On Two Types Of Degree-based Topological Indices

Chemical Graph Theory is an important research field of Applied Graph Theory,in which the topological invariants and topological properties of(molecular)graphs,as well as their relationships with the physical and chemical properties of compounds are mainly investigated.In Chemical Graph Theory,topological indices,as graph invariants used to describe the molecular structure of compounds,not only realize the numeralization of information in molecular structure,but also reflect the physical and chemical properties and the biological activities of compounds.So,in the study of quantitative structure-property relationship(QSPR)and quantitative structure-activity relationship(QSAR),topological indices play an important role.In this thesis,we mainly study two types of degree-based topological indices-the generalized ABC index and ISI index,and obtain the following results:In Chapter 2,by using the property that the generalized ABC index increases with the addition of edges and some graph transformations,when 0<??1/2,the maximal values of the generalized ABC index of connected graphs with given number of pendent vertices are determined,and the extremal graphs achieving this maximal values are also characterized.In Chapter 3,we first show that the BFS graph has the maximal ISI index among the connected graphs with a given degree sequence,from which we may conclude that the greedy tree has the maximal ISI index among the trees with a given degree sequence.Second,by means of this result we construct a search algorithm to search the trees of order up to 150 and then determine the trees of order up to 150 which have the maximal ISI index.Finally,based on the analysis of search outputs,we further prove that the(general)trees with maximal ISI index possess the following structural features:(1)having no internal paths of length at least 2;(2)having no pendent paths of length at least 3;(3)having at most one pendent path of length 2.