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 - (dn-4/d) for an odd prime, n.

• Type: Journal
• Year: 2006
• Where: JCT
• Authors: Joel Friedman, Ram Murty, Jean-Pierre Tillich
• Keywords: Abelian Group, D-regular Cayley Graph, JCT 2006, Short Proofs