Graph Hausdorff dimension, Kolmogorov complexity and construction of fractal graphs

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.