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.
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Keywords: chip-firing, Brill–Noether theory, banana graphs, special divisors, tropical geometry
Pflueger, Nathan 1; Solomon, Noah 2
CC-BY 4.0
@article{ALCO_2025__8_5_1415_0,
author = {Pflueger, Nathan and Solomon, Noah},
title = {Twice-marked banana graphs & {Brill{\textendash}Noether} generality},
journal = {Algebraic Combinatorics},
pages = {1415--1457},
year = {2025},
publisher = {The Combinatorics Consortium},
volume = {8},
number = {5},
doi = {10.5802/alco.443},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.443/}
}
TY - JOUR AU - Pflueger, Nathan AU - Solomon, Noah TI - Twice-marked banana graphs & Brill–Noether generality JO - Algebraic Combinatorics PY - 2025 SP - 1415 EP - 1457 VL - 8 IS - 5 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.443/ DO - 10.5802/alco.443 LA - en ID - ALCO_2025__8_5_1415_0 ER -
%0 Journal Article %A Pflueger, Nathan %A Solomon, Noah %T Twice-marked banana graphs & Brill–Noether generality %J Algebraic Combinatorics %D 2025 %P 1415-1457 %V 8 %N 5 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.443/ %R 10.5802/alco.443 %G en %F ALCO_2025__8_5_1415_0
Pflueger, Nathan; Solomon, Noah. Twice-marked banana graphs & Brill–Noether generality. Algebraic Combinatorics, Volume 8 (2025) no. 5, pp. 1415-1457. doi: 10.5802/alco.443
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