On the Wiener indices of trees ordering by diameter-growing transformation relative to the pendent edges (Q2298935)

Summary: 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.