On The Wiener Index Of Trees

The Wiener index was named after Harry Wiener and was proposed in 1947.At the time,the Wiener index was called the "path number".In chem-ical graph theory,the Wiener index is the molecular topological index,which defines is the sum of the lengths of the shortest paths between all vertices of non-hydrogen atoms in the molecule in the chemical graph.In the mathemat-ical domain,the Wiener index is defined as the sum of the distances between all pairs of vertices in the simple graph.In the field of graph theory,everyone knows that a tree is a connected acyclic graph.Any two vertices u and v in a tree T have one and only one path between them.So,between two vertices u and v in tree T,the distance is equal to the length of the path between u and v.Therefore,using this property of trees,this paper discusses the Wiener index of trees.The first chapter of this paper first introduces the research background of the Wiener index.Next,it briefly describes the core issues,research progress and some main results of this paper.The second chapter of this paper mainly introduces the algorithm and extremal graphs of the Wiener index.In the first section,we give the algorit,hm of the Wiener index of trees and the time complexity is O(n)in the worst case.In the second section,we describe the extreme tree and the extreme chemical tree of the Wiener index.Among them,the maximal graph of the Wiener index corresponding to the tree is the path,and the minimal graph is the star;the maximal graph of the Wiener index corresponding to the chemical tree is still the path,and the minimal graph is a chemical tree with the largest 4 degree vertex and the shortest diameter.Since the extremal graphs can only depict the graph class corresponding to the maximal and minimal graphs of the Wiener index,we cannot know the graph class corresponding to other values.Therefore,we need to calculate more indices of the corresponding trees and chemical trees.However,as the number of graphs continues to increase,it is impossible to calculate the index of all graphs one by one.So,in Chapter 3,we will present a random tree model and a random chemical tree model.And under two models,we show the global distribution of the Wiener index.In addition,through multiple sets of data analysis,we conclude that the Wiener index follows a normal distribution under above two models.