Weighted Tutte-Grothendieck Polynomials of Graphs

In this paper, we introduce the notion of weighted chromatic polynomials of a graph associated to a weight function f of a certain degree, and discuss some of its properties. As a generalization of this concept, we define the weighted Tutte-Grothendieck polynomials of graphs. When f is harmonic, we notice that there is a correspondence between the weighted Tutte-Grothendieck polynomials of graphs and the weighted Tutte polynomials of matroids. Moreover, we present some constructions of the weighted Tutte-Grothendieck invariants for graphs as well as the weighted Tutte invariants for matroids. Finally, we give a remark on the categorification of the weighted chromatic polynomials of graphs and weighted Tutte polynomials of matroids.