Weighted Tutte-Grothendieck polynomials of graphs

TL;DR

This paper introduces weighted harmonic chromatic and Tutte-Grothendieck polynomials for graphs, explores their properties, and generalizes the relationship between these polynomials and matroid invariants, including categorification aspects.

Contribution

It presents a new framework for weighted harmonic polynomials of graphs and matroids, extending classical invariants and establishing a generalized recipe theorem.

Findings

Defined weighted harmonic chromatic polynomials for graphs

Established properties and constructions of weighted harmonic Tutte-Grothendieck invariants

Proposed a categorification perspective for these polynomials

Abstract

In this paper, we introduce the concept of the weighted (harmonic) chromatic polynomials of graphs and discuss some of its properties. We also present the notion of the weighted (harmonic) Tutte--Grothendieck polynomials of graphs and give a generalization of the recipe theorem between the harmonic Tutte--Grothendieck polynomials graphs and the harmonic Tutte polynomials of matroids. Moreover, we give some constructions of the weighted (harmonic) Tutte--Grothendieck invariants for graphs and the weighted (harmonic) Tutte invariants for matroids. Finally, we give a remark on the categorification of the harmonic chromatic polynomials of graphs and harmonic Tutte polynomials of matroids.