  We define a new algebraic invariant of a graph $G$ called the Ceresa-Zharkov
class and show that it is trivial if and only if $G$ is of hyperelliptic type,
equivalently, $G$ does not have as a minor the complete graph on 4 vertices or
the loop of 3 loops. After choosing edge-lengths, this class specializes to an
algebraic invariant of a tropical curve with underlying graph $G$ that is
closely related to the Ceresa cycle for an algebraic curve defined over
$\mathbb{C}(\!(t)\!)$.
