Eigenvalue distribution of families of regular graphs

Abstract

This thesis presents an exposition of a result of Serre about the asymptotic distribution of eigenvalues of families of regular graphs. This result is part of a paper published by Serre in 1997 titled \the equidistribution of eigenvalues of Hecke operators". Then, we discuss a speci c example of a family of Ramanujan graphs given by Lubotzky, Phillips and Sarnak in their 1988 paper on Ramanujan graphs, and calculate this limiting distribution measure of the eigenvalues of that family using Serre's result. We also give an alternate way of computing the measure using a result published by B.D.McKay in 1981 about the limiting distribution measure of the eigenvalues of a family of regular graphs satisying certain properties. We then discuss a similar result for a family of cycle graphs.