The Turán number of a graph H, ex(n,H), is the maximum number of edges in a graph on n vertices which does not have H as a subgraph. Let Pk be the path with k vertices, the square Pk2 of Pk is obtained by joining the pairs of vertices with distance one or two in Pk. The powerful theorem of Erdős, Stone and Simonovits determines the asymptotic behavior of ex(n,Pk2). In the present paper, we determine the exact value of ex(n,P52) and ex(n,P62) and pose a conjecture for the exact value of ex(n,Pk2).

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.