The Ceresa period from tropical homology

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 $L_3$.