  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.
