A finite Hausdorff dimension for graphs

TL;DR

This paper introduces a new finite Hausdorff dimension for graphs, providing a non-trivial measure that distinguishes different graph structures beyond classical zero-dimensional metrics.

Contribution

It extends the concept of finite Hausdorff dimension to connected graphs, exploring its behavior in coarse and fine metric cases.

Findings

Finite Hausdorff dimension is non-trivial for graphs.

Different cases show varied dimension behaviors.

Application bridges graph theory and metric space analysis.

Abstract

The classical Hausdorff dimension of finite or countable metric spaces is zero. Recently, we defined a variant, called \emph{finite Hausdorff dimension}, which is not necessarily trivial on finite metric spaces. In this paper we apply this to connected simple graphs, a class that provides many interesting examples of finite metric spaces. There are two very different cases: one in which the distance is coarse (and one is doing Graph Theory), and another case in which the distance is much finer (and one is somewhere between graphs and finite metric spaces).