An extending result on spectral radius of bipartite graphs

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.