Eigenvalue bounds for independent sets

We derive bounds on the size of an independent set based on eigenvalues. This generalizes a result due to Delsarte and Hoffman. We use this to obtain new bounds on the independence number of the Erd\H{o}s-R'{e}nyi graphs. We investigate further properties of our bounds, and show how our results on the Erd\H{o}s-R'{e}nyi graphs can be extended to other polarity graphs.