Graphes de Ramanujan et applications

Valette, Alain

Séminaire Bourbaki : volume 1996/97, exposés 820-834, Astérisque, no. 245 (1997), Exposé no. 829, 30 p.

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Valette, Alain. Graphes de Ramanujan et applications, dans Séminaire Bourbaki : volume 1996/97, exposés 820-834, Astérisque, no. 245 (1997), Exposé no. 829, 30 p. http://www.numdam.org/item/SB_1996-1997__39__247_0/

Bibliographie
Cité par

[1] N. Alon, Eigenvalues and expanders, Combinatorica, 6 (1986), pp. 83-96. | DOI | MR | Zbl

[2] N. Alon and V. Milman, λ1, isoperimetric inequalities for graphs, and superconcentrators, J. Combin. Theory, ser. B, 38 (1985), pp. 73-88. | DOI | Zbl

[3] J. Angel, N. Celniker, S. Poulos, A. Terras, C. Trimble, and E. Velasquez, Special functions on finite upper half planes, Contemporary Maths., 138 (1992), pp. 1-26. | DOI | MR | Zbl

[4] J. Angel, S. Poulos, A. Terras, C. Trimble, and E. Velasquez, Spherical functions and transforms on finite upper half planes : eigenvalues of the combinatorial Laplacian, uncertainty, traces, Contemporary Math., 173 (1994), pp. 15-70. | DOI | MR | Zbl

[5] W. Ballmann and J. Swiatkowski, On $L^{2}$ -cohomology and property $(T)$ for automorphism groups of polyhedral cell complexes. Preprint, 1996. | MR | Zbl

[6] F. Bien, Constructions of telephone networks by group representations, Notices Amer. Math. Soc., 36 (1989), pp. 5-22. | MR | Zbl

[7] N. Biggs, Algebraic graph theory (2nd ed.), Cambridge University Press, 1993. | MR | Zbl

[8] N. Biggs and A. Boshier, Note on the girth of Ramanujan graphs, J. Combinatorial Theory, ser. B, 49 (1990), pp. 190-194. | DOI | MR | Zbl

[9] A. Borel, Cohomologie de certains groupes discrets et Laplacien $p$ -adique, in Séminaire Bourbaki, exposé 437, Springer, pp. 12-34, 1975. | EuDML | Numdam | MR | Zbl

[10] M. Burger, Cheng's inequality for graphs. Preprint, 1987.

[11] P. Bürgisser, M. Clausen, and M. Shokrollahi, Algebraic complexity theory, Springer-Verlag, 1997. | DOI | MR | Zbl

[12] D. Cartwright, W. Mlotkowski, and T. Steger, Property $(T)$ and ${\tilde{A}}_{2}$ groups, Ann. Inst. Fourier, Grenoble, 44 (1993), pp. 213-248. | DOI | EuDML | Numdam | MR | Zbl

[13] P. Chiu, Cubic Ramanujan graphs, Combinatorica, 12 (1992), pp. 275-285. | DOI | MR | Zbl

[14] F. Chung, Diameters and eigenvalues, Journal Amer. Math. Soc., 2 (1989), pp. 187-196. | DOI | MR | Zbl

[15] -, Spectral graph theory, CBMS reg. conf. ser. in Math. 92, Amer. Math. Soc., 1997. | MR

[16] P. De La Harpe and A. Valette, La propriété $(T)$ de Kazhdan pour les groupes localement compacts, Astérisque 175, Soc. Math. France, 1989. | Numdam | Zbl

[17] Y. C. De Verdière, Distribution de points sur une sphère [d'après Lubotzky, Phillips et Sarnak], in Séminaire Bourbaki, exposé 703, Astérisque 177-178, p. 83-93, 1989. | Numdam | Zbl

[18] P. Deligne, La conjecture de Weil I, Publ. Math. IHES, 43 (1974), pp. 273-308. | DOI | EuDML | Numdam | Zbl

[19] V. Drinfeld, Finitely additive measures on $S^{2}$ and $S^{3}$ , invariant with respect to rotations, Funct. Anal. and its Appl., 18 (1984), pp. 245-246. | DOI | Zbl

[20] -, The proof of Petersson's conjecture for $G L (2)$ over a global field of characteristic $p$ , Funct. Anal. Appl., 22 (1988), pp. 28-43. | DOI | Zbl

[21] M. Eichler, Quaternäre quadratische Formen und die Riemannsche Vermutung für die Kongruenzzetafunktion, Arch. Math., 5 (1954), pp. 355-366. | DOI | Zbl

[22] P. Erdös and H. Sachs, Reguläre Graphen gegebener Taillenweite mit minimaler Knollenzahl, Wiss. Z. Univ. Halle-Willenberg Math. Nat. R., 12 (1963), pp. 251-258. | Zbl

[23] K. Feng and W. Li, Spectra of hypergraphs and applications, Journal of number theory, 60 (1996), pp. 1-22. | DOI | Zbl

[24] J. Friedman, Some graphs with small second eigenvalues, Combinatorica, 15 (1995), pp. 31-42. | DOI | Zbl

[25] O. Gabber and Z. Galil, Explicit constructions of linear-sized superconcentrators, J. Comp. and Syst. Sci., 22 (1981), pp. 407-420. | DOI | Zbl

[26] H. Garland, $p$ -adic curvature and the cohomology of discrete subgroups of $p$ -adic groups, Ann. of Math., 97 (1973), pp. 375-423. | DOI | Zbl

[27] L. Gerritzen and M. Van Der Put, Schottky groups and Mumford curves, Springer Lect. Notes in Math. 817, 1980. | Zbl

[28] Y. Greenberg, Thèse, PhD thesis, Hebrew Univ., Jerusalem, 1995.

[29] D. Husemoller, Elliptic curves, Springer, 1987. | DOI | MR | Zbl

[30] Y. Ihara, Discrete subgroups of $P L (2, k_{p})$ , in Algebraic groups and discontinuous subgroups, Proc. Symp. pure Math. IX, Amer. Math. Soc., pp. 272-278, 1966. | MR | Zbl

[31] -, On discrete subgroups of the two by two projective linear group over $p$ -adic fields, J. Math. Soc. Japan, 18 (1966), pp. 219-235. | DOI | MR | Zbl

[32] M. Junge and G. Pisier, Bilinear forms on exact operator spaces and $B (H) \otimes B (H)$ , Geometric and Functional Analysis, 5 (1995), pp. 329-363. | DOI | EuDML | Zbl

[33] N. Katz, Estimates for Soto-Andrade sums, J. reine angew. Math., 438 (1993), pp. 143-161. | EuDML | MR | Zbl

[34] F. Lazebnik, V. Ustimenko, and A. Woldar, A new series of dense graphs of high girth, Bull. Amer. Math. Soc., 32 (1995), pp. 73-79. | DOI | MR | Zbl

[35] W. Li, Eigenvalues of Ramanujan graphs. Preprint, 1996. | Zbl

[36] -, Character sums and abelian Ramanujan graphs, J. Number Theory, 41 (1992), pp. 199-214. | DOI | MR

[37] -, Number theory with applications, World Scientific, 1996. | MR

[38] -, A survey of Ramanujan graphs, in Arithmetic, geometry and coding theory (R. Pellikaan, M. Perret, S.G. Vladut eds), W. De Gruyter, pp. 127-143, 1996. | MR | Zbl

[39] W. Li and P. Solé, Spectra of regular graphs and hypergraphs, and orthogonal polynomials, European J. Combinatorics, 17 (1996), pp. 461-477. | DOI | MR | Zbl

[40] B. J. R. Livne, Ramanujan local systems on finite graphs. Preprint, 1997.

[41] A. Lubotzky, Discrete groups, expanding graphs and invariant measures, Birkhäuser, 1994. | DOI | MR | Zbl

[42] A. Lubotzky, R. Phillips, and P. Sarnak, Hecke operators and distributing points on $S^{2}$ , I, Comm. pure and applied Math., 39 (1986), pp. 149-186. | DOI | MR | Zbl

[43] -, Ramanujan conjectures and explicit constructions of expanders, Proc. Symp. on Theo. of Comp. Sci. (STOC), 86 (1986), pp. 240-246.

[44] -, Ramanujan graphs, Combinatorica, 8 (1988), pp. 261-277. | DOI | MR | Zbl

[45] G. Margulis, Explicit construction of concentrators, Problems Inform. Transmission, 9 (1973), pp. 325-332. | MR | Zbl

[46] -, Explicit constructions of graphs without short cycles and low density codes, Combinatorica, 2 (1982), pp. 71-78. | DOI | MR | Zbl

[47] -, Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and concentrators, J. Problems of Information Transmission, 24 (1988), pp. 39-46. | MR | Zbl

[48] A. Medrano, P. Myers, H. Stark, and A. Terras, Finite analogues of Euclidean space, J. comput. applied maths., 68 (1996), pp. 221-238. | DOI | MR | Zbl

[49] J.-F. Mestre, La méthode des graphes. exemples et applications, in Proc. int. Conf. on class numbers and fund. units of alg. number fields, Katata, Japan, 217-242, 1986. | MR | Zbl

[50] T. Miyake, Modular forms, Springer, 1989. | DOI | MR | Zbl

[51] M. Morgenstern, Ramanujan graphs and diagrams : function field approach, in Expanding graphs, Amer. Math. Soc, DIMACS ser. 10, pp. 111-117, 1993. | MR | Zbl

[52] -, Existence and explicit construction of $q + 1$ regular Ramanujan graphs for every prime power $q$ , J. Combinatorial Theory, ser. B, 62 (1994), pp. 44-62. | DOI | MR | Zbl

[53] S. Mozes, A zero entropy, mixing of all orders tiling system, Contemporary Math., 135 (1992), pp. 319-325. | DOI | MR | Zbl

[54] A. Nevo and Y. Shalom, Explicit Kazhdan constants for representations of semisimple groups and their lattices. Preprint, 1996. | MR

[55] A. Nilli, On the second eigenvalue of a graph, Discrete Math., 91 (1991), pp. 207- 210. | DOI | MR | Zbl

[56] A. Ogg, Modular forms and Dirichlet series, Benjamin, 1969. | MR | Zbl

[57] P. Pansu, Formules de Matsushima, de Garland, et propriété $(T)$ pour des groupes agissant sur des espaces symétriques ou des immeubles. Preprint, 1995. | Numdam | MR | Zbl

[58] -, Sous-groupes discrets des groupes de Lie : rigidité, arithméticité, in Séminaire Bourbaki, exposé 778, Astérisque 227, pp. 69-105, 1995. | Numdam | MR | Zbl

[59] G. Pisier, Quadratic forms in unitary operators. A paraître dans Linear Algebra and Appl. | MR | Zbl

[60] -, Espaces d'opérateurs : une nouvelle dualité, in Séminaire Bourbaki, exposé 814, Février, 1996. | Numdam

[61] A. Pizer, Ramanujan graphs and Hecke operators, Bull. (New Ser.) Amer. Math. Soc., 23 (1990), pp. 127-137. | DOI | MR | Zbl

[62] P. Sarnak, Some applications of modular forms, Cambridge University Press, 1990. | DOI | MR | Zbl

[63] J.-P. Serre, Lettre à Winnie Li. 8 octobre 1990.

[64] -, Lettre à Winnie Li. 5 Novembre 1990.

[65] -, Arbres, amalgames, $S L_{2}$ , Astérisque 46, Soc. Math. France, 1977.

[66] -, Répartition asymptotique des valeurs propres de l'opérateur de Hecke Tp, J. Amer. Math. Soc., 10 (1997), pp. 75-102. | DOI | MR | Zbl

[67] J. Silverman, The arithmetic of elliptic curves, Springer, 1986. | DOI | MR | Zbl

[68] M. Takesaki, Theory of operator algebras I, Springer-Verlag, 1979. | DOI | MR | Zbl

[69] A. Terras, Survey of spectra of Laplacians on finite symmetric spaces, Experimental Maths., 5 (1996), pp. 15-32. | DOI | EuDML | MR | Zbl

[70] A. Valette, An application of Ramanujan graphs to $C *$ -algebra tensor products, II, in Sém. théorie spectrale et géométrie, pp. 105-107, Institut Fourier, Grenoble, 1996. | DOI | EuDML | Numdam | MR | Zbl

[71] -, An application of Ramanujan graphs to $C *$ -algebra tensor products, Discrete Math., 167 (1997), pp. 597-603. | MR | Zbl

[72] A. VenkovandA. Nitikin, The Selberg trace formula, Ramanujan graphs, and some problems of mathematical physics, St. Petersburg Math. J., 5 (1994), pp. 419- 484. | MR | Zbl

[73] M.-F. Vignéras, Arithmétique des algèbres de quaternions, Springer LNM 800, 1980. | MR | Zbl

[74] A. Weil, On some exponential sums, Proc. Nat. Acad. Sci. U.S.A., 34 (1948), pp. 204-207. | DOI | MR | Zbl

[75] -, Sur les courbes algébriques et les variétés qui s'en déduisent, Hermann, 1948. | Zbl

[76] A. Zuk, La propriété $(T)$ de Kazhdan pour les groupes agissant sur les polyèdres, C.R. Acad. Sci. Paris, sér. I, 323 (1996), pp. 453-458. | MR | Zbl