Wiener index of quadrangulation graphs

Abstract

The Wiener index of a graph $G$, denoted $W\left(G\right)$, 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, $n\ge 4$,

$$W\left(G\right)\le \left\{\begin{array}{cc}\frac{1}{12}{n}^3+\frac{7}{6}n-2,\hfill & n\equiv 0\left(mod2\right)\text{, }\hfill \\ \frac{1}{12}{n}^3+\frac{11}{12}n-1,\hfill & n\equiv 1\left(mod2\right).\hfill \end{array}\right.$$

In this paper, we confirm this conjecture.