The Wiener Index for Weighted Trees

The Wiener index of a graph is the sum of the distances between all pairs of vertices. In fact, many mathematicians have study the property of the sum of the distances for many years. Then later, we found that these problems have a pivotal position in studying physical properties and chemical properties of chemical molecules and many other fields. Fruitful results have been achieved on the Wiener index in recent years. Most of the research focus on the extreme values and the corresponding graphs for the non-weighted simple graphs. In this paper, we consider the edge-weighted graphs. Firstly, we give the exact definition of the distances in edge-weighted graphs. Secondly, we get a useful variant formula of the Wiener index. Then, we take our attention on edge-weighted trees of order n. We get the minimum, the second minimum, the third minimum, the maximum, the second maximum values of the Wiener index, and characterize the corresponding extremal trees. Key–Words: Weighted graph; Tree; Wiener index; Minimum value; Maximum value; Bound