Upper bounds of orders of automorphism groups of leafless metric graphs

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.

12 pages, 3 figures