The Ceresa period from tropical homology

TL;DR

This paper introduces the Ceresa period as a new invariant for finite graphs to study tropical Ceresa cycles, revealing its connection to hyperelliptic types and minor-closed properties.

Contribution

It defines the Ceresa period for graphs and establishes its characterization of hyperelliptic graphs and minor-closed properties.

Findings

Ceresa period equals zero if and only if the graph is hyperelliptic.

Zero Ceresa period condition is minor-closed with specific forbidden minors.

Provides a new algebraic tool for analyzing tropical cycles in graph theory.

Abstract

Given a finite graph $G$, we define the Ceresa period $\alpha(G)$ as a tool for studying algebraic triviality of the tropical Ceresa cycle introduced by Zharkov. We show that $\alpha(G) = 0$ if and only if $G$ is of hyperelliptic type; then a theorem of Corey implies that having $\alpha(G) = 0$ is a minor-closed condition with forbidden minors $K_4$ and $L_3$.