Chromatic Polynomial, Colered Jones Function and q-Binomial Counting

We define a q-chromatic function on graphs, list some of its properties and provide some formulas in the class of general chordal graphs. Then we relate the q-chromatic function to the colored Jones function of knots. This leads to a curious expression of the colored Jones function of a knot diagram as a 'defected chromatic operator' applied to a power series whose coefficients are linear combinations of chord diagrams constructed from 'flows' on the reduced knot diagram.