Research On The Graph Parameters And Graph Structures For Simple Connected Graph Of Diameter 2

Studying the inner structure contained in the graph is the main work of Graph theory.By studying some graph parameter,one may characterize the structure of the graph.The study between the distance-based parameters and the structures of graphs is an important research field in modern graph theory.Let G =(VG,EG)be a simple connected graph.The eccentric resistance-distance sum(ξR)of G is defined aswhere δG(·)is the eccentricity of the corresponding vertex and RvuG(shorted for Ruv)is the effective resistance-distance between u and v in G.This novel graph invariant was first proposed by Li et al.in 2016,it attracts more and more researchers’ attention.In this thesis,according to the knowledge of electrical networks,we obtain the internal relation between the resistance distance-based graph parameter and graph structure.The specific contents of this paper are as follows:In Chapter 1,we introduce the background of the paper,the study of meaning,and the current state of the situation.And through the in-depth analysis of the research background and the current status of the study,it fully shows a good indication of the necessity and innovation of our work.In Chapter 2.we offer the basic concepts and symbols involved in this paper.In Chapter 3,we consider the simple connected graph of diameter 2,in which,the largest,second largest,third largest,smallest,second smallest and third smallest eccentric resistance-distance sums are identified,respectively.And the corresponding extremal graphs with the upper and lower bounds are character-ized as well.In Chapter 4,we summarize the full text and make relevant outlook.