Torelli theorem for graphs and tropical curves

TL;DR

This paper establishes a Torelli theorem for finite graphs and tropical curves, linking graph contractions and 2-isomorphism to their Jacobians, and explores related poset structures and Delaunay decompositions.

Contribution

It proves a Torelli theorem for graphs and tropical curves, connecting graph contractions and 2-isomorphism to their Jacobians, and characterizes associated posets and decompositions.

Findings

Graphs with the same Albanese torus are related by contraction of separating edges.

Strong Torelli theorem holds for 3-connected graphs.

Posets associated to graphs characterize Delaunay decompositions.

Abstract

Algebraic curves have a discrete analogue in finite graphs. Pursuing this analogy we prove a Torelli theorem for graphs. Namely, we show that two graphs have the same Albanese torus if and only if the graphs obtained from them by contracting all separating edges are 2-isomorphic. In particular, the strong Torelli theorem holds for 3-connected graphs. Next, using the correspondence between compact tropical curves and metric graphs, we prove a tropical Torelli theorem giving necessary and sufficient conditions for two tropical curves to have the same principally polarized tropical Jacobian. Finally we describe some natural posets associated to a graph and prove that they characterize its Delaunay decomposition.