| The graph theory is closely related to our practical life as an important branch of combinatorial mathematics. The topology index of graph also plays a role of a springboard and a link in the course of solving practical problems, especially in the chemistry field, it can be showed that the architectural feature and character from different molecules which we can attain help to analyze and solve the problems beneficially. Since 1947, the Wiener index was first derived, it has been used to study the structure of molecules. What the Wiener index describes is the distance between different vertexes, and a new valuable field was created by this concept with Network Transport theory. In the basis of its widespread application and development in mathematics, Ovidiu Ivanciuc and others came up with two new concepts——Wiener-1 index and Wiener-2 index (they were called Similar Wiener index together). Both of them were received basing on the Wiener index and the parity of length, and a new view was raised for graph theory.This paper mainly solves several important problems as follows:It first gives us the important computational formulas of caterpillar binary trees and double star graphs based on parity of the similar wiener index;The second part mainly proposes the changing rules of similar wiener index of single edge subdivision star graphs and double edges subdivision star graphs and provide the relation of them based on analyzing the calculation results;The third part analyzes the computational formulas of similar wiener index of vertices of paths and give us the extreme positions and the changing rules. This paper also calculates the similar wiener index of pending vertices and vertices of the main path of a caterpillar binary tree by separating the similar wiener index of it, at last it summarizes the changing rules and extreme points of them. |
