Research On The Extremal Graphs With Given Number Of Cut Vertices Concerning On The Reduced Second Zagreb Index

As an interdisciplinary subject of mathematics and chemistry,chemical graph theory studies the molecular structure of compounds with some physical or chemical properties(such as melting point,boiling point,biological activity,etc.).Molecular graph is an important kind of graph in chemistry.In this kind of graph,the vertex represents the atom,while the edge represents the chemical bond.Molecular graph can be used to predict the physical and chemical properties of compounds,which is the main reason for the development of chemical graph theory.In order to quantitatively describe the molecular structure,the basic method of chemical graph theory is to study the topological index of molecular graph and calculate its value,and then establish the database of molecular graph with this index value.In recent years,a large number of new indicators have been proposed,which reflect a large amount of information of molecular structure and greatly promote the study of molecular graphs.Among these descriptors,the first and second Zagreb indices are well-known and thay are extensively studied,which reflect the molecular energy and the branch number of molecular skeleton.It has been widely used in molecular design,molecular complexity,energy and so on.Let G=(V(G),E(G))be a simple graph.The first Zagreb index M1 of graph G is defined as the sum of the square of every vertex degree,whereas the second Zagreb index M2(G)is defined as the sum of the product of vertex degrees of each pair of adjacent vertices of G.In 2014,Furtula,Gutman and Ediz[12]first proposed the reduced second Zagreb index as(?)where ?M(G)is the difference of M2(G)and M1(G)and m(G)is the size of G.In this thesis,some novel graph transformations are introduced.We study the monotonicity of ?M(G)with respect to these graph transformations.Then based on these properties,we present sharp upper and lower bounds on ?M(G)of connected n-vertex cyclic graphs with given number of cut vertices.All the corresponding extremal graphs are identified.The concrete content of our thesis is in the following:·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.Our main results are given in this chapter.·In Chapter 2,we give some necessary definitions and lemmas.·In Chapter 3,we present sharp upper bound on the reduced second Zagreb index of connected n-vertex cyclic graphs with given number of cut vertices.All the corresponding extremal graphs are identified.·In Chapter 4,we present sharp lower bound on the reduced second Zagreb index of connected n-vertex cyclic graphs with given number of cut vertices.All the corresponding extremal graphs are identified.·In Chapter 5,we summarize the main results in this paper and give some prospects for further research in the future.