Let ${\rm ex}_{\mathcal{P}}(n,T,H)$ denote the maximum number of copies of $T$ in an $n$-vertex planar graph which does not contain $H$ as a subgraph. When $T=K_2$, ${\rm ex}_{\mathcal{P}}(n,T,H)$ is the well-studied function, the planar Turán number of $H$, denoted by ${\rm ex}_{\mathcal{P}}(n,H)$. The topic of extremal planar graphs was initiated by Dowden [ J. Graph Theory, 83 (2016), pp. 213--230]. He obtained a sharp upper bound for both ${\rm ex}_{\mathcal{P}}(n,C_4)$ and ${\rm ex}_{\mathcal{P}}(n,C_5)$. Later on, Lan, Shi, and Song continued this topic and proved that ${\rm ex}_{\mathcal{P}}(n,C_6)\leq \frac{18(n-2)}{7}$. In this paper, we give a sharp upper bound ${\rm ex}_{\mathcal{P}}(n,C_6) \leq \frac{5}{2}n-7$, for all $n\geq 18$, which improves Lan, Shi, and Song's result. We also pose a conjecture on ${\rm ex}_{\mathcal{P}}(n,C_k)$, for $k\geq 7$.

The Wiener index of a graph G, denoted W(G), is the sum of the distances between all non-ordered pairs of vertices in G.É. Czabarka, et al. conjectured that for a simple quadrangulation graph G on n vertices, n≥4, W(G)≤112n3+76n−2,n≡0(mod2), 112n3+1112n−1,n≡1(mod2).In this paper, we confirm this conjecture.