This book provides a timely overview of fuzzy graph theory, laying the foundation for future applications in a broad range of areas. In the open literature, there are many papers written on the subject of fuzzy graph theory. For example, the vehicle travel time or vehicle capacity on a road network may not be known exactly. If there is a vertex which is adjacent to vertices with distinct neighbourhood degrees, then g is called an irregular bipolar intuitionistic fuzzy graph.
Chapter 2 fuzzy graph structures basic concepts in this chapter, we introduce the concept of fuzzy graph structures as an extension to that of graph structures of e. Center for the mathematics of uncertainty an introduction to the mathematics of uncertainty including set theory, logic, probability, fuzzy sets, rough sets, and evidence theory. However, there are relatively books available on the very same topic. Fuzzy graph theory is growing rapidly, with numerous applications in many domains, including networking, communication, data mining, clustering, image capturing, image segmentation, planning, and scheduling.
In such cases, it is natural to deal with the uncertainty using the methods of fuzzy. Bhattacharya 10 has extended the definitions of eccentricity and center based on the metric in fuzzy graphs defined in 5, and the inequality rg. Recently rashmanlou and pal defined irregular interval valued fuzzy graphs. After that, zimmermann 1993 introduced recent application of fuzzy set theory. Fuzzy set theoryand its applications, fourth edition. In this paper, our aim is to introduce the notion of anti fuzzy graph and some properties and. Fuzzy graph coloring is one of the most important problems of fuzzy graph theory. Applicationof ifgraphsandifrelationmethodsarealsodeveloped. Thenotionsoffuzzysoftgraph,union,intersectionoftwo. Following the usual convention of graph theory, a symmetric graph is simply a weighted undirected graph such that the edge corresponding to a pair of symmetric arcs u, v and v, u, and the weight between two nodes u and v is.
It started in 1965 after the publication of lotfi asker zadehs seminal work fuzzy sets. The application of fuzzy magic graph is illustrated with suitable example. The fuzzy graph theory as a generalization of eulers graph theory was first. In this paper we consider the problem of scheduling n jobs on a single machine and obtain the minimum value of the job completion times which is equivalent to finding. One way is to define the distance disx,y between x and y as the length of the shortest strongest path between them. Graph theory has numerous application to problems in systems analysis, operations research, economics, and transportation. The 2dominating set d of a graph is defined as if for every node v. Bipolar fuzzy graphs, irregular bipolar fuzzy graphs, totally irregular bipolar fuzzy graphs. A fuzzy graph g v, r is called symmetric iff r is symmetric. He has introduced another elaborated definition of fuzzy graphs and also proved many results on the fuzzy graph as an analog of graph theory. A first look at graph theory world scientific publishing. In application to basic fields such as set theory, arith metic, topology, graph theory, proba bility theory, and logic, fuzzification leads to fuzzy set theory, fuzzy arith metic, fuzzy topology, fuzzy graph the ory, and fuzzy logic in its narrow sense. Some problems in graph theory studies on fuzzy graphs thesis submitted to the cochin university of science and technology for the award of the degree of doctor ofphilosophy under the faculty of science by m.
Ma 8151 fuzzy graph theory and applications prerequisite. It gives an introduction to the subject with sufficient theory for students at those levels, with emphasis on algorithms and applications. As mentioned in the introduction this definition is more close to a graph rather than a fuzzy graph since, in a fuzzy graph the concept of strength of connectedness plays a crucial role. This is a list of graph theory topics, by wikipedia page. Introduction first, euler established the concept of graph theory, in 1736. The solution of the konigsberg bridge problem given by euler is appraised to be the first theorem in the history of graph theory. Recall that a fuzzy relation on a set s 2 is a fuzzy subset of s x s, that is, a map. This procedure generalizes an ordinary mapping of a function f to a mapping between fuzzy sets. Myna, abstract in this paper, we use a fuzzy graph model to represent a traffic network of a city and discuss a method to find the different type of accidental zones in a traffic flows using edge coloring of a fuzzy graph. Aymeric perchant and isabella bloch 37 introduced a generic definition of. Fuzzy mathematics forms a branch of mathematics related to fuzzy set theory and fuzzy logic. Fuzzy graph, intuitionistic fuzzy graph, intuitionistic fuzzy set, energy ams classification. Every fuzzy magic graph is a fuzzy labeling graph,but the converse is not true.
This concept of obtaining fuzzy sum of fuzzy colorings problem has a natural application in scheduling theory. In this section, we defined anti fuzzy graph as following. Professors mordeson and nair have made a real contribution in putting together a very com prehensive book on fuzzy graphs and fuzzy hypergraphs. This article gives a survey of the fundamentals of fuzzy set theory and describes potential applications. Pdf recent developments on the basics of fuzzy graph theory. On generalised fuzzy graph structures 1 introduction. Complement properties of tensor product of strong fuzzy. It is clear that g is an irregular bipolar intuitionistic fuzzy graph. After that fuzzy graph theory becomes a vast research area. Operations on intuitionistic anti fuzzy graphs ijrte. In this paper, strongly regular fuzzy graph which is analogous to the concept of strongly regular graph in crisp graph theory is introduced and examples are presented, necessary and sufficient condition for a cycle to be strongly regular fuzzy graph is provided and some properties of strongly regular fuzzy graph are studied.
The reference set sometimes denoted by or is called universe of discourse, and for each. It introduces readers to fundamental theories, such as craines work on fuzzy interval graphs, fuzzy analogs of marczewskis theorem, and the gilmore and hoffman characterization. Further, we define what we mean by an ordered fuzzy graph, a complete fuzzy graph and show that every fuzzy group can be embedded in a fuzzy group of the group of automorphisms of some fuzzy graph. In this chapter, the authors introduce some basic definitions related to fuzzy graphs like directed and undirected fuzzy graph, walk, path and circuit of a fuzzy graph, complete and strong fuzzy. We now provide two popular ways of defining the distance between a pair of vertices. Pdf basic definitions and concepts of graph theory. Pdf the concept of an antipodal bipolar fuzzy graph of a given bipolar. This function is also called a membership function. This distance is symmetric and is such that disx,x 0 since by our definition of a fuzzy graph, no path from x to x can have strength.
Theory, reference may be made to 1, for fuzzy graph theory, to 2 and for. This book is intended to be an introductory text for mathematics and computer science students at the second and third year levels in universities. Sc fuzzy set theory introduction fuzzy set theory fuzzy set theory is an extension of classical set theory where elements have varying degrees of membership. See glossary of graph theory terms for basic terminology examples and types of graphs. A crisp set consisting of a subset of ordered points is a crisp relation in the cartesian product x 1 x 1 and xx 22 xx 12, xx 12. It took a hundred years before the second important contribution of kirchhoff 9 had been made for the analysis of electrical networks. A logic based on the two truth values, true and false, is sometimes inadequate when describing human reasoning. Vijaya department of mathematics, marudupandiyar college, thanjavur, tamil nadu, india 6403 abstract in this work we introduce the complement of strong fuzzy graph, tensor product of fuzzy graphs and strong fuzzy graph. A fuzzy set a defined on a non empty set x is the family ax, a x. Let g be a bipolar intuitionistic fuzzy graph on g. Furthermore, in the years 1973 and 1975, zadeh, explain the theory of fuzzy sets that result from the extension as well as a fuzzy logic based on the set theory. However, in many cases, some aspects of a graph theoretic problem may be uncertain. The cardinality of a fuzzy set a, with finite universe x, is defined as. Kauffman 3 introduced fuzzy graphs using zadehs fuzzy relation 4.
Chapter 2 fuzzy graph structures basic concepts in this chapter, we introduce the concept of fuzzy graph structures as an extension to that of graph structures of. Rosenfeld 16 first introduced the concept of fuzzy graphs. Some basic theorems related to the stated graphs have also been presented. In 3, the notion of edge connectivity of a fuzzy graph if defined as given below.