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. be a graph of order n, average degree d, independence number α ( G ) , and clique number ω ( G ) . μ n is the smallest eigenvalue of G, then ω ( G ) ⩾ 1 + d n ( n − d ) ( d − μ n ) . Equality holds if and only if G is a complete regular ω-partite graph. f μ n ¯ is the smallest eigenvalue of the complement of G, and 2 ⩽ d < n − 1 , then α ( G ) > ( n d + 1 − 1 ) ( ln d + 1 − μ n ¯ − ln ln ( d + 1 ) ) . For d sufficiently large this inequality is tight up to factor of 4 for almost all d-regular graphs.