Twice-Marked Banana Graphs & Brill-Noether Generality

TL;DR

This paper studies banana graphs with two marked vertices using Hurwitz-Brill-Noether theory, constructing new examples of finite graphs that are Brill-Noether general, and analyzing conditions that prevent certain banana graphs from having this property.

Contribution

It introduces explicit new examples of Brill-Noether general graphs among chains of loops and theta graphs, expanding the understanding of graph Brill-Noether theory.

Findings

Constructed explicit Brill-Noether general graphs of genus 2.

Identified obstructions for banana graphs of genus ≥ 3 to be Brill-Noether general.

Demonstrated limitations due to submodularity failure and permutation inversions.

Abstract

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.