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, then ω (G) ≥ 1 + dn (n− d) (d− μn) . Equality holds if and only if G is a complete regular ω-partite graph. (ii) if μn is the smallest eigenvalue of the complement of G, and 2 ≤ d < n− 1, then α (G) > (