  Let $b(k,\ell,\theta)$ be the maximum number of vertices of valency $k$ in a
$(k,\ell)$-semiregular bipartite graph with second largest eigenvalue $\theta$.
We obtain an upper bound for $b(k,\ell,\theta)$ for $0 < \theta < \sqrt{k-1} +
\sqrt{\ell-1}$. This bound is tight when there exists a distance-biregular
graph with particular parameters, and we develop the necessary properties of
distance-biregular graphs to prove this.
