Weighted spanning trees on some self-similar graphs

TL;DR

This paper calculates the number of spanning trees in specific self-similar graphs, including Sierpiński graphs and Schreier graphs of the Hanoi Towers group, using weighted generating functions.

Contribution

It provides explicit computations of spanning tree complexities for two families of self-similar graphs, expanding understanding of their combinatorial structures.

Findings

Derived formulas for spanning tree counts in Sierpiński graphs.

Computed weighted generating functions for Schreier graphs of the Hanoi Towers group.

Enhanced understanding of graph complexity in self-similar structures.

Abstract

We compute the complexity of two infinite families of finite graphs: the Sierpi\'{n}ski graphs, which are finite approximations of the well-known Sierpi\'nsky gasket, and the Schreier graphs of the Hanoi Towers group $H^{(3)}$ acting on the rooted ternary tree. For both of them, we study the weighted generating functions of the spanning trees, associated with several natural labellings of the edge sets.