Spectral estimates for Abelian Cayley graphs

We give two short proofs that for fixed d, a d-regular Cayley graph on an Abelian group of order n has second eigenvalue bounded below by d - O(dn-4/d), where the implied constant is absolute. We estimate the constant in the O(dn-4/d) notation. We show that for any fixed d, then for a large odd prime, n, the O(dn-4/d) cannot be improved; more precisely, most d-regular graphs on prime n vertices have second eigenvalue at most d - Ω(dn-4/d) for an odd prime, n.

[1]  J. Friedman.  Some geometric aspects of graphs and their eigenfunctions , 1993 .

[2]  Joel Friedman,et al.  Generalized Alon--Boppana Theorems and Error-Correcting Codes , 2005, SIAM J. Discret. Math..

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

[4]  G. Nemes.  Asymptotic Expansions of Integrals , 2004 .

[5]  W. Fischer,et al.  Sphere Packings, Lattices and Groups , 1990 .

[6]  Maria M. Klawe,et al.  Non-existence of one-dimensional expanding graphs , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

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