The Study On The Weighted Szeged Index Of Some Graphs

Chemical graph theory is a widely applied mathematical discipline that focuses on graphs as its research object and can usually be used to describe specific relationships between certain things.A topological index is a numerical quantity that is derived in an unambiguous manner from the structural graph of a molecule.Among them,the weighted Szeged index proposed by Ili′(8 et al.in 2013 is a very important topological index.In this paper,we mainly study the problem of the weighted Szeged index of some graphs.Firstly,based on the relationship between the weighted Szeged index and other topological indices and diameters,we determine the upper and lower bounds of the weighted Szeged index of a connected graph and characterize the corresponding extremal graphs.Secondly,by using some graph transformations,we obtain the upper and lower bounds of the weighted Szeged index of two types of trees and characterize the corresponding extremal graphs.Then,we provide expressions of the weighted Szeged indices of hexagonal systems by using the cut method.We also obtain expressions of the weighted Szeged indices of the complete cobweb graph and the cobweb graph.Finally,based on the structural characteristics of double graph,joint graph and the edge corona of two graphs,we give expressions of their weighted Szeged indices.The paper introduces the cut method for the first time in the study of weighted Szeged index problems of special graphs.Then,it imposes conditional restriction on tree and determines the upper bound of the weighted Szeged index among all trees with a given diameter.Thus,it provides a new approach for the study of weighted Szeged index problems of other graph classes.According to the results studied in this paper,we can not only find the extremal graph with the maximum or minimum weighted Szeged index in some complex graphs,but also quickly calculate the corresponding the weighted Szeged index based on the expression.This paper is divided into five chapters for discussion:In chapter 1,we mainly give the research background,research status and some basic concepts of the weighted Szeged index.In chapter 2,we determine the upper and lower bounds of the weighted Szeged index of a connected graph and characterize the extremal graphs that attain the upper and lower bounds.In chapter 3,we give the upper and lower bounds of the weighted Szeged index of blossomed stars and the upper bound of the weighted Szeged index among all trees with a given diameter.Meanwhile,we characterize their corresponding extremal graphs.In chapter 4,we study the weighted Szeged indices of hexagonal system,the complete cobweb graph and the cobweb graph.In chapter 5,we study the weighted Szeged indices of double graph,joint graph and the edge corona of two graphs.