On The Irregularity Indices Of Graphs

The irregularity indices of graphs are measures of the defect of regularity of graphs.Moreover,as descriptors of molecular structure,they can reflect the physicochemical properties and the biological activities of compounds.Thus,they are used widely in the study of quantitative structure-property relationship(QSPR)and quantitative structure-activity relationship(QSAR).In this thesis,two types of irregularity indices—the Albertson index and the misbalance indeg index are studied.They are defined,respectively,as(?),where E(G)is the edge set of G and du is the degree of vertex u in G.These two indices characterize the defect of regularity of graphs from different as-pects.And they are significant predictors of standard enthalpy of vaporisation for octane isomers.The main contents of this thesis are as follows:In Chapter 2,extremal problems of the Albertson index of connected graph-s with given number of pendant vertices are studied.The maximal and minimal values of the Albertson index are determined and the corresponding extremal graphs are characterized.In Chapter 3,extremal problems of the misbalance indeg index across sev-eral classes of graphs are studied.Firstly,sharp upper bounds on the misbalance indeg index of connected graphs are given.Secondly,by using these obtained bounds,the maximal values and corresponding extremal graphs of the misbal-ance indeg index are determined across several graph classes,including con-nected graphs,trees,unicyclic graphs,chemical graphs,chemical trees,chemi-cal unicyclic graphs,connected graphs and trees with given number of pendan-t vertices.Finally,the minimal values and corresponding extremal graphs of the misbalance indeg index are determined for trees,chemical trees,unicyclic graphs,chemical unicyclic graphs,connected graphs and trees with given num-ber of pendant vertices.