On The Bounds Of Multiplicative Zagreb Indexes In Given Cut Vertex Graphs

Some topological indexes of graphs have been applied in computer science,medicine and chemistry.In the simple connected graph G,the first multiplicative Zagreb index ?1 is considered to be the product of square of the degree of its ver-tices,while the second multiplicative Zagreb index ? is expressed as the product of end vertex degree of each over all edges.In this paper,we explore simple and connect graphs in terms of given number of cut vertices.Denote Bn,k by a set of graphs on n vertices and k cut vertices.The largest and smallest values of ?1 and ?2 of graphs in Bn,k are obtained.In addition,the corresponding extremal graphs are characterized,and our results extend and enrich some known conclusions.The full text is divided into four chapters,mainly as follows:In chapter 1,we introduce the research background and expound the correspond-ing conclusions of the paper.In chapter 2,we introduce some symbols,concepts and propositions in the article.In chapter 3,we mainly study the maximum value of the Multiplicative Zagreb index in the class of the graph Bn,k.In addition,we give the corresponding extremal graphs.In the first subsection,We show the upper bound of the first Multiplicative Zagreb index and its corresponding the structure of the extremal maximal graph.In the second subsection,we show the upper bound of the second Multiplicative Zagreb index and its corresponding the structure of the extremal maximal graph.In chapter 4,we mainly study the minimum value of the Multiplicative Zagreb index in the class of the graph Bn,k.In addition,we give the corresponding extremal graphs.In the first subsection,we show the lower bound of the first Multiplicative Zagreb index and its corresponding the structure of the extremal minimal graph.In the second subsection,we show the lower bound of the second Multiplicative Zagreb index and its corresponding the structure of the extremal minimal graph.