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.

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), 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.