Brill-Noether Theory of Maximally Symmetric Graphs

TL;DR

This paper investigates the Brill-Noether properties of highly symmetric graphs, revealing counterexamples to a conjecture and expanding understanding of the relationship between symmetry and algebraic properties in graph theory.

Contribution

It provides new examples of symmetric graphs that are Brill-Noether special, challenging existing conjectures and extending the theory to various classes of graphs.

Findings

Existence of maximally symmetric Brill-Noether special graphs in certain genera

Counterexamples to Caporaso's conjecture in large genus cases

Symmetry influences Brill-Noether properties in trivalent and multigraphs

Abstract

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.