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, theO(dn−4/d ) cannot be improved; more precisely, most d-regular graphs on prime n vertices have second eigenvalue at most d − (dn−4/d ) for an odd prime, n. © 2005 Elsevier Inc. All rights reserved.