Eigenvalues, expanders and gaps between primes

We consider several problems regarding the eigenvalues of regular graphs; their connection with expansion and gaps between primes.; Using non-elementary methods, J.-P. Serre has proved several theorems regarding the extreme eigenvalues of regular graphs. In the first part, we present new and elementary proofs of some of Serre's results. We also discuss the eigenvalues of claw free regular graphs and answer a question of Linial.; In the second part, we improve a result of Greenberg regarding the behaviour of the extreme eigenvalues of irregular graphs.; The third part of the thesis is concerned with the Abelian Cayley graphs. We show that these graphs contain a large number of closed walks of even length. Using this result, we prove the nontrivial eigenvalues of Abelian Cayley graphs are large.; In the last part, we present a simple method of constructing new expanders from old. This method has connections with the study of gaps between consecutive primes. We show that for almost all the degrees, one can construct regular graphs with small nontrivial eigenvalues by modifying previous constructions of expanders.