# Eigenvalues of Ramanujan Graphs

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

## Abstract

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.

## Keywords

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.

## References

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.
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.
4. [C]
F. R. K. Chung, Diameters and eigenvalues, J. Amer. Math. Soc. 2 (1989), 187–196.
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.
7. [E1]
R. Evans, Spherical functions for finite upper half planes with characteristic 2, Finite Fields and Their Applications 1, (1995), 376–384.
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.
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.
12. [K2]
N. Katz, A note on exponential sums, Finite Fields and Their Applications, vol. 1, No. 3 (1995), 395–398.
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.
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.
17. [L1]
W. C. W. Li, Character sums and abelian Ramanujan graphs, J. Number Theory, vol. 41 (1992), 199–217.
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.
24. [Lps]
A. Lubotzky, R. Phillips and P. Sarnak, Ramanujan graphs, Combinatorica 8 (1988), 261–277.
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.
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.
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).
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.
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.
34. [W2]
A. Weil, Dirichlet Series and Automorphic Forms, Lecture Notes in Math. 189, Springer-Verlag, Berlin-Heidelberg-New York, 1971.Google Scholar