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

From Alon and Boppana, and Serre, we know that for any given integer $k\geq 3$andrealnumber$\lambda<2\sqrt{k-1}$,therearefinitelymany$k$-regular graphs whose second largest eigenvalue is at most $\lambda$. In this paper, we investigate the largest number of vertices of such graphs.