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?