Spectral bounds for the clique and independence numbers of graphs

We obtain a sequence k1(G) ≤ k2(G) ≤ … ≤ kn(G) of lower bounds for the clique number (size of the largest clique) of a graph G of n vertices. The bounds involve the spectrum of the adjacency matrix of G. The bound k1(G) is explicit and improves earlier known theorems. The bound k2(G) is also explicit, and is shown to improve on the bound from Brooks' theorem even for regular graphs. The bounds k3,…, kr are polynomial-time computable, where r is the number of positive eigenvalues of G.