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

Let $\bar{G}=(G,\omega )$ be a vertex-weighted graph, and $\delta$ a divisor class on $G$. Let $r_{\bar{G}}(\delta )$ denote the combinatorial rank of $\delta$. Caporaso has introduced the algebraic rank $r_{\bar{G}}^{\mathrm{alg}}(\delta )$ of $\delta$, by using nodal curves with dual graph $\bar{G}$. In this paper, when $\bar{G}$ is hyperelliptic or of genus $3$, we show that $r_{\bar{G}}^{\operatorname{alg}}(\delta) \geq r_{\bar{G}}(\delta)$ 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.