Abstract
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.
Highlights
All graphs considered in this paper are finite, simple and connected
Let G be a graph, the vertex and edge sets of G are denoted by V (G) and E(G) respectively
The Wiener index of the graph G is denoted by W (G) and is defined as, W (G) =
Summary
All graphs considered in this paper are finite, simple and connected. Let G be a graph, the vertex and edge sets of G are denoted by V (G) and E(G) respectively. Wiener in 1947, while studying its correlations with boiling points of paraffin considering its molecular structure [14] Since it has been one of the most frequently used topological indices in chemistry, as molecular structures are usually modelled as undirected graphs. The most basic upper bound of W (G) states that, if G is a connected graph of order n, . E. Czabarka, et al [3], gave an asymptotic upper bound for the Wiener index of triangulation and quadrangulation graphs. Et al proved the following asymptotic upper bound. In the same paper [3], based on the following constructions Qn, see Figure 1, they conjectured that W (Qn) is the upper bound for the Wiener index of quadrangulation graphs. First we need some notations and preliminaries
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