More spectral bounds on the clique and independence numbers
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.
Journal of Combinatorial Theory. Series B
Nikiforov, V. (2009). More spectral bounds on the clique and independence numbers. Journal of Combinatorial Theory. Series B, 99 (6), 819-826. https://doi.org/10.1016/j.jctb.2009.01.003