Several Classes Of(i,k)-step Competition Graphs Of Fuzzy Digraphs

The American biologist Cohen proposed the concept of competition graph to investigate the food web in the ecosystem.Because of its important research value in both theory and application,the competition graph has become a hot topic in graph theory research.The fuzzy digraph is used to reflect the competition relation among things that are unclear,uncertain and indistinct.Because of the complexity and diversity of competition relation in real life,fuzzy competition graph has attracted a lot of researchers' attention.Subsequently,the concept and application of many classes of fuzzy competition graphs are proposed.In this thesis,the intuitionistic fuzzy competition graph,the bipolar fuzzy competi-tion graph,the bipolar single valued neutrosophic competition graph and the p-competition fuzzy graph in a fuzzy competition graph are generalized to the(i,k)-step intuitionistic fuzzy competition graph,(i,k)-step bipolar fuzzy competition graph,(i,k)-step bipolar sin-gle valued neutrosophic competition graph,m-step p-competition fuzzy graph and(i,k)-step p-competition fuzzy graph.We study the relationship among them and give examples of their application in real life.The thesis consists of six chapters.Chapter 1 introduces the application background,the research status and related basic concepts of competition graphs and fuzzy graphs,and propose the main content of this thesis.In chapter 2,we introduce the concept of(i,k)-step intuitionistic fuzzy competition graph,give and prove a condition of edge independence in(i,k)-step intuitionistic fuzzy competition graph,and prove that for an intuitionistic fuzzy digraph D,its competition graph of(i,k)-step intuitionistic fuzzy digraph CC(Di,k)is equal to its(i,k)-step intuitionistic fuzzy competition graph Ci,k(D).An example of(i,k)-step intuitionistic fuzzy competition graph in ecosystem is described.In chapter 3,we introduce the concept of(i,k)-step bipolar fuzzy competition graphs,and prove that for a bipolar fuzzy digraph D,its competition graph of(i,k)-step bipolar fuzzy digraph CC(Di,k)is equal to its(i,k)-step bipolar fuzzy competition graph Ci,k(D),and describes an example of(i,k)-step bipolar fuzzy competition graphs in social groups.In chapter 4,we introduce the concepts of(i,k)-step bipolar single valued neutrosophic digraphs and competition graphs,give the sufficient condition for an edge in an(i,k)-step bipolar single valued neutrosophic competition graph to be independent strong,and prove that for a bipolar single valued neutrosophic digraph D,its competition graph of(i,k)-step bipolar single valued neutrosophic digraph CC(Di,k)is equal to its(i,k)-step bipolar single valued neutrosophic competition graph Ci,k(D).Also an example of(i,k)-step bipolar single valued neutrosophic competition graphs in market competition is described.In chapter 5,we introduce the concepts of m-step p-competition fuzzy graph and(i,k)-step p-competition fuzzy graph,and then study some properties of them,including some conditions of edge being strong,the relationship of the edge membership values between Cmp(D)and Cmj(D).the relationship of the edge membership values between Ci,k p(D)and Ci,k j(D).the relationship between the fuzzy digraph and the fuzzy subdigraph and so on,and an example in food web is illustrated.In chapter 6,we introduce the domination sets and domination numbers in fuzzy di-graphs,and give the bounds of domination numbers for fuzzy digraphs with different restric-tions.We prove the conclusions:(1)??p-?N+(D).Here,p is the cardinality of the fuzzy set V in the digraph D=(V,?,?),?N+(D)is the maximum fuzzy outneighborhood degree in D;(2)If a fuzzy digraph D has a total of m arcs(x1,y1),(x2,y2),...,(xm,ym)satisfies the(xi,yi}=?(xi)??(yi),i=1,2,...,m,z1,z2...,zn0 are the isolated vertices in D,then?j=1 n0 ?(zj)???n0+m where xi,yi repeatable;(3)For any fuzzy digraph D,if the membership values of all vertices in the minimum domination set S are all 1,then p/?N+(D)+???p/?N+()D+1.