  We analyze the Brill-Noether theory of trivalent graphs and multigraphs
having largest possible automorphism group in a fixed genus. For trivalent
multigraphs with loops of genus at least 3, we show that there exists a graph
with maximal automorphism group which is Brill-Noether special. We prove
similar results for multigraphs without loops of genus at least 6, as well as
simple graphs of genus at least 7. This analysis yields counterexamples, in any
sufficiently large genus, to a conjecture of Caporaso.
