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)α(G), and clique number ω(G)ω(G).(i) If μnμn is the smallest eigenvalue of G, thenω(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(nd+1−1)(lnd+1−μn¯−lnln(d+1)). For d sufficiently large this inequality is tight up to factor of 4 for almost all d-regular graphs.