  We analyze a family of graphs known as banana graphs, with two marked
vertices, through the lens of Hurwitz-Brill-Noether theory. As an application,
we construct explicit new examples of finite graphs which are Brill-Noether
general. These are the first such examples since the analysis of chains of
loops by Cools, Draisma, Payne and Robeva. The graphs constructed are chains of
loops and "theta graphs," which are banana graphs of genus 2. We also
demonstrate that almost all banana graphs of genus at least 3 cannot be used
for this purpose, due either to failure of a submodularity condition or to the
presence of far too many inversions in certain permutations associated to
divisors called transmission permutations.
