Canonical representatives for divisor classes on tropical curves and the Matrix-Tree Theorem

Let $Gamma$ be a compact tropical curve (or metric graph) of genus $g$. Using the theory of tropical theta functions, Mikhalkin and Zharkov proved that there is a canonical effective representative (called a break divisor) for each linear equivalence class of divisors of degree $g$ on $Gamma$. We present a new combinatorial proof of the fact that there is a unique break divisor in each equivalence class, establishing in the process an integral version of this result which is of independent interest. As an application, we provide a geometric proof of (a dual version of) Kirchhoffs celebrated Matrix-Tree Theorem. Indeed, we show that each weighted graph model $G$ for $Gamma$ gives rise to a canonical polyhedral decomposition of the $g$-dimensional real torus ${rm Pic}^g(Gamma)$ into parallelotopes $C_T$, one for each spanning tree $T$ of $G$, and the dual Kirchhoff theorem becomes the statement that the volume of ${rm Pic}^g(Gamma)$ is the sum of the volumes of the cells in the decomposition.