This is an interesting question! It seems that the corresponding problem even with "integer" replaced by "real" is hard (see Thomas J. Laffey, _Inverse Eigenvalue Problems for Matrices_, [JSTOR](https://www.jstor.org/stable/20490189)), 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.