On the order of regular graphs with fixed second largest eigenvalue

Let $v(k,\lambda )$ be the maximum number of vertices of a connected $k$-regular graph with second largest eigenvalue at most $\lambda$. The Alon-Boppana Theorem implies that $v(k,\lambda )$ is finite when $k > \frac{\lambda^2 + 4}{4}$\mathrm{.}Inthispaper,weshowthatforfixed$\lambda \geq1$,thereexistsaconstant$C(\lambda)$suchthat$2k+2 \leq v(k, \lambda) \leq 2k + C(\lambda)$when$k > \frac{\lambda^2 + 4}{4}$.