Shannon capacity and the Lovász Theta function

Abstract

This thesis focuses on the study of two graph parameters known as the Shannon capacity and the Lovász number. The first was introduced by Claude Elwood Shannon on \cite{shannon} and describes the maximum rate at which information can be transmitted through a noisy channel of comunication, where the noise of the channel is encoded by a graph. The second one was introduced by László Lovász as an attempt to determine the first. We start by introducing all the required concepts in order to formally state the problem by Shannon. We then move on to develop some theory regarding the spectrum of a set of matrices associated with graphs, providing an algebraic aproach to the matter in question. Then we prove some results due to Lovász on \cite{Lovasz} that allow us to effectively compute the Shannon capacity for specific families of graphs, including the Kneser graphs and the perfect graphs. We end up by studying further properties of the Lovász number, of particular interest for the case of perfect graphs.