambridge University Press, Cambridge, 1990.

    Google Scholar 

  16. B. Schoeneberg:Elliptic modular functions, Springer Verlag, 1974.

  17. J. P. Serre:Trees, Springer Verlag, 1980.

  18. M. Vigneras:Arithmetique de algebras de quaternions, Springer-Verlag Lecture Notes in Math. 800 1980.

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Authors and Affiliations

  1. Stanford University, 94305, Stanford, CA, U.S.A.

    Patrick Chiu

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  1. Patrick Chiu
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Chiu, P. Cubic Ramanujan graphs. Combinatorica 12, 275–285 (1992). https://doi.org/10.1007/BF01285816

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  • DOI: https://doi.org/10.1007/BF01285816

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