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 ≥ 2 has at most 12 automorphisms when g = 2 and 2 g g ! automorphisms when g ≥ 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.