Ordering trees with given matching number by their Wiener indices

Abstract

The Wiener index of a connected graph is the sum of distances between all pairs of vertices in the graph. Let \(\Gamma (n,i)\) be the set of all trees with order \(n\) and matching number \(i\). In this article, we give five graphic transformations that change the Wiener index of graphs, then with them we determine the second to sixth trees in \(\Gamma (2i+1,i)\) and the third to eighth trees in \(\Gamma (n,i)\) for \(n\ge 2i+2\) having the smallest Wiener indices.