Nonnegative realization of spectra having negative real parts

The nonnegative inverse eigenvalue problem (NIEP) is the problem of determining necessary and sufficient conditions for a list of complex numbers σ to be the spectrum of a nonnegative matrix. In this paper the problem is completely solved in the case when all numbers in the given list σ except for one (the Perron eigenvalue) have real parts smaller than or equal to zero.