Spectral bounds for the clique and independence numbers of graphs
Journal of Combinatorial Theory Series B
On the independence number of random graphs
Discrete Mathematics
On the independence and chromatic numbers of random regular graphs
Journal of Combinatorial Theory Series B
Some Inequalities for the Largest Eigenvalue of a Graph
Combinatorics, Probability and Computing
Laplacian spectral bounds for clique and independence numbers of graphs
Journal of Combinatorial Theory Series B
Eigenvalue bounds for independent sets
Journal of Combinatorial Theory Series B
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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 @a(G), and clique number @w(G). (i) If @m"n is the smallest eigenvalue of G, then@w(G)=1+dn(n-d)(d-@m"n). Equality holds if and only if G is a complete regular @w-partite graph. (ii) if @m"n@? is the smallest eigenvalue of the complement of G, and 2=(nd+1-1)(lnd+1-@m"n@?-lnln(d+1)). For d sufficiently large this inequality is tight up to factor of 4 for almost all d-regular graphs.