Geodesic
The structure of the characteristic polynomials of matrices with
Izvestiya. Mathematics , Tome 71 (2007) no. 1, pp. 3-13

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We obtain conditions on a real polynomial necessary and sufficient for this polynomial to be the characteristic polynomial of a matrix with non-negative elements. 
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Yu. S. Kolesov. The structure of the characteristic polynomials of matrices with. Izvestiya. Mathematics , Tome 71 (2007) no. 1, pp. 3-13. http://geodesic.mathdoc.fr/item/IM2_2007_71_1_a1/