6 deleted 136 characters in body

Let me ask a more specific question.

Suppose P(x) $P(x)$ is a monic integer polynomial with roots r1, r2$r_1, ... rnr_n$ such that pk$p_k = r1k + r2kr_1^k + ... + rnkr_n^k$ 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, $r_1, ... rnr_n$ to be the eigenvalues of a directed multigraph.)

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

is a non-negative integer for all d?

5 edited tags
4 Changed wording.

Let me ask a more specific question. Suppose P(x) is a monic integer polynomial with roots r1, r2 ... rn such that pk = r1k + r2k + ... + rnk is a non-negative integer for all k. Is P(x) necessarily the characteristic polynomial of a non-negative integer matrix?

(The motivation here is that I want r1, ... rn to be the eigenvalues of a graph.directed multigraph.)

Edit: If that condition isn't strong enough, how about the additional condition that \sumd | n μ(d) pn/d is a non-negative integer for all d?

3 Fixed wording.