The Equivalence of Two Graph Polynomials and a Symmetric Function

TL;DR

This paper demonstrates that the U-polynomial, polychromate, and Stanley's symmetric function generalization of the Tutte polynomial are equivalent, and extends this equivalence to their generalized forms, addressing a question by Welsh.

Contribution

It proves the equivalence of these graph polynomials and their generalizations, providing a unified framework and answering an open question in the field.

Findings

The three graph polynomials are equivalent in coefficients.

The equivalence extends to their generalized forms.

Addresses Welsh's question on the generalization of these functions.

Abstract

The U-polynomial, the polychromate and the symmetric function generalization of the Tutte polynomial due to Stanley are known to be equivalent in the sense that the coefficients of any one of them can be obtained as a function of the coefficients of any other. The definition of each of these functions suggests a natural way in which to generalize them which also captures Tutte's universal V-functions as a specialization. We show that the equivalence remains true for the extended functions thus answering a question raised by Dominic Welsh.