Let $b(k,\ell,θ)$ be the maximum number of vertices of valency $k$ in a $(k,\ell)$-semiregular bipartite graph with second largest eigenvalue $θ$. We obtain an upper bound for $b(k,\ell,θ)$ for $0 < θ< \sqrt{k-1} + \sqrt{\ell-1}$. 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.

25 pages