Isomorphism classification of infinite Sierpinski carpet graphs

TL;DR

This paper classifies the isomorphism types of infinite Sierpinski carpet graphs constructed from infinite words, revealing uncountably many distinct classes.

Contribution

It introduces a method to generate and classify infinite Sierpinski carpet graphs based on infinite words, demonstrating the uncountable diversity of their isomorphism classes.

Findings

Uncountably many isomorphism classes of the limit graphs

A construction linking infinite words to graph structures

Insights into the diversity of fractal-like graph limits

Abstract

For each infinite word over a given finite alphabet, we define an increasing sequence of rooted finite graphs, that can be thought as approximations of the famous Sierpinski carpet. These sequences naturally converge to an infinite rooted limit graph. We show that there are uncountably many classes of isomorphism of such limit graphs, regarded as unrooted graphs.