The Atomic Bond Connectivity Index Of The Graph

This thesis studies ABC index and two of its generalizations,namelyABC_GG index and3 ABC index.By properties of graph transformations,some special functions and corresponding number theory,and combing some known methods,we determine some extremal value of these indices and characterize the corresponding graphs.The first chapter introduces basic graph theory corresponding to this study,including some graph concepts and symbols,the background of some topological indices,and the main results of this thesis.The second chapter determines the upper bounds for ABC index of direct product,disjunction and symmetric difference of two graphs;and the upper bounds forABC_GG index of join,Cartesian product and corona product of two graphs.Those bounds are shown tight,the corresponding extremal graphs are also characterized.The main results are: Theorem 2.1.1、2.1.2、2.1.3、2.2.1、2.2.2 and 2.2.3.The third chapter determines the maximalABC_GG index among all bicyclic graphs and characterizes the extremal graphs,the main results are: Theorem 3.2.9.The forth chapter determines the maximal and second maximal3 ABC index of unicyclic graphs with any given girth and order;and the maximal and second maximal3 ABC index of trees with any given diameter and order,the main results are:Theorem 4.2.3、4.3.3 and 4.4.2,and characterizes the corresponding extremal graphs.