Wiener Index of Quadrangulation Graphs

TL;DR

This paper proves a conjecture that provides an upper bound for the Wiener index in simple quadrangulation graphs, confirming a specific cubic growth rate depending on the parity of the number of vertices.

Contribution

The paper confirms a conjecture giving an exact upper bound for the Wiener index of quadrangulation graphs based on vertex count and parity.

Findings

Confirmed the conjectured upper bound for Wiener index in quadrangulation graphs.

Derived explicit formulas for the Wiener index depending on the parity of the number of vertices.

Validated the cubic growth rate of the Wiener index in these graphs.

Abstract

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.