Ramanujan graphs

Combinatorica - Tập 8 Số 3 - Trang 261-277 - 1988
Alexander Lubotzky1, Ralph S. Phillips2, Peter Sarnak2
1Institute of Mathematics and Computer Science, Hebrew University, Jerusalem, Israel
2Department of Mathematics, Stanford University, 94305, Stanford, California, USA

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Tài liệu tham khảo

N.Alon,Private communication 1986.

N. Alon, Eigenvalues, geometric expanders, sorting in rounds and Ramsey theory,Combinatorica,6 (1986), 207–219.

N. Alon, Eigenvalues and expanders,Combinatorica,6 (1986), 83–96.

B. Bollobás,Extremal graph theory, Academic Press, London 1978.

L. E. Dickson, Arithmetic of quaternions,Proc. London Math. Soc. (2)20 (1922), 225–232.

M. Eichler, Quaternäre quadratische Formen und die Riemannsche Vermutung für die kongruentz Zeta Funktion,Archiv. der Math. Vol. V, (1954), 355–366.

P. Erdős andH. Sachs, Reguläre Graphen gegenebener Teillenweite mit Minimaler Knotenzahl, Wiss. Z. Univ. Halle-Wittenberg,Math. Nat. R. 12 (1963), 251–258.

L.Gerritzen and N.Van der Put,Schottky groups and Mumford curves, Springer-Verlag, L. N. in Math. 817 (1980).

G.Hardy and E.Wright,An introduction to number theory, Oxford University Press 1978 (Fifth Edition).

E.Hecke, Analytische arithmetik der positiven quadratic formen,Collected works pp. 789–898, Göttingen, 1959.

A.Hofmann, On eigenvalues and colorings of graphs,in Graph theory and its applications (ed. B. Harris) Academic Press (1970), 79–91.

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

Y.Ihara, Discrete subgroups of PL (2, k p ),Proc. Symp. in Pure Math. IX, AMS (1968), 272–278.

W. Imrich, Explicit construction of regular graphs with no small cycles,Combinatorica 4 (1984), 53–59.

H. Kesten, Symmetric random walks on groups,Trans. AMS 92 (1959), 336–354.

M. Knesser, Strong approximation in:Algebraic Groups and Discontinuous Subgroups, Proc. Symp. Pure Math. Vol. IX, (1966), 187–196.

A. Lubotzky, R. Phillips, P. Sarnak, Ramanujan conjecture and explicit construction of expanders,Proc. Stoc. 86 (1986), 240–246.

A. Lubotzky, R. Phillips, P. Sarnak, Hecke operators and distributing points onS 2 I, IIComm. Pure and Applied Math. 39 (1986), 149–186,40 (1987), 401–420.

G. A. Margulis, Graphs without short cycles,Combinatorica 2 (1982), 71–78.

Mališev, On the representation of integers by positive definite forms,Mat. Steklov 65 (1962).

A. Ogg,Modular forms and Dirichlet series, W. A. Benjamin Inc., New York 1969.

S. Ramanujan, On certain arithmetical functions,Trans. Camb. Phil. Soc. 22 (1916), 159–184.

J. P. Serre,Trees, Springer Verlag, Berlin-Heidelberg-New York, (1980).

M. F.Vignéras,Arithmetique dè Algebras de Quaternions, Springer Lecture Notes; V. 800, (1980).

G. L. Watson, Quadratic diophantine equations,Royal Soc. of London, Phil. Trans., A 253, 227–2 (1960).

A.Weil, Sur les courbes algébriques et les varétés qui s'en déduisent,Actualites Sci. Et ind. No. 1041 (1948).

A. Weiss, Girths of bipartite sextet graphs,Combinatorica 4 (1984), 241–245.