Lower bound of the asymptotic complexity of self-similar fractal graphs

TL;DR

This paper investigates the asymptotic complexity of self-similar fractal graphs, establishing a lower bound and confirming the existence of the asymptotic complexity constant for symmetric structures, thus addressing prior conjectures.

Contribution

It proves the existence of the asymptotic complexity constant for symmetric fractal graphs and provides a sharp lower bound, resolving two conjectures by Anema.

Findings

Existence of the asymptotic complexity constant for symmetric self-similar fractal graphs

A sharp lower bound for the complexity constant

Resolution of two conjectures by Anema

Abstract

We study the asymptotic complexity constant of the sequence of approximating graphs to a fully symmetric self-similar structure on a finitely ramified fractal $K$. We show how full symmetry implies existence of the asymptotic complexity constant and obtain a sharp lower bound thereby answering two conjectures by Anema.