  The spectral radius {\rho}(G) of a digraph G is the maximum modulus of the
eigenvalues of its adjacency matrix. We present bounds on {\rho}(G) that are
often tighter and are applicable to a larger class of digraphs than previously
reported bounds. Calculating the final bound pair is particularly suited to
sparse digraphs.
  For strongly connected digraphs, we derive equality conditions for the
bounds, relating to the outdegree regularity of the digraph. We also prove that
the bounds hold with equality only if {\rho}(G) is the r-th root of an integer,
where r divides the index of imprimitivity of G.
