"More spectral bounds on the clique and independence numbers" by Vladimir Nikiforov
 

More spectral bounds on the clique and independence numbers

Abstract

We give some new bounds for the clique and independence numbers of a graph in terms of its eigenvalues. In particular we prove the following results. Let G be a graph of order n, average degree d, independence number α (G), and clique number ω (G). (i) If μn is the smallest eigenvalue of G, thenω (G) ≥ 1 + frac(d n, (n - d) (d - μn)) . Equality holds if and only if G is a complete regular ω-partite graph. (ii) if over(μn, -) is the smallest eigenvalue of the complement of G, and 2 ≤ d < n - 1, thenα (G) > (frac(n, d + 1) - 1) (ln frac(d + 1, - over(μn, -)) - ln ln (d + 1)) . For d sufficiently large this inequality is tight up to factor of 4 for almost all d-regular graphs. © 2009 Elsevier Inc. All rights reserved.

Publication Title

Journal of Combinatorial Theory. Series B

Plum Print visual indicator of research metrics
PlumX Metrics
  • Citations
    • Citation Indexes: 39
  • Usage
    • Abstract Views: 4
  • Captures
    • Readers: 13
see details

Share

COinS