Upper bounds of orders of automorphism groups of leafless metric graphs

TL;DR

This paper establishes optimal upper bounds on the number of automorphisms of leafless metric graphs of genus g, extending Hurwitz's theorem to a tropical setting and characterizing extremal graphs.

Contribution

It provides the first tropical analogue of Hurwitz's theorem for leafless metric graphs, with explicit bounds and classifications of extremal cases.

Findings

Maximum automorphisms for genus 2 is 12.

Maximum automorphisms for genus g ≥ 3 is 2^g g!.

Explicit graphs achieving maximum automorphisms are characterized.

Abstract

We prove a tropical analogue of the theorem of Hurwitz: a leafless metric graph of genus $g \ge 2$ has at most $12$ automorphisms when $g = 2$; $2^g g!$ automorphisms when $g \ge 3$. These inequalities are optimal; for each genus, we give all metric graphs which have the maximum numbers of automorphisms. The proof is written in terms of graph theory.