TITLE: When is a monic integer polynomial the characteristic polynomial of a non-negative integer matrix?
QUESTION [27 upvotes]: Suppose $P(x)$ is a monic integer polynomial with roots $r_1, ... r_n$ such that $p_k = r_1^k + ... + r_n^k$ is a non-negative integer for all positive integers $k$.  Is $P(x)$ necessarily the characteristic polynomial of a non-negative integer matrix?
(The motivation here is that I want $r_1, ... r_n$ to be the eigenvalues of a directed multigraph.)
Edit:  If that condition isn't strong enough, how about the additional condition that 
$$\frac{1}{n} \sum_{d | n} \mu(d) p_{n/d}$$
is a non-negative integer for all d?

REPLY [33 votes]: This question is completely answered, and the result is that the condition involving
the Moebius inversion you mention is both necessary and sufficient! See

K. H. Kim, N. Ormes, F. Roush. The spectra of nonnegative integer matrices via formal power series, J. Amer. Math. Soc. 13 (2000) 773–806. https://doi.org/10.1090/S0894-0347-00-00342-8

This is really a remarkable and
beautiful theorem.