On the rooted Tutte polynomial

TL;DR

This paper introduces the rooted Tutte polynomial, extending the classical Tutte polynomial to graphs with prescribed vertex colors, and explores its properties, duality relations, and connections to the Potts model.

Contribution

It presents the definition of the rooted Tutte polynomial and establishes new duality relations and connections to statistical physics models.

Findings

Duality relation for rooted Tutte polynomial in planar graphs

Connection between rooted Tutte polynomial and Potts model

Extension of Tutte polynomial to graphs with prescribed vertex colors

Abstract

The Tutte polynomial is a generalization of the chromatic polynomial of graph colorings. Here we present an extension called the rooted Tutte polynomial, which is defined on a graph where one or more vertices are colored with prescribed colors. We establish a number of results pertaining to the rooted Tutte polynomial, including a duality relation in the case that all roots reside around a single face of a planar graph. The connection with the Potts model is also reviewed.