Graph Hausdorff dimension, Kolmogorov complexity and construction of fractal graphs

TL;DR

This paper introduces discrete analogues of Lebesgue and Hausdorff dimensions for graphs, linking fractal properties to graph characteristics and Kolmogorov complexity, and explores their implications across graph theory and metric spaces.

Contribution

It defines fractal graphs using new dimension concepts and connects these to Kolmogorov complexity, advancing interdisciplinary understanding of graph fractality.

Findings

Established formal definitions of fractal graphs.

Linked Hausdorff dimension to Kolmogorov complexity.

Estimated Prague dimension for most graphs using information theory.

Abstract

In this paper we introduce and study discrete analogues of Lebesgue and Hausdorff dimensions for graphs. It turned out that they are closely related to well-known graph characteristics such as rank dimension and Prague (or Ne\v{s}et\v{r}il-R\"odl) dimension. It allows us to formally define fractal graphs and establish fractality of some graph classes. We show, how Hausdorff dimension of graphs is related to their Kolmogorov complexity. We also demonstrate fruitfulness of this interdisciplinary approach by discovering a novel property of general compact metric spaces using ideas from hypergraphs theory and by proving an estimation for Prague dimension of almost all graphs using methods from algorithmic information theory.