Ramanujan graphs and zeta functions

This is largely a survey paper on the theory of Ramanujan graphs. It also includes a simplified proof of the celebrated Alon-Bopanna theorem and discusses the future direction of the theory of zeta functions of regular graphs.

[1]  Maria M. Klawe,et al.  Limitations on Explicit Constructions of Expanding Graphs , 1984, SIAM J. Comput..

[2]  P. Diaconis,et al.  Generating a random permutation with random transpositions , 1981 .

[3]  K. Hashimoto.  Zeta functions of finite graphs and representations of p-adic groups , 1989 .

[4]  F. Chung.  Diameters and eigenvalues , 1989 .

[5]  V. Drinfel'd,et al.  The proof of peterson's conjecture for GL(2) over a global field of characteristic p , 1988 .

[6]  A. Terras,et al.  Zeta functions of finite graphs and coverings, III , 1996 .

[7]  Moshe Morgenstern,et al.  Existence and Explicit Constructions of q + 1 Regular Ramanujan Graphs for Every Prime Power q , 1994, J. Comb. Theory, Ser. B.

[8]  Heinz Adolf Jung,et al.  The connectivities of locally finite primitive graphs , 1989, Comb..

[9]  Alexander Lubotzky,et al.  Discrete groups, expanding graphs and invariant measures , 1994, Progress in mathematics.

[10]  Giuliana P. Davidoff,et al.  Elementary number theory, group theory, and Ramanujan graphs , 2003 .

[11]  László Babai,et al.  Spectra of Cayley graphs , 1979, J. Comb. Theory B.

[12]  W. Li,et al.  Number theory with applications , 1996 .

[13]  Y. Ihara.  On discrete subgroups of the two by two projective linear group over p-adic fields , 1966 .

[14]  Noga Alon,et al.  Random Cayley Graphs and Expanders , 1994, Random Struct. Algorithms.

[15]  P. Deligne.  La conjecture de Weil. I , 1974 .

[16]  Yves Colin de Verdière,et al.  On a new graph invariant and a criterion for planarity , 1991, Graph Structure Theory.