In this paper, we study the spectral radius of bipartite graphs. Let $G$ be 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 $(e-1,e+1)$ is not a pair of twin primes. We describe 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.
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