On Extremal Cacti With Respect To Balaban Index And Sum-Balaban Index

It is well known that the appearance and the development of graph theory are closely connected with the research of chemical molecular graph.Chem-ical graph theory is a significant branch of graph theory.It investigates the physical-chemical and mathematical properties of the chemical molecular graph and some special graph through topological index.Since Winener-index which was based on discovery of certain relationship between boiling point of paraffin wax and its molecular structure was introduced in 1947 by Harold Wiener,researchers have put forward a large number of topological index to study the structure and chemical property,and many important achievements and applications have been obtained.The Balaban index of a connected graph G is defined asIt was introduced by Balaban which is also called the average distance-sum connectivity or J index.It appears to be a very useful molecular descriptor with attractive properties.In 2010,Balaban et al.also proposed the sum-Balaban index SJ(G)of a connected graph G,which is defined asIn this thesis,we focus on the extremal graphs respect to Balaban index and Sum-Balaban index,we will mainly discuss the cacti,polyphenyl chains and spiro hexagonal chains respectively.Here is the main work in this paper:In chapter two,we study the maximum Balaban index and Sum-Balaban index of cacti by some transformations of graphs,auxiliary graphs.We show that CO(n,k)is the graph with the maximum Balaban index and Sum-Balaban index among all cacti with n vertices and k cycles,where CO(n,k)denotes a bundle of k triangles with n-2k-1 pendent vertices attached at the common vertex.In chapter three,firstly,we we focus on the extremal graphs of polyphenyl and spiro hexagonal chains with respect to the Balaban index and the sum-Balaban index by some transformations of graphs,auxiliary graphs.We show thatJ(Pn)<J(PPCn)<J(On),SJ(Pn)<SJ(PPCn)<SJ(On)if PPCn is a polyphenyl chain of length n.And we get thatJ(SPn)<J(SPCn)<J(SOn),SJ(SPn)<SJ(SPCn)<SJ(SOn)if SPCn is a six-membered ring spiro chain of length n.