The Wiener Index For Trees With Given Degree Sequences

The Wiener index is the sum of distances between all pairs of vertices in a (connected) graph. It is an important topological index in Chemistry since Harold Wiener defined it in 1947. It is used for the structure of molecule. Scientists found that there is a very close relation between the physical, chemical characteristics of many compounds and the topological structure of that. The Wiener index is such a topological index and it has been widely used in Chemistry fields. Several years later, Mathematician began to pay attention to the Wiener index and study it from the mathematical point of view.In this paper, based on previous studies, we make in-depth study on the Wiener index for trees with given degree sequences. First, we summed up the previous studies of the Wiener Index, and then highlight the latest developments in extreme optimal field. This article was inspired by the study of the Wiener minimum optimal tree. So we decide to study the Wiener maximum optimal tree. And some previous studies pointed out that the caterpillar tree is the maximum optimal tree in trees with fixed degree sequence. But the problem is the number of caterpillar tree is not only one. In order to find the maximum caterpillar tree for Wiener index, we have done a lot of works. We found that the composition of the tree depends on the degree of sequence values. We have studied some cases that the number of the non-suspension vertex of the tree are less than or equal to six.