  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.
