  With a main tool is signed graphs, we give a full description of the
characteristic quasi-polynomials of ideals of classical root systems ($ABCD$)
with respect to the integer and root lattices. As a result, we obtain a full
description of the characteristic polynomials of the toric arrangements defined
by these ideals. As an application, we provide a combinatorial verification to
the fact that the characteristic polynomial of every ideal subarrangement
factors over the dual partition of the ideal in the classical cases.
