Tutte polynomial of a fractal scale-free lattice

TL;DR

This paper derives recursive formulas and exact expressions for the Tutte polynomial of a self-similar, scale-free lattice, linking combinatorics and statistical physics.

Contribution

It introduces a recursive method to compute the Tutte polynomial for an infinite family of fractal scale-free lattices, providing exact solutions at specific points.

Findings

Recursive formulas for Tutte polynomials of the lattice

Exact analytical expressions at special points

Link between graph invariants and physical models

Abstract

The Tutte polynomial of a graph, or equivalently the $q$-state Potts model partition function, is a two-variable polynomial graph invariant of considerable importance in both combinatorics and statistical physics. The computation of this invariant for a graph is NP-hard in general. In this paper, based on their self-similar structures, we recursively describe the Tutte polynomials of an infinite family of scale-free lattices. Furthermore, we give some exact analytical expressions of the Tutte polynomial for several special points at $(X,Y)$-plane.