Eigenvalues of Ramanujan Graphs

  • Wen-Ching Winnie Li
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 109)


The adjacency matrix of a (directed or undirected) finite graph X is a square matrix A = A X with rows and columns parametrized by the vertices of X such that the xy entry of A records the number of edges from the vertex x to the vertex y. It may be regarded as a linear operator on the space of functions on vertices of X which sends a function f to Af whose value at a vertex x is given by
$$(Af)(x) = \sum\limits_{y} {f(y),}$$
where y runs through all ending points of the edges starting from x. The eigenvalues of A are called the eigenvalues of X. Call X a k-regular graph if at every vertex of X there are k edges going out and k edges coming in. If X is undirected and k-regular, then the eigenvalues of X are real and bounded between k and -k. The multiplicity of k as an eigenvalue of X is equal to the number of connected components of X, and -k is an eigenvalue if and only if X is a bipartite graph. In general, the eigenvalues λ i of X provide lots of information about the graph G. An eigenvalue of X with absolute value k is called a trivial eigenvalue, we are interested in the nontrivial eigenvalues. For this matter, define
$$\lambda = \lambda (X){{ = }_{{\lambda i}}}\begin{array}{*{20}{c}} {\max } \\ {nontrivial} \\ \end{array} |{{\lambda }_{i}}|.$$
Assume X connected. It turns out that the smaller λ(X) is, the larger the magnifying constant X has, and hence the more ”expanding” X is. It is also known that the smaller λ(X) is, the smaller the diameter of X is. In the case that X represents a communication network, the transmission efficiency is measured by the magnifying constant, and the transmission delay is measured by the diameter. Thus regular graphs with small λ have important applications in communication networks and in computer science.


Cayley Graph Automorphic Form Congruence Subgroup Double Coset Quaternion Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [A]
    N. Alon, Eigenvalues, geometric expanders, sorting in rounds, and Ramsey theory, Combinatorica 6, No. 3 (1986), 207–219.Google Scholar
  2. [ACPTTV]
    J. Angel, N. Celniker, S. Poulos, A. Terras, C. Trimble, and E. Velasquez, Special functions on finite upper half planes, Contemp. Math., 138 (1992), 1–26.MathSciNetCrossRefGoogle Scholar
  3. [CPTTV]
    N. Celniker, S. Poulos, A. Terras, C. Trimble, and E. Velasquez, IS there life on finite upper half planes?, Contemp. Math., 143 (1993), 65–88.MathSciNetCrossRefGoogle Scholar
  4. [C]
    F. R. K. Chung, Diameters and eigenvalues, J. Amer. Math. Soc. 2 (1989), 187–196.MathSciNetMATHCrossRefGoogle Scholar
  5. [De]
    P. Deligne, Cohomologie Etale, Lecture Notes in Math. 569, Springer-Verlag, Berlin-Heidelberg-New York (1977).Google Scholar
  6. [Dr]
    V. G. Drinfeld, The proof of Petersson’s conjecture for GL(2) over a global field of characteristic p, Functional Anal. Appl. 22 (1988), 28–43.MathSciNetCrossRefGoogle Scholar
  7. [E1]
    R. Evans, Spherical functions for finite upper half planes with characteristic 2, Finite Fields and Their Applications 1, (1995), 376–384.MathSciNetMATHCrossRefGoogle Scholar
  8. [E2]
    R. Evans, Character sums as orthogonal eigenfunctions of adjancency operators for Cayley graphs, Contemporary Math., vol. 168, Amer. Math. Soc., Providence, RI. (1994), 33–50.Google Scholar
  9. [FL]
    K. Feng and W.-C. W. Li, Spectra of hypergraphs and applications, J. Number Theory, vol. 60 (1996), 1–22.MathSciNetCrossRefGoogle Scholar
  10. [JL]
    H. Jacquet and R. P. Langlands, Automorphic Forms on GL(2), Lecture Notes in Math. 114, Springer-Verlag, Berlin-Heidelberg-New York (1970).Google Scholar
  11. [K1]
    N. Katz, Estimates for Soto-andrade sums, J. reine angew. Math. 438 (1993), 143–161.MathSciNetMATHGoogle Scholar
  12. [K2]
    N. Katz, A note on exponential sums, Finite Fields and Their Applications, vol. 1, No. 3 (1995), 395–398.MathSciNetMATHCrossRefGoogle Scholar
  13. [K3]
    N. Katz, Gauss Sums, Kloosterman Sums, and Monodromy Groups, Annals of Math. Studies 116, Princeton Univ. Press, Princeton, 1988.Google Scholar
  14. [Ku]
    J. Kuang, Eigenfunctions on the finite Poincare planes, MSRI preprint no. 015-95, 1995.Google Scholar
  15. [LR1]
    J. Lafferty and D. Rockmore, Fast Fourier analysis for SL 2 Over a finite field and related numerical experiments, Experimental Math., vol 1 (1992), No. 2, 115–139.MathSciNetMATHCrossRefGoogle Scholar
  16. [LR2]
    J. Lafferty and D. Rockmore, Numerical investigation of the spectrum for certain families of Cayley graphs, DIMACS Series in Discrete Math, and Theoretical Comp. Sci., vol. 10 (1993), 63–73.MathSciNetGoogle Scholar
  17. [L1]
    W. C. W. Li, Character sums and abelian Ramanujan graphs, J. Number Theory, vol. 41 (1992), 199–217.MathSciNetMATHCrossRefGoogle Scholar
  18. [L2]
    W. C. W. Li, Estimates of character sums arising from finite upper half planes, In: Finite Fields and Applications, Proc. of the third international conference, Glasgow, 11–14 July, 1995, S. Cohen and H. Niederreiter eds., Lecture Notes in Math., vol. 233, London Math. Soc, Cambridge Univ. Press, 219–228, 1996.Google Scholar
  19. [L3]
    W. C. W. Li, Elliptic curves, Kloosterman sums and Ramanujan graphs, In: Computational Perspectives on Number Theory: Proceeding of a Conference in Honor of A.O.L. Atkin, D.A. Buell and J.T. Teitelbaum edited, AMS/IP Studies in Advanced Math., vol. 7, Amer. Math. Soc, Providence, R.I, 1998.Google Scholar
  20. [L4]
    W. C. W. Li, Number-theoretic constructions of Ramanujan graphs, In: Columbia University Number Theory Seminar, New York 1992, Astérisque, vol. 228 (1995), Soc. Math. de France, 101–120.Google Scholar
  21. [L5]
    W. C. W. Li, A survey of Ramanujan graphs, In: Arithmetic, Geometry and Coding Theory, Proceedings of International Conference held at Luminy, France, June 28–July 2, 1993, de Gruyter, Berlin-New York, 127–143, 1996.Google Scholar
  22. [L6]
    W. C. W. Li, Number Theory with Applications, World Scientific, Singapore, 1996.Google Scholar
  23. [LS]
    W. C. W. Li and P. Solé, Spectra of regular graphs and hypergraphs and orthogonal polynomials, Europ. J. Combinatorics, vol. 17 (1996), 461–477.MATHCrossRefGoogle Scholar
  24. [Lps]
    A. Lubotzky, R. Phillips and P. Sarnak, Ramanujan graphs, Combinatorica 8 (1988), 261–277.MathSciNetMATHCrossRefGoogle Scholar
  25. [Ma]
    G. Margulis, Explicit group theoretic constructions of combinatorial schemes and their application to the design of expanders and concentrators, J. Prob. of Info. Trans. (1988), 39–46.Google Scholar
  26. [Mc]
    B. D. Mckay, The expected eigenvalue distribution of a large regular Graph, Linear Alg. and Its Appli. 40 (1981), 203–216.MathSciNetMATHCrossRefGoogle Scholar
  27. [Me]
    J. F. Mestre, La méthode des graphes, Exemples et applications, Proc. Int. Conf. on Class Numbers and Fundamental Units of Algebraic Number Fields, June 24–28, 1986, Katata, Japan, 217-242.Google Scholar
  28. [Mo]
    C. Moreno, The Kloosterman conjecture in characteristic two, preprint.Google Scholar
  29. [Mn]
    M. Morgenstern, Existence and explicit constructions of q+1 regular Ramanujan graphs for every prime power q, J. Comb. Theory, series B, vol. 62 (1994), 44–62.MathSciNetMATHCrossRefGoogle Scholar
  30. [Sa]
    P. Sarnak, Statistical properties of eigenvalues of the Hecke operators, In: Analytic Number Theory and Diophantine Problems, A. Adolphson, J. Conrey, A. Ghosh, and R. Yager edited, Progress in Math. 70, Birkhäuser, Boston-Basel-Stuttgart, 321–331 (1987).CrossRefGoogle Scholar
  31. [Se]
    J-P. Serre, Répartition Asymptotique des Valeurs Propres de L’opéra-teur de Hecke Tp, J. Amer. Math. Soc, vol. 10 (1997), 75–102.MathSciNetMATHCrossRefGoogle Scholar
  32. [TZ]
    J. P. Tillich and G. ZéMor, Optimal cycle codes constructed from Ramanujan graphs, preprint, 1996.Google Scholar
  33. [W1]
    A. Weil, On some exponential sums, Proc. Nat. Aca. Sci., 34 (1948), 204–207.MathSciNetMATHCrossRefGoogle Scholar
  34. [W2]
    A. Weil, Dirichlet Series and Automorphic Forms, Lecture Notes in Math. 189, Springer-Verlag, Berlin-Heidelberg-New York, 1971.Google Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Wen-Ching Winnie Li
    • 1
  1. 1.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA

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