A tight bound on $\{C_3,C_5\}$-free connected graphs with positive Lin-Lu-Yau Ricci curvature

In this paper, we prove that any simple $\{C_3,C_5\}$-free non-empty connected graph $G$ with LLY curvature bounded below by $\kappa >0$ has the order at most $2^{\frac{2}{\kappa }}$. This upper bound is achieved if and only if $G$ is a hypercube $Q_d$ and $\kappa =\frac{2}{d}$ for some integer $d\ge 1$.