On the second eigenvalue and linear expansion of regular graphs.

Abstract. The spectral method is the best currently known technique to prove lower bounds on expansion. Ramanujan graphs, which have asymptotically optimal second eigenvalue, are the best-known explicit expanders. The spectral method yielded a lower bound of k\4 on the expansion of Iinear-sized subsets of k-regular Ramanujan graphs. We improve the lower bound ontheexpansion of Ramanujan graphs to approximately k/2, Moreover. we construct afamilyof k-regular graphs with asymptotically optimal second eigenvalue and linear expansion equal to k/2. This shows that k/2 is the best bound one can obtain using the second eigenwdue method. We also show an upper bound of roughly 1 +~on the average degree of linear-sized induced subgraphs of Ramanujan graphs. This compares positively with the classical bound 2~. As a byproduct, we obtain improved results on random walks on expanders and construct selection networks (respectively, extrovert graphs) of smaller size (respectively, degree) than was previously known