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On spectral radius of strongly connected digraphs

We determine the digraphs which achieve the second, the third and the fourth minimum spectral radii respectively among strongly connected digraphs of order $n\ge 4$, and thus we answer affirmatively the problem whether the unique digraph which achieves the minimum spectral radius among all strongly connected bicyclic digraphs of order $n$ achieves the second minimum spectral radius among all strongly connected digraphs of order $n$ for $n\ge 4$ proposed in [H. Lin, J. Shu, A note on the spectral characterization of strongly connected bicyclic digraphs, Linear Algebra Appl. 436 (2012) 2524--2530]. We also discuss the strongly connected bicyclic digraphs with small and large spectral radii respectively.