Realizations of automorphism groups of metric graphs induced by rational maps

TL;DR

This paper explores how automorphism groups of metric graphs can be realized via rational maps into tropical projective spaces, linking graph symmetries with tropical linear groups.

Contribution

It provides a method to realize automorphism groups of metric graphs as subgroups of tropical linear groups through rational maps.

Findings

Automorphisms induce permutations of coordinates in tropical projective space.

Automorphism groups can be embedded into tropical linear groups.

The approach connects graph symmetries with tropical algebraic structures.

Abstract

For a rational map $\phi$ from a metric graph $\varGamma$ to a tropical projective space $\boldsymbol{TP^n}$ defined by a ratio of rational functions $f_1, \ldots, f_{n + 1}$, an automorphism $\sigma$ of $\varGamma$ induces a permutation of the coordinates of $\boldsymbol{TP^n}$ if $\{ f_1, \ldots, f_{n + 1} \}$ is $\langle \sigma \rangle$-invariant. Through this description, we can realize the automorphism group of $\Gamma$ as ambient automorphism group such as tropical projective general linear group, tropical general linear group and $\boldsymbol{Z}$-linear transformation group of Euclidean space.