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Algebraic rank on hyperelliptic graphs and graphs of genus $3$

Let $\bar{G} = (G, ω)$ be a vertex-weighted graph, and $δ$ a divisor class on $G$. Let $r_{\bar{G}}(δ)$ denote the combinatorial rank of $δ$. Caporaso has introduced the algebraic rank $r_{\bar{G}}^{\operatorname{alg}}(δ)$ of $δ$, by using nodal curves with dual graph $\bar{G}$. In this paper, when $\bar{G}$ is hyperelliptic or of genus $3$, we show that $r_{\bar{G}}^{\operatorname{alg}}(δ) \geq r_{\bar{G}}(δ)$ holds, generalizing our previous result. We also show that, with respect to the specialization map from a non-hyperelliptic curve of genus $3$ to its reduction graph, any divisor on the graph lifts to a divisor on the curve of the same rank.