On The Topics Related To Rupture Degree And Edge-Rupture Degree Of Graphs

In the design of computer and communication networks,in order to avoid or decrase the loss caused by network interruptions,invulunerability is an important factor must be considered.Therefore,one of the basic ideas in network design is that they do not easily be destroyed by outside attacks;further,once the damage occurred,it is also relatively easy to repair or reconstruct.A network is often modeled by a connected graph and its invulunerability is described quantitatively by introducing appropriate parameters.Therefore,a network invulunerability is the invulunerability of the corresponding graph.Connectivity and edge-connectivity are the earliest invulunerability parameters.However,these two parameters have a certain degree of limitations in the measure of invulunerability.Toughness and edge-toughness,integrity and edge-integrity,tenacity and edge-tenacity,scattering number,rupture degree and edge-rupture degree and other parameters introduced afterward have been studied deeply,since they mesaure the difficulty of breaking a network and the extent to which the network is destroyed under the “worst” case.Considering the deficiency of rupture degree and edge-rupture degree,this thesis introduces a new parameter---weak-rupture degree.The parameter value of several types of basic graphs and operation graphs,the boundary and the relationship with other parameters are given.Based on these work,the Nordhaus-Gaddum type problems of weak-rupture degree are discussed.The general conclusion of the relationship between the parameters and the invulnerability is obtained by the methods of constructing graph examples and comparing.For the invulnerability measure of certain graphs,the effect of weak-rupture degree is better than tenacity,rupture degree and weak-integrity.This thesis is divided into five chapters and the struction is organized as follows.The first chapter mainly introduces the related concepts of networks with invulnerability,the research background,significance and research situation of invulnerability parameters.In Chapter 2,the main results of the research on rupture degree and edge-rupture degree are summarized.In Chapter 3,a new concept called weak-rupture degree is introduced,the formula for calculating the weak-rupture degree of several types of special graphs,the boundary of weak-rupture degree of certain general graphs are given,and the relationships with other parameters are discussed preliminarily.In Chapter 4,the formulas and boundary of the weak-rupture degree of several types of operation graphs are studied and given.In Chapter 5,on the summary of this thesis,some further research questions of invulnerability parameters are proposed.