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For a projective curve $C$ and the canonical divisor $K_C$ on $C$, it isclassically known that the canonical ring $R(C) = \oplus_{m=0}^\infty H^0(C, mK_C)$ is finitely generated in degree at most three. In this article, we studywhether analogous statements hold for finite graphs and tropical curves. Forany finite graph $G$, we show that the canonical semi-ring $R(G)$ is finitelygenerated but that the degree of generators are not bounded by a universalconstant. For any hyperelliptic tropical curve $\Gamma$ with integeredge-length, we show that the canonical semi-ring $R(\Gamma)$ is not finitelygenerated, and, for tropical curves with integer edge-length in general, wegive a sufficient condition for non-finite generation.

Canonical semi-rings of finite graphs and tropical curves