  In this paper, we obtain the sharp upper and lower bounds for the spectral
radius of a nonnegative irreducible matrix. We also apply these bounds to
various matrices associated with a graph or a digraph, obtain some new results
or known results about various spectral radii, including the adjacency spectral
radius, the signless Laplacian spectral radius, the distance spectral radius,
the distance signless Laplacian spectral radius of a graph or a digraph.

Combinatorics