I can possibly offer a counterexample, from James McKee and Chris Smyth's Integer symmetric matrices of small spectral radius and small Mahler measure.
If $P=x^7-8x^5+19x^3-12x+1$ were the characteristic polynomial of a matrix corresponding to a graph, then it would be the char.poly of a matrix corresponding to a charged signed graph (symmetric, all entries $0$,$1$ or $-1$). For such matrices we define the associated reciprocal polynomial to be $(z^d)X(z+1/z)$, where $X$ is the characteristic polynomial and d its degree. In this case, the associate reciprocal polynomial would be $z^14-z^12+z^7-z^2+1$. For any integer polynomial we can find a Mahler measure, and the Mahler measure of this polynomial is $1.20261\!\ldots$ However, Smyth and McKee determined the Mahler measures less than $1.3$ that arise from associated reciprocal polynomials of charged signed graphs, and this quantity is not attained.
So $P$ cannot be the characteristic polynomial of a charged signed graph, of which graphs are a special case. Does $P$ satisfy your non-negativity conditions on the roots? The sums of odd powers seem to be zero.
