Extremal Values Of Vertex-degree-based Exponential Topological Index

Topological index is the mathematical descriptors of molecular structure.It is one of the most active fields in mathematical chemistry,because it is easy to calculate,objective and not easily affected by experience and experiment.Among them,the study of topological index based on vertex degree(VDB topological index for short)in graph is favored by mathematical chemistry researchers.Specially,in 2018,J.Rada discussed the discrimination property of VDB topological indices over Gn,the set of graphs with nonisolated vertices,and puted forward the exponential VDB topological index eφ(G)=∑(i,j)∈Kmi,j(G)eφi,j,proved that the exponential VDB topological index has higher discrimination and resolution and better mathematical properties than the ordinary VDB topological index.In this paper,we first solve the open problem about the maximum value of the exponential second Zagreb index of trees,From this,we further study two kinds of exponential VDB index that for trees with n vertices,i.e.φi,j=(ij)α or φi,j=iα+jα.Finally,we study the structure properties of the tree when the exponential ABC index reaches the minimum and the exponential AZ index reaches the maximum.Here is the main work in this dissertation:In chapter 2,we study the extremal value and extremal graph of the exponential second Zagreb index.Firstly,we find the distance relationship between the pendent vertices and the maximum degree vertex in the corresponding extremal graph.Then adjust the pendent vertices on the different branch vertices to obtain the maximum value of the exponential second Zagreb index.Finally,we get the extremal graph when φi,j=iα+jα.In chapter 3,we study the exponential forgotten index and the exponential inverse forgotten index of trees.Firstly,using edge lifting transformation,we consider the extremal value and extremal graph of the exponential forgotten index of the trees,furthermore,we get the extremal graph when φi,j=(ij)α.Then,using edge-lifting inverse transformation,slide transformation,we determine the extremal value of the exponential inverse forgotten index,and describe the structural characteristics of the extremal graph.In chapter 4,we study the structural properties of the trees with the minimum exponential ABC index and with the maximum exponential AZ index.