  For integer q>1, we derive edge q-colouring models for (i) the Tutte
polynomial of a graph G on the hyperbola H_q, (ii) the symmetric weight
enumerator of the set of group-valued q-flows of G, and (iii) a more general
vertex colouring model partition function that includes these polynomials and
the principal specialization order q of Stanley's symmetric monochrome
polynomial. In the second half of the paper we exhibit a family of
non-symmetric edge q-colouring models defined on k-regular graphs, whose
partition functions for q >= k each evaluate the number of proper edge
k-colourings of G when G is Pfaffian.
