Algebraic curves have a discrete analog 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 -isomorphic. In particular, the strong Torelli theorem holds for -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. By contrast, we prove that, in a suitably defined sense, the tropical Torelli map has degree one. Finally, we describe some natural posets associated to a graph and prove that they characterize its Delaunay decomposition.
Lucia Caporaso. Filippo Viviani. "Torelli theorem for graphs and tropical curves." Duke Math. J. 153 (1) 129 - 171, 15 May 2010. https://doi.org/10.1215/00127094-2010-022