Let D be a digraph with vertex set V (D) and A be the adjacency matrix of D. The largest
eigenvalue of A, denoted by ρ (D), is called the spectral radius of the digraph D. In this
paper, we establish some sharp upper or lower bounds for digraphs with some given graph
parameters, such as clique number, girth, and vertex connectivity, and characterize the
corresponding extremal graphs. In addition, we give the exact value of the spectral radii of
those 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 …