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    On the second eigenvalue and linear expansion of regular graphs.

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    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
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