Question

Let me ask a more specific question.

Suppose P(x) $P(x)$ is a monic integer polynomial with roots r₁, r_{2$r_1, ... rnr_n$ such that pk$p_k = r1^k + r2^{kr_1^k + ... + rnkr_n^k$ is a non-negative integer for all k. positive integers $k$. Is P(x) $P(x)$ necessarily the characteristic polynomial of a non-negative integer matrix?}}

(The motivation here is that I want r₁, $r_1, ... r_{nr_n$ to be the eigenvalues of a directed multigraph.)}

Edit: If that condition isn't strong enough, how about the additional condition that $$\frac{1}{n} \sum_{d sum_{d | n} μ(d) p_{n/d} \mu(d) p_{n/d}$$}

is a non-negative integer for all d?