Research On The Eccentric Distance Sum Of Unicycle Graphs

The topological index of graph theory is the invariant of graphs,which plays an important role in the fields of physics,chemistry and biology.There are many topological indexes of graph,and this thesis mainly focuses on the sum of eccentric distance.For a simple connected graph G,the definition of the eccentric distance sum of the graph is ξd(G)=∑v∈VG εG(v)DG(v),where the eccentricity εg(v)denotes the maximum distance from the vertex v to other vertices and DG(v)=∑u∈VG dG(u,v)is the sum of all distances from the vertex v.This thesis mainly discusses three aspects in the way of graphic transformation.Let Un,γ be all unicycle graphs of order n with domination number γ.The unicycle graph having the maximal eccentric distance sum among Un,2 and Un,3 are characterized,and the minimal eccentric distance sum of Un,γ is determined.And the extremal graph is also characterized.The research methods are including Classified discussion,Exclusive method and Reduction to absurdity.The results are as follows:In Chapter 1,it mainly introduces the background and the research status.In Chapter 2,it illustrates the basic definitions,commonly used graph transformations and related lemmas.In chapter 3,the extremal unicycle graph among Un,2 and Un,3 having the maximal EDS is characterized.Let U3l(a,b)be the graph obtained by connecting an endpoint of the path Pl with any vertex of the cycle C3 and connecting a and b pendent vertices at the two endpoints of the path Pl respectively.The following conclusion is obtained:1)For any graph G ∈Un,2,satisfying:2)For any graph G ∈Un,3(n≥11),satisfying:In chapter 4,the extremal unicycle graph among Un,γ having the minimal EDS is characterized.Let Un,2 denotes the unicycle graph obtained by attaching n-3 pendent vertices at any vertex of C3;U’n,2 denotes the unicycle graph obtained by connecting 1 and n-4 pendent vertices on any two vertices of C3;U*n,γ refers to the unicycle graph obtained by connecting γ-1 paths of length 2 and n-2γ pendent vertices on any vertex of C3.The following conclusion is obtained:for any graph G∈Un,γ,satisfying:In chapter 5,we summarize the main results of this thesis and give the prospects.There are totally 25 figures,82 references in this thesis.