Spectral estimates for Abelian Cayley graphs

  • Authors:

  • Joel Friedman;Ram Murty;Jean-Pierre Tillich

  • Affiliations:

  • Department of Computer Science, University of British Columbia, Vancouver, BC, Canada and Department of Mathematics, University of British Columbia, Vancouver, BC, Canada;Department of Mathematics and Statistics, Queen's University, Kingston, Ontario, Canada;Inria Projet Codes, Domaine de Voluceau BP., France

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2006

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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.