Research On The Extended Wiener Index Of Bipartite Graph

In 1947,the chemist Harold Wiener proposed an index named Wiener index,which is an important topological index for the study of quantum chemistry.The related problems of Wiener index can be solved by graph theory which has attracted a lot of graph theorists' attention and many achievements have been achieved in recent decades.In this paper,the extended Wiener index Sk(G)is defined and more general conclusions are obtained.Sk(G)is the sum of k-power of all distances between all pairs of vertices of G,which is denoted byWhere dG(u,v)is the distance between vertices u and v in the graphG;L(k)(v)denotes the sum of k-power of all distances from v to all other vertices in G.In particular,if k=1,L(1)(v)= D(v),this index is called the Wiener index,which can be represented as The third to sixth chapters of this paper,the new conclusions fordifferent graphs and its corresponding extremal graphs are obtained.Theeighth chapter is in the case that k is an integer with no zero.The generalstructure of the paper is as follows:In the first chapter,we introduce the background of the topic,the current developments and the main structure of the paper.In the second chapter,we introduce the basic knowledge of theconcepts and theorems as well as its proofs used in the paper,and so on.In the third chapter,considering the bipartite graphs with the givenmatching number and diameter,the extremal values of S2(G)are depicted and the corresponding extremal graphs are characterized.In the fourth chapter,the extremal graphs with the minimal Sk(G)of bipartite graphs are identified by graph operations when the given vertex connectivity and edge connectivity are given.In the fifth chapter,we classify the trees with fixed diameterd,also,we discuss and identify the corresponding extremal graphs with minimum and second-minimum Sk(G)among all the trees by using the formula of the extended Wiener index.In the sixth chapter,the trees with fixed diameter d are redivided,and the corresponding extremal graphs with third-minimum Sk(G)among all the trees are discussed.In the seventh chapter,the extremal graph of bipartite graphs about the index Sk(G)is characterized with given matching number q.Also,the corresponding extremal graphs of bipartite graphs with fixed diameter d about the index Sk(G)are characterized.The eighth chapter,we summarize the main contents of the paper,and put forward the future study goals and research directions.