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.

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.

I can possibly offer a counterexample, from here .

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... 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.

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.