The Research On Properties And Applications Of Zero-divisor Graphs And Fuzzy Graphs

This paper mainly studies some properties and applications of zero-divisor graphs and fuzzy graphs.In Chapter 2,some graph-theoretic properties about the zero-divisor graph ?(Q)of a finite quasi-ordered set Q with a least element 0 and its line graph L(?(Q))are studied.First,we offer a method to find all the minimal prime ideals of a quasi-ordered set Q.Especially,this method is applicable for a partially ordered set.Then,we completely characterize the diameter and girth of zero-divisor graph ?(Q)by studying properties of the minimal prime ideals of Q.Besides,we perfectly classify all finite quasi-ordered sets whose zero-divisor graphs are complete graphs,star graphs,complete bipartite graphs,complete k partite graphs,respectively.We also characterize the planarity of ?(Q).Finally,we study some properties about the line graph L(?(Q)in terms of its diameter,girth and planarity.In Chapter 3,we first recall the concepts of fuzzy cut-sets and fuzzy arc cuts in a fuzzy graph.In order to generalize edge cuts from classic graph theory to the fuzzy settings,Yeh and Bang introduced the concept of fuzzy cut-sets.As the reduction of strength of connectedness between two vertices in a fuzzy graph appears more frequent-ly than the disconnection of entire graph,Mathew and Sunitha modified the definition of fuzzy cut-sets and introduced the definition of fuzzy arc cuts.Mathew and Sunitha argued that non-strong edges need not be considered in a fuzzy arc cut as the removal of such edges from a fuzzy graph would not change the strength of connectedness between any vertices.However,we give a counterexample,which illustrates that this viewpoint may contain some flaws.Based on this observation,we give a more general definition of fuzzy arc cuts(called fuzzy edge cuts)and study the types of edges in a minimal fuzzy edge cut.Moreover,we define fuzzy edge connectivity which is decided by two parameters,this definition allows us to talk about edge connectivity more precisely.The fuzzy edge connectivity of fuzzy trees,complete fuzzy graphs,fuzzy cycles are obtained and related algorithms are presented.In contrast to the research of local edge connectivity in classic graph theory,local edge connectivity in fuzzy settings has not been studied yet.In Chapter 4,we introduce the concepts of fuzzy local edge cuts and fuzzy local edge connectivity.And we inves-tigate the relations between fuzzy edge cuts and fuzzy local edge cuts and the relations between minimal fuzzy edge cuts and minimal fuzzy local edge cuts.A characterization for the weight of a minimal fuzzy local edge cut is obtained.The calculation formu-las about fuzzy local edge connectivity of any two vertices in special graphs are given.Finally,we present some applications of fuzzy edge connectivity and fuzzy local edge connectivity in communication networks.In Chapter 5,we introduce the definition of fuzzy average edge connectivity,which generalizes average edge connectivity from classic graph theory to the fuzzy settings.The calculation formulas about fuzzy average edge connectivity of fuzzy trees,com-plete fuzzy graphs,saturated fuzzy cycles are given.We discuss fuzzy average edge connectivity of edge deleted fuzzy subgraphs.The upper and lower bounds for fuzzy average edge connectivity of fuzzy graphs are obtained.We also characterize M-strong fuzzy graphs whose fuzzy average edge connectivity can arrive the lower bound.