The Wiener Index Of Trees With Given Degree Sequences

The Wiener index is the sum of topological distance between all pairs of vertices in an acyclic graph(tree) representing the structural formula of a molecule.Firstly,we investigate some properties of the partial ordered set of all vectors whose component corresponding to each vertex of a rooted tree is equal to the number of its vertex and its successors with respect to majorization. Then these results are used to characterize the trees which minimize the Wiener index among all trees with given degree sequence.Consequently,all extremal trees with the maximum degree,the leave number and the matching number respectively.Further,using similar method,we characterize trees which maximize the Wiener index with maximum degree,as well as given degree sequence with degree types less than 4. Base on the conclusion,we could suppose the structure of tree which maximize the Wiener index with given degree sequence.