Introduction one of the remarkable mathematical inventions of the 20 th century is that of fuzzy sets by lotfi. One of the usages of graph theory is to give a unified formalism for many very different. Connectivity, paths, trees, networks and flows, eulerian and hamiltonian graphs, coloring problems and complexity issues, a number of applications, large scale problems in graphs, similarity of nodes in large graphs, telephony problems and graphs. We now provide two popular ways of defining the distance between a pair of vertices. Rosenfeld7 considered fuzzy relations on fuzzy sets and developed the theory of fuzzy graphs in 1975. Many problems of practical interest can be modeled and solved by using graph algorithms. Introduction in 1736, euler first introduced the concept of graph theory.
Graphtheoretic applications and models usually involve connections to the real. Rosenfeld 46 considered fuzzy relations on fuzzy sets and developed the theory of fuzzy graphs in. Professors mordeson and nair have made a real contribution in putting together a very com prehensive book on. In these algorithms, data structure issues have a large role, too see e.
Also, jgj jvgjdenotes the number of verticesandeg jegjdenotesthenumberofedges. Fuzzy set theoryand its applications, fourth edition. On matrices associated with lfuzzy graphs 1801 definition 2. It is illustrated that the operations lexicographic products are not commutative. Ma 8151 fuzzy graph theory and applications prerequisite. This is the background to introduce the new concept fuzzy topological graph and some of its properties are discussed. This document pdf may be used for research, teaching and private study purposes.
This concept of obtaining fuzzy sum of fuzzy colorings problem has a. Keywords cartesian product, degree of an edge, composition, total edge degree. Graph theory with applications to engineering and computer science pdf. G to denote the numbers of vertices and edges in graph g.
New concepts of intervalvalued intuitionistic s, t. Fuzzy graph and relation based on the concepts of fuzzy relation described in the previous chapter, we introduce fuzzy graph and its related topics. Jacobson, ndomination in graphs, graph theory with applications to algorithms and computer science, wiley, new york 1985 282300. A relationship between the direct sum and the strong product of two fuzzy graphs is obtained. Bipolar fuzzy graph, level graph, cross product, lexicographic product of fuzzy graphs. Index terms fuzzy graph, direct sum, strong product, effective fuzzy graph, connectedness, upper and lower truncations. Diestel is excellent and has a free version available online. Research article novel properties of fuzzy labeling graphs. This book provides a timely overview of fuzzy graph theory, laying the foundation for future applications in a broad range of areas. Fuzzy graphs to the problem concerning group structure 219 group using these disjoint connectedness categories. In this section, fuzzy graphs will be analyzed from the connectedness viewpoint. In general, graph theory has a wide range of applications in diverse fields.
In this paper, firstly different kinds of interval valued intuitionistic s, tfuzzy graphs are defined. The concepts of fuzzy labeling and fuzzy magic labeling graph are introduced. The notion of complement of a fuzzy graph is modified and some of its properties are studied. Drawing a simple graph from known degrees stack exchange. Topological graph theory deals with ways to represents the geometric realization of graphs. 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. We will also develop characteristics of fuzzy relation and study various types of fuzzy relations. The elements of v are thought of as vertices of the graph and the elements of r are thought of as the edges similarly, any fuzzy relation. The degree of a vertex in the lexicographic products of two fuzzy graphs is obtained.
Here we consider fuzzy graph by taking fuzzy set of vertices and fuzzy set of edges. When the two fuzzy hypergraphs and are same the weak isomorphism between them becomes an isomorphism and similarly the coweak isomorphism between them also becomes isomorphism. Regular fuzzy graphs, irregular fuzzy graphs, antipodal fuzzy graphs, bipolar fuzzy graphs, complementary fuzzy graphs, bipolar fuzzy hypergraph, fuzzy dual graph etc. We write vg for the set of vertices and eg for the set of edges of a graph g.
One way is to define the distance disx,y between x and y as the length of the shortest strongest path between them. While typically many approaches have been mainly mathematics focused, graph theory has become a tool used by scientists, researchers, and engineers in using modeling techniques to solve realworld problems. Let b be any point of a digraph g, and let gb be the sub graph obtained from g by the removal of b. After that fuzzy graph theory becomes a vast research area. The concepts of fuzzy homomorphism and strong homomorphism are also introduced. Imparts developments in various properties of fuzzy topology viz. An automorphism of a fuzzy hypergraph is an isomorphism of to itself. Arc analysis of fuzzy graph structures, cycles in fuzzy graphs, blocks in fuzzy graphs, cycle connectivity of fuzzy graphs are discussed in the subsequent chapters. In the open literature, there are many papers written on the subject of fuzzy graph theory. A point b is said to be of the type i,j if g is in ci, while gb is in cj. The connectivity parameter only discusses the number of sub graphs. It is proved that every fuzzy magic graph is a fuzzy labeling graph, but the converse is not true. 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.
Graph theory, narosa addison wesley, indian student edition, 1988. We have shown that the removal of a fuzzy bridge from a fuzzy magic cycle with odd nodes reduces the strength of a fuzzy magic cycle. Fuzzy graphs and fuzzy hypergraphs studies in fuzziness. Graph theory for operations research and management. E wherev isasetofvertices andeisamultiset of unordered pairs of vertices. After introducing and developing fuzzy set theory, a lot of studies have been done in this field and then a result appeared as a fuzzy graph combination of graph theory and fuzzy set theory. Organization of the thesis and the necessary basic definitions required for the development of the thesis are given. Fuzzy graphs graph theory is proved to be tremendously useful in modeling the essential features of systems with finite components. What are some good books for selfstudying graph theory. The results will be applied to clustering analysis and modelling of information networks. The degree of a vertex in the strong product of two fuzzy graphs is obtained.
The first textbook on graph theory was written by denes konig, and published in 1936. Fuzzy magic labeling for some graphs like path, cycle, and star graph is defined. The origins of graph theory can be traced back to eulers work on the konigsberg. At the end of each chapter, there is a section with. Graph theory has wide range of applications in the eld of computer networks, chemical structures, biological models, and real life problems. A planar embedding g of a planar graph g can be regarded as a graph isomorphic to g. We introduce some definitions for fuzzy graphs and provide examples to explain various notions introduced.
In this book, youll learn about the essential elements of graph the. Graph and sub graphs, isomorphic, homomorphism graphs, 2 paths, hamiltonian circuits, eulerian graph, connectivity 3 the bridges of konigsberg, transversal, multi graphs, labeled graph 4 complete, regular and bipartite graphs, planar graphs 5 graph colorings, chromatic number, connectivity, directed graphs 6 basic definitions, tree graphs, binary trees, rooted trees. 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. In this paper, the intuitionistic fuzzy organizational and neural network models, intuitionistic. 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. Bhutani department of mathematics, the catholic university of america, washington, dc 20064, usa received 24 august 1988 abstract. Chandrasekaran, domination in fuzzy graph, advances in. We introduced several types of arcs in the interval valued intuitionistic s, tfuzzy graphs and studied their. A fuzzy graph g is a pair v, r, where v is a set of vertices, and r is a fuzzy relation on v. Handbook of graph theory history of graph theory routledge. Let g be connected fuzzy graph and have n vertices m edges if n 4, m n and g is not a circuit then. In crisp hyper graphs when two hypergraphs are isomorphic they are of same order. Somasundaram 9 presented the concepts of domination in fuzzy graphs.
In this chapter the progress of the field, fuzzy graph is presented. Connectivity plays a vital role in all these models. Hedetniemi, towards a theory of domination in graphs, networks, 7 1977 247261. In the history of mathematics, the solution given by euler of the well known konigsberg. 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. On irregular fuzzy graphs 519 regular fuzzy graph of total degree k or a ktotally regular fuzzy graph. In early 1987, the frontiers of topological graph theory are advancing in numerous di erent directions. Pdf cs6702 graph theory and applications lecture notes. In this work we discussed about fuzzyversion of classical graph theory.
It has many applications in fuzzy control and the most computationally intensive part of fuzzy control is defuzzification. During the same time yeh and bang 10 have also introduced various connectedness concepts in fuzzy graph. Novel applications of intuitionistic fuzzy digraphs in. Nagoor gani and radha6 introduced regular fuzzy graphs. In mathematics, graph theory is the study of graphs, which are mathematical structures used to. The connected, effective and complete properties of the operations lexicographic products are studied. Applications of fuzzy graph include data mining, image segmentation, clustering, image capturing, networking, communication, planning, scheduling, etc. A graph is a pair v, r, where v is a set and r is a relation on v. We believe that this book will help students, researchers and faculty of different institutes around the world to do fruitful research in fuzzy graph theory and related areas.
Till now fuzzy graphs has been witnessing a tremendous growth and finds applications in many branches. A nontrivial connected graph g is called even if for each vertex v of g there is a unique vertex v. In this thesis an attempt to develop the properties of basic concepts in fuzzy graphs such as fuzzy bridges, fuzzy cutnodes, fuzzy trees and blocks in fuzzy graphs have been made. However, there are relatively books available on the very same topic. The study of dominating sets in graphs was begun by orge and berge. Application of fuzzy ifthen rule in a fuzzy graph with. 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.
Here introduced ifthen rules with modus ponens method and related results are proved. The latter appeared in the book vorstudien zur topologie. Crisp graph and fuzzy graph both are structurally similar. Pattern recognition letters 9 1989159162 april 1989 northholland on automorphisms of fuzzy graphs kiran r. Fuzzy graph theory, a combination of graph theory and fuzzy set theory have been applied in various elds of science and engineering. The basis of graph theory is in combinatorics, and the role of graphics is only in visualizing things. But when there is an uncertainty on vertices andor edges. Free graph theory books download ebooks online textbooks. Research article novel properties of fuzzy labeling graphs a. The fuzzy graph theory as a generalization of eulers graph theory was.