The Wiener index of a graph $G$, denoted $W(G)$, is the sum of the distances between all pairs of vertices in $G$. E. Czabarka, et al. conjectured that for an $n$-vertex, $n\geq 4$, simple quadrangulation graph $G$, \begin{equation*}W(G)\leq \begin{cases} \frac{1}{12}n^3+\frac{7}{6}n-2, &\text{ $n\equiv 0~(mod \ 2)$,}\\ \frac{1}{12}n^3+\frac{11}{12}n-1, &\text{ $n\equiv 1~(mod \ 2)$}. \end{cases} \end{equation*} In this paper, we confirm this conjecture.

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四边形图的维纳指数

图$G$ 的维纳指数，表示为$W(G)$，是$G$ 中所有顶点对之间距离的总和。E. Czabarka 等人。推测对于 $n$-顶点，$n\geq 4$，简单四边形图 $G$，\begin{equation*}W(G)\leq \begin{cases} \frac{1}{12}n ^3+\frac{7}{6}n-2, &\text{ $n\equiv 0~(mod \ 2)$,}\\ \frac{1}{12}n^3+\frac{ 11}{12}n-1, &\text{ $n\equiv 1~(mod \ 2)$}。\end{cases} \end{equation*} 在本文中，我们证实了这个猜想。