Realizations of automorphism groups of metric graphs induced by rational maps

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.