Sharp bounds on the spectral radius of nonnegative matrices and digraphs

The spectral radius ρ(A) of a matrix A is the maximum modulus of its eigenvalues. We present bounds on ρ(A) that are often tighter and are applicable to a larger class of nonnegative matrices than previously reported. The bounds are particularly suited to matrices which are sparse.We complete the paper by applying these bounds to digraphs, deriving the associated equality conditions which relate to the outdegree regularity of the digraph. Finally, we show that the equality conditions may be achieved only for very specific values of the digraph’s spectral radius.