Characteristic polynomials, associated to the energy graph of the non-linear Schrodinger equation

We study the irreducibility and the separation of characteristic polynomials, associated to the energy graph of the non-linear Schrodinger equation. This fact will be useful in the study of stability of a class of normal forms of the completely resonant non-linear Schrodinger equation on a torus, described in a recent work by M. Procesi and C. Procesi. The problem can be also considered as an independent interesting algebraic combinatorial problem.