Eigenvalues and expansion of regular graphs

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 linear-sized subsets of k -regular Ramanujan graphs. We improve the lower bound on the expansion of Ramanujan graphs to approximately k /2. Moreover, we construct a family of 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 eigenvalue method. We also show an upper bound of roughly 1 + √k - 1 on the average degree of linear-sized induced subgraphs of Ramanujan graphs. This compares positively with the classical bound 2√k - 1. 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.