Hypergraphs

This book provides an introduction to hypergraphs, its aim being to overcome the lack of recent manuscripts on this theory. In the literature hypergraphs have many other names such as set systems and families of sets. This work presents the theory of hypergraphs in its most original aspects, while also introducing and assessing the latest concepts on hypergraphs. The variety of topics, their originality and novelty are intended to help readers better understand the hypergraphs in all their diversity in order to perceive their value and power as mathematical tools. This book will be a great asset to upper-level undergraduate and graduate students in computer science and mathematics. It has been the subject of an annual Master's course for many years, making it also ideally suited to Master's students in computer science, mathematics, bioinformatics, engineering, chemistry, and many other fields. It will also benefit scientists, engineers and anyone else who wants to understand hypergraphs theory.

Combinatorial designs represent an important area of contemporary discrete mathematics closely related to such fields as finite geometries, regular graphs and multigraphs, factorisations of graphs, linear algebra, number theory, finite fields, group and quasigroup theory, Latin squares, and matroids. It has a history of more than 150 years when it started as a collection of unrelated problems. Nowadays the field is a well-developed theory with deep mathematical results and a wide range of applications in coding theory, cryptography, computer science, and other areas. In the most general setting, a combinatorial design consists of a ground set of elements and a collection of subsets of these elements satisfying some specific restrictions; the latter are often expressed in the language of graphs. On the other side, hypergraph theory is a relatively new field which started in early 60s of the last century as a generalization of graph theory. A hypergraph consists of a ground set of elements and a collection of subsets of these elements without any specific restrictions. In this sense the concept of hypergraph is more general than the concept of combinatorial design. While it started as a generalization of graph theory, hypergraph theory soon became a separate subject because many new properties have been discovered that miss or degenerate in graphs. Compared to graph theory, the language of hypergraphs not only allows us to formulate and solve more general problems, it also helps us to understand and solve several graph theory problems by simplifying and unifying many previously unrelated concepts. The main feature of this book is applying the hypergraph approach to the theory of combinatorial designs. An alternative title of it could be "Combinatorial designs as hypergraphs". There is no analogue to this book on the market. Its primary audience is researchers and graduate students taking courses in design theory, combinatorial geometry, finite geometry, discrete mathematics, graph theory, combinatorics, cryptography, information and coding theory, and similar areas. The aim of this book is to show the connection and mutual benefit between hypergraph theory and design theory. It does not intend to give a survey of all important results or methods in any of these subjects.

Graph Theory has proved to be an extremely useful tool for solving combinatorial problems in such diverse areas as Geometry, Algebra, Number Theory, Topology, Operations Research and Optimization. It is natural to attempt to generalise the concept of a graph, in order to attack additional combinatorial problems. The idea of looking at a family of sets from this standpoint took shape around 1960. In regarding each set as a ``generalised edge'' and in calling the family itself a ``hypergraph'', the initial idea was to try to extend certain classical results of Graph Theory such as the theorems of Turán and König. It was noticed that this generalisation often led to simplification; moreover, one single statement, sometimes remarkably simple, could unify several theorems on graphs. This book presents what seems to be the most significant work on hypergraphs.

The theory of graph coloring has existed for more than 150 years. Historically, graph coloring involved finding the minimum number of colors to be assigned to the vertices so that adjacent vertices would have different colors. From this modest beginning, the theory has become central in discrete mathematics with many contemporary generalizations and applications. Generalization of graph coloring-type problems to mixed hypergraphs brings many new dimensions to the theory ofcolorings. A main feature of this book is that in the case of hypergraphs, there exist problems on both the minimum and the maximum number of colors. This feature pervades the theory, methods, algorithms, and applications of mixed hypergraph coloring. The book has broad appeal. It will be of interest to bothpure and applied mathematicians, particularly those in the areas of discrete mathematics, combinatorial optimization, operations research, computer science, software engineering, molecular biology, and related businesses and industries. It also makes a nice supplementary text for courses in graph theory and discrete mathematics. This is especially useful for students in combinatorics and optimization. Since the area is new, students will have the chance at this stage to obtain results that maybecome classic in the future.

In the course of fuzzy technological development, fuzzy graph theory was identified quite early on for its importance in making things work. Two very important and useful concepts are those of granularity and of nonlinear ap proximations. The concept of granularity has evolved as a cornerstone of Lotfi A.Zadeh's theory of perception, while the concept of nonlinear approx imation is the driving force behind the success of the consumer electronics products manufacturing. It is fair to say fuzzy graph theory paved the way for engineers to build many rule-based expert systems. In the open literature, there are many papers written on the subject of fuzzy graph theory. However, there are relatively books available on the very same topic. Professors' Mordeson and Nair have made a real contribution in putting together a very com prehensive book on fuzzy graphs and fuzzy hypergraphs. In particular, the discussion on hypergraphs certainly is an innovative idea. For an experienced engineer who has spent a great deal of time in the lab oratory, it is usually a good idea to revisit the theory. Professors Mordeson and Nair have created such a volume which enables engineers and design ers to benefit from referencing in one place. In addition, this volume is a testament to the numerous contributions Professor John N. Mordeson and his associates have made to the mathematical studies in so many different topics of fuzzy mathematics.

This book presents the fundamental and technical concepts of fuzzy hypergraphs and explains their extensions and applications. It discusses applied generalized mathematical models of hypergraphs, including complex, intuitionistic, bipolar, m-polar fuzzy, Pythagorean, complex Pythagorean, and q-rung orthopair hypergraphs, as well as single-valued neutrosophic, complex neutrosophic and bipolar neutrosophic hypergraphs. In addition, the book also sheds light on real-world applications of these hypergraphs, making it a valuable resource for students and researchers in the field of mathematics, as well as computer and social scientists.

View the abstract.

We introduce certain concepts, including single-valued neutrosophic hyper-graph, line graph of single-valued neutrosophic hypergraph, dual single-valued neutrosophic hypergraph and transversal single-valued neutrosophic hypergraph.

Combinatorics is a book whose main theme is the study of subsets of a finite set. It gives a thorough grounding in the theories of set systems and hypergraphs, while providing an introduction to matroids, designs, combinatorial probability and Ramsey theory for infinite sets. The gems of the theory are emphasized: beautiful results with elegant proofs. The book developed from a course at Louisiana State University and combines a careful presentation with the informal style of those lectures. It should be an ideal text for senior undergraduates and beginning graduates.