Subgraph Entropy

TL;DR

This paper proves a lower bound on the metric entropy of proper subgraphs in metric graphs of fixed rank, connecting graph theory with hyperbolic geometry and the pressure metric on Outer Space.

Contribution

It establishes a new entropy lower bound for subgraphs of metric graphs of fixed rank, answering a previously open question.

Findings

Existence of a positive entropy lower bound for subgraphs of metric graphs

Connection between graph entropy and hyperbolic geometry concepts

Interpretation of results via the Bers Lemma and pressure metric

Abstract

Given $r \geq 3$, we prove that there exists $\lambda >0$ depending only on $r$ so that if $G$ is a metric graph of rank $r$ with metric entropy $1$, then there exists a proper subgraph $H$ of $G$ with metric entropy at least $\lambda$. This answers a question of the second two authors together with Rieck. We interpret this as a graph theoretic version of the Bers Lemma from hyperbolic geometry, and explain some connections to the pressure metric on the Culler-Vogtmann Outer Space.