The Partial Order Relation Of Trees By Their Edge Division Vectors

There are many applications of graph theory in the field of chemistry,the most active research is the study of molecular topological indices or invariants of graphs,which can be used to describe the physicochemical properties of compounds.As early as 1947,the chemist H.Wiener discovered that the boiling point of paraffin has a special relationship with its molecular structure,and thus came up with the concept of Wiener index,which is very helpful for chemist-s to study the physical,chemical and mathematical properties of molecular structure diagrams.Recently,D.Vukicevic and J.Sedlar introduced the edge division vectors of trees to study the topological index related to distance,and further studied the extremum of Wiener type index and anti-Wiener type index.On this basis,we study the partial order relation of the trees with respect to the edge division vectors,and give the maximal elements and the minimal elements in this poset.Using the partial order,we order trees on n vertices with respect to Wiener index and Steiner k-Wiener index.The main contents are as follows.In chapter 2,we study the partial order structure of trees with respect to the edge division vectors,and obtain the maximal elements of the partial order relation with respect to edge division vectors of trees on n vertices.In chapter 3,using graph transformations,we obtain the minimal elements of the partial order relation with respect to edge division vectors of trees on n vertices by classifying trees according to the number of non-pendent edges.In chapter 4,using these maximal elements and minimal elements in this poset,we study the ordering of trees on n vertices by their Wiener indices and Steiner k-Wiener indices,and determine the trees with the first 15 maximal values and the first 12 minimal values of Wiener index,and the trees with the first four extremal values of Steiner k-Wiener index.