  We introduce two notions of quandle polynomials for G-families of quandles:
the quandle polynomial of the associated quandle and a G-family polynomial with
coefficients in the group ring of G. As an application we define image
subquandle polynomial enhancements of the G-family counting invariant for
trivalent spatial graphs and handlebody-links. We provide examples to show that
the new enhancements are proper.
