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From Alon and Boppana, and Serre, we know that for any given integer $k\geq3$ and real number $\lambda<2\sqrt{k-1}$, there are finitely many $k$-regulargraphs whose second largest eigenvalue is at most $\lambda$. In this paper, weinvestigate the largest number of vertices of such graphs.

Maximizing the order of a regular graph of given valency and second eigenvalue