Research On The ABC Estrada Index Of Graphs

The internal structure of graphs is the main research content of graph theory.An important research field of algebraic graph theory and combinatorial graph theory is the index theory of graphs.It mainly uses to character the structural properties of graphs through the index parameter of matrices associated to their graph representa-tions,and studies the intrinsic relations between the index parameters and structure of graphs.In this thesis,according to the eigenvalues(resp.ABC matrix's eigenvalue)of a class of graph matrices,we obtain the upper and lower bounds of graphs concerning the number of vertices,the number of edges,the maximum degree and the minimum degree,and some bounds for ABC Estrada index involving the ABC energy of graphs are also presented.All the corresponding extremal graphs are characterized.·In Chapter 1,we introduce the background and significance of the research,including the development of a representative at home and abroad regarding this aspect.Based on this research background and profound discussion,by using deep-going analysis,it fully shows the main work's necessity and innovation.·In Chapter 2,we first give some.necessary notations and terminologies,then introduce some preliminary lemmas and theorems.·In Chapter 3,according to the number of vertices,the number of edges,the maximum degree and the minimum degree,we establish some upper and lower bounds for the ABC Estrada index of graph and describe the corresponding polar diagrams in terms of some theorems.·In Chapter 4,based on the characteristic of the ABC energy of graphs,we study the relationship between the ABC Estrada index and the ABC energy of graphs.·In Chapter 5,we summarize the main results in this paper and give some prospects for further research in the future.