Spectral bounds for the clique and independence numbers of graphs

We obtain a sequence k,(G) Q k,(G) < *. . < k,(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 k,(G) is explicit and improves earlier known theorems. The bound k,(G) is also explicit, and is shown to improve on the bound from Brooks’ theorem even for regular graphs. The bounds k 3 Ye.*, k, are polynomial-time computable, where r is the number of positive eigenvalues of G.