Research On Relations Between Related Atom-bond Connectivity Indices And Invariant Of Graphs

Graph theory has gradually become foundation and pillar of the subjects such as computer science,combinatorial optimization and so on.Today,random graph theory,algebraic graph theory and algorithm graph theory are the branches of graph theory.The occurrence of these branches enrich the research contents of graph theory and extends the research method of graph theory.In fact,related knowledge of graph theory are used for the analysis and research work in fluid dynamics,information,telecommunications and transportation.In addition,the wide application background of graph theory is also embodied in chemistry.After a period of study,scientists found that physical and chemical properties of many compounds have connection with their topological properties,such as boiling point,water-soluble,molecular volume and surface area,energy level,electronic distribution and so on.Topological indices have numerous good properties in physical and chemical field-s.The relationship among degree-based indices is obvious and there are many results about this aspect.From another view,we can also consider the relation between a degree-based index and a distance-based index,it is certainly a valuable topic.In par-ticular,diameter is an invariant,based on distance.The relationship between related atom-bond connectivity indices and diameter in trees and unicyclic graphs are studied.Second atom-bond connectivity index of a graph G is ABC₂?G?=???,where n_ustands for the number of vertices of G whose distance to the vertex u is shorter than the distance to the vertex v.In this paper,we firstly introduce a new transfor-mation by which we can give a short proof of the theorems in^[1].Secondly,by the same transformation,we also obtain the trees with the second maximum and minimum ABC₂-index.The full text is divided into five parts.In the first part,we briefly introduce the research background,definitions and current situation of related atom-bond connectivity indices,show the main conclusion of this paper.In the second part,we present the notations,symbol marks and theorems of this paper.In the third part,we get the maximum,the second maximum,the minimum,the second minimum of the second atom-bond connectivity for trees,and characterize the corresponding extremal trees respectively.In the forth part,we give a discussion about the relationship between related atom-bond connectivity indices and diameter in trees and unicyclic graphs respectively,obtain the lower bounds of the difference of indices and diameter.In the fifth part,we summarize the main work of this paper and outlook of the future.