Two-vertex generators of Jacobians of graphs
David Brandfonbrener, Pat Devlin, Netanel Friedenberg, Yuxuan Ke,, Steffen Marcus, Henry Reichard, and Ethan Sciamma
TL;DR
This paper characterizes when the Jacobian of a graph is generated by a divisor formed by two vertices, answering a specific open question and exploring properties of such divisors with empirical support.
Contribution
It provides necessary and sufficient conditions for two-vertex divisors to generate the Jacobian, addressing an open problem and proposing conjectures supported by empirical data.
Findings
Conditions for two-vertex divisors to generate Jacobians
Proven propositions about divisor orders
Empirical evidence supporting conjectures on random graphs
Abstract
We give necessary and sufficient conditions under which the Jacobian of a graph is generated by a divisor that is the difference of two vertices. This answers a question posed by Becker and Glass and allows us to prove various other propositions about the order of divisors that are the difference of two vertices. We conclude with some conjectures about these divisors on random graphs and support them with empirical evidence.
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