This is an interesting question! It seems that the corresponding problem even with "integer" replaced by "real" is hard (see http://www.jstor.org/pss/20490189, Inverse eigenvalue problems for matrices, T. Laffey), i.e.,  there are "further" inequalities satisfied by the eigenvalues of non-negative real matrices. I do not know what extra complexity is induced by passing to integers but I suspect it must be very hard to give exact conditions.