A spectral approach to Szemerédi’s Regularity Lemma

Abstract

Szemerédi s Regularity Lemma says that for any graph there is a partition of the vertices into a bounded number of parts such that edges between most different parts behave almost randomly. Recently, Tao gave a spectral version of the Regularity Lemma which originated on work of Frieze and Kannan which applies to self adjoint operators. Its application to adjacency matrices provides a spectral proof of Szemerédi s Regularity Lemma. This thesis has two main purposes. The first one is to discuss in detail the spectral proof and the decomposition of the adjacency matrix used to describe the partition. The second one is to study the natural extension of the notion of regularity and the Regularity Lemma itself for self adjoint matrices. The associated Counting and Removal Lemmas are also discussed.