Spectral estimates for Abelian Cayley graphs

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Abstract

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.

Keywords

Abelian Cayley graphs
Eigenvalue bounds

Cited by (0)

1

Research supported in part by an NSERC grant.

2

Research partially supported by an NSERC grant.

3

Research supported in part by France Telecom.