Research On Wiener Index Of Graphs

The Wiener index is the sum of distances all pairs of a (connected) graph.It is an important topological index in Chemistry since Harold Wiener defined it in his paper in 1947 and it is used for the structure of molecule. After many years'reseach,scientists found that there is a very close relation between the physical characteristics and chemical characteristics of many compounds and the topological index that has close relation with the physical characteristics and chemical characteristics of many compouds and its characteristics has been applied widely to many fields of Chemisty.After several years,Mathematician began to pay attention to this index and discribed it in Mathemarical words. In his papers,Wiener did not use the math-theoretical language.Graph-Theory can be used to discribe the structure of molecule,as an useful mathmatical language,so it is a good tool that is for the Wiener index. We will use the standard language of graph theory and study this important index.Firstly,we study the index of the graph with only cycles (the number of points and edges is equal),We give that adding an edge to the star can det the graph with minimum Wiener index,and the graph of G₂with maximum index; secondly,We studythe structure of the trees with the order n and the constant diameter; Then,we give the trees with second smallest and the secondlargest Wiener index; Finally, we give the conclusions.