The Study On Steiner K-(hyper) Wiener Index Problems Of Special Graphs

Graph theory is a very widely used mathematical discipline,which takes graph as its research object and focuses on the expression of the connection between points through lines.It can usually be used to describe a specific relationship between some things.Topological index refers to a mapping from the molecular graph to the set of real numbers in a certain way,which can reflect the physical,chemical and pharmaceutical properties of molecules.Therefore,topological index is widely used in real life.Among them,Steiner k-Wiener index and Steiner k-hyper Wiener index are very important topological indexes.In 2016,Li et al.determined the calculation formula of Steiner k-wiener index of some special graph classes,and determined the upper and lower bounds of Steiner k-wiener index of trees.In the same year,Mao et al.obtained the calculation formula of Steiner k-wiener index of graph product.In 2018,for trees with a given diameter,Lu et al.gave a lower bound of Steiner k-wiener index and characterized the graph reaching this lower bound.In the same year,Niko Tratnik first proposed the Steiner k-hyper Wiener index,and gave the relationship between the Steiner k-hyper Wiener index and the Steiner kHosoya polynomial.In this paper,we study the extreme problem of Steiner k-(hyper)wiener index of special graphs.The main contents include:the Steiner k-Wiener index of unicyclic graphs,on the second minimum Steiner k-Wiener index for trees with given diameter,Steiner k-hyper Wiener index of graph products and on Steiner k-hyper Wiener index for trees with given diameter.There are six chapters in this paper:In Chapter one,we mainly give the research background and status of Steiner k(hyper)Wiener index as well as some basic concepts involved.In Chapter two,we determine the lower bound of Steiner k-Wiener index among all unicyclic graphs and characterise the extremal graphs that attains the bounds.In Chapter three,we determine the second minimum Steiner k-Wiener index among all the given diameter trees and characterise the extremal graphs that attains the bounds.In Chapter four,we give expressions or lower bounds of Steiner k-hyper Wiener indices of five kinds of graph product,respectively.In Chapter five,we determine the lower bound of Steiner k-hyper Wiener index among all the given diameter trees and characterise the extremal graphs that attains the bounds.In Chapter six,it is the summary and the prospect of future work.