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.