Research On Topological Indices Of Several Graphs

A topological index is a numerical value related to the structure of the graph.It is a topo-logical invariant of the molecular graph.A corresponding relationship is established between the molecular topological index and the corresponding physical property,chemical reaction or biological activity of the compound.In order to establish a good correlation with the physi-cal and chemical properties of molecules,many scholars have defined a variety of topological indices on the basis of distance,degree,count,and spectrum.At present,there are hundreds of topological indices proposed.The first molecular topological indices was proposed by the chemist Wiener in 1947.This topological index is the Wiener index.Let G=(V(G),E(G))be a simple connected graph,the Wiener index of G is the sum of all distances between all pairs of vertices in G,then Wiener index is equal to(?)d(u,v),where d(u,v)is the distance between the vertices u and v of V(G).In 1993,Mlian Randic introduce hyper-Wiener,it is cqual(?),It contains the square of the distance。Similarly,in 2009,Iranmanesh[16]and Dankelmann et.al[7]introduced the edge-Wiener indices,which is equivalent to the Wiener indices of the line graph,it is the sum of the distances between all pairs of edges.That is,it is equal to(?)d(e,f),where d(e,f)is the distance between any two edges e and f of the graph G.In addition,in 2011,Iranmanesh et.al[17]proposed the edge-hyper-Wiener indices,which is the sum of the distances and the square of distances between all pairs of edges.It is equal to#12This thesis mainly studies the topological index which is the distance-based index,that is the edge-Wiener index and the edge-hyper-Wiener index.The main work of this thesis are as follows:In chapter 1,we introduce the research background,basic concepts and research status of topological indicators,and finally introduces the main work of this article.In chapter 2,we study the edge-Wiener index of the zigzag nanotubes.In order to calculate it,we first find the line graph of zigzag nanotubes,than layer the line graph,calculate the sum of all distances from a fixed vertex to all other vertices in the line graph,and use the symmetry of the zigzag nanotube structure to calculate the sum of all distances between all pairs of vertices in the line graph.Using a similar calculation method,we can calculate the edge-Wiener index of armchair nanotubesIn chapter 3,we study the edge-hyper-Wiener index of the zigzag nanotubes.This indicator not only includes the sum of the distances between all the vertex pairs of the line graph of the zigzag nanotube,but also the sum of the squares of the distances between all the vertex pairs.Similar to chapter 2,we can obtain it.In chapter 4,we study the extremum of the edge-Wiener index of trees with given segment sequence.We select the starlike tree with a given segment sequence,determine the maximum and minimum value of its edge-Wiener index,and then determine the minimum value of the edge-Wiener index of any tree with a given segment sequence.