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

Twice-Marked Banana Graphs & Brill-Noether Generality