The Tutte Polynomial of the Schreier graphs of the Grigorchuk group and the Basilica group

TL;DR

This paper computes the Tutte polynomial for Schreier graphs of the Grigorchuk and Basilica groups, revealing their combinatorial structures and special properties through explicit evaluations.

Contribution

It provides the first detailed description and computation of the Tutte polynomial for these two families of infinite graphs associated with self-similar groups.

Findings

Explicit formulas for the Tutte polynomial of Schreier graphs

Evaluation of the Tutte polynomial at key points

Insights into the combinatorial structure of the graphs

Abstract

We study the Tutte polynomial of two infinite families of finite graphs. These are the Schreier graphs associated with the action of two well-known self-similar groups acting on the binary rooted tree by automorphisms: the first Grigorchuk group of intermediate growth, and the iterated monodromy group of the complex polynomial $z^2-1$ known as the Basilica group. For both of them, we describe the Tutte polynomial and we compute several special evaluations of it, giving further information about the combinatorial structure of these graphs.