  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.
