Study On Distance Parameters And Inertia Index Of Graphs

Wiener index is a kind of distance index proposed by Harold Wiener in 1947,this index is an important topological index for studying quantum chemistry.Graph theory can be used to solve the problem of Wiener index.In recent decades,many graph theory experts have paid attention to this kind of problem,and many achievements have been made.Based on the previous research results,this paper discusses the properties of the extreme graph of Wiener index.We study the Wiener index properties of unicyclic graphs,give the lower bound of Wiener index of unicyclic graphs with given pendant vertices,and characterize all extremely graph that attains the lower bound.Spectral theory is an important branch of graph theory,which describes the structural properties of graphs by studying their spectral properties.In spectral theory,adjacency spectral theory occupies an important position.Among them,the study of the inertial index of graph is one of the hot issues in the theory of adjacency spectrum in recent years.Graph of the positive and negative inertia index is defined as the number of positive and negative eigenvalues in the spectrum,the order of the graph is its adjacency matrix rank,which is the sum of zero eigenvalue multiplicity(i.e.nullity)and positive inertia index and negative inertia index.Inspired by the relation between rank and diameter,this paper discusses in detail the extremely graph structure with given inertia index achieving the maximum diameter.In this paper,new conclusions of Wiener index and inertial index are obtained.The full text is divided into five chapters:Chapter 1:introduces the background of the thesis,the current development and the main structure of the thesis.Chapter 2:introduces the concepts and theorems used in the paper and their proof and other basic knowledge.Chapter 3:the lower bound of Wiener index of unicyclic graph with given pendant vertices is considered,and characterization all extremal graphs attaining the lower bound are depicted.Chapter 4:considering the relation between the positive and negative inertial index and diameters,the inequality relation between them is obtained,and the necessary and sufficient conditions for the equality are described in detail.Chapter 5:summarize the main content of the article,and put forward the future learning objectives and research direction.Figure[10]reference[59]...