  Let $G$ denote a bipartite graph with $e$ edges without isolated vertices. It
was known that the spectral radius of $G$ is at most the square root of $e$,
and the upper bound is attained if and only if $G$ is a complete bipartite
graph. Suppose that $G$ is not a complete bipartite graph, and $e-1$ and $e+1$
are not twin primes. We determine the maximal spectral radius of $G$. As a
byproduct of our study, we obtain a spectral characterization of a pair $(e-1,
e+1)$ of integers to be a pair of twin primes.
