Note: Extreme eigenvalues of nonregular graphs

  • Authors:

  • Sebastian M. Cioabă;David A. Gregory;Vladimir Nikiforov

  • Affiliations:

  • Department of Mathematics, University of California, San Diego, La Jolla, CA 92093-0112, USA;Department of Mathematics, Queen's University at Kingston, Ontario, K7L 3N6, Canada;Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2007

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Abstract

Let @l"1 be the greatest eigenvalue and @l"n the least eigenvalue of the adjacency matrix of a connected graph G with n vertices, m edges and diameter D. We prove that if G is nonregular, then@D-@l"1n@D-2mn(D(n@D-2m)+1)=1n(D+1), where @D is the maximum degree of G. The inequality improves previous bounds of Stevanovic and of Zhang. It also implies that a lower bound on @l"n obtained by Alon and Sudakov for (possibly regular) connected nonbipartite graphs also holds for connected nonregular graphs.