Outerplanar graphs with positive Lin-Lu-Yau curvature

In this paper, we show that all simple outerplanar graphs $G$ with minimum degree at least $2$ and positive Lin-Lu-Yau Ricci curvature on every edge have maximum degree at most $9$. Furthermore, if $G$ is maximally outerplanar, then $G$ has at most $10$ vertices. Both upper bounds are sharp.