Centrosymmetric nonnegative realization of spectra

A list $\Lambda =\{\lambda _{1},\lambda _{2},\ldots ,\lambda _{n}\}$ of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix. In this paper we intent to characterize those lists of complex numbers, which are realizable by a centrosymmetric nonnegative matrix. In particular, we show that lists of nonnegative real numbers, and lists of complex numbers of Suleimanova type (except in one particular case), are always the spectrum of some centrosymmetric nonnegative matrix. For the general lists we give sufficient conditions via a perturbation result. We also show that for $n=4,$ every realizable list of real numbers is also realizable by a nonnegative centrosymmetric matrix.