On the Wiener Indices of Trees Ordering by Diameter-Growing Transformation Relative to the Pendent Edges

Abstract EN

The Wiener index of a graph is defined as the sum of distances between all unordered pairs of its vertices.

We found that finite steps of diameter-growing transformation relative to vertices can not always enable the Wiener index of a tree to increase sharply.

In this paper, we provide a graph transformation named diameter-growing transformation relative to pendent edges, which increases Wiener index W(T) of a tree sharply after finite steps.

Then, twenty-two trees are ordered by their Wiener indices, and these trees are proved to be the first twenty-two trees with the first up to sixteenth smallest Wiener indices.