Kombinatorik og grafteori

"This volume contains nine survey articles by the invited speakers of the 30th British Combinatorial Conference, one of the major international events in combinatorics. Written by leading experts in the field, these articles provide a snapshot of current developments in combinatorics for researchers and graduate students in discrete mathematics"--

This book provides a comprehensive introduction to the foundations and frontiers of graph neural networks. In addition, the book introduces the basic concepts and definitions in graph representation learning and discusses the development of advanced graph representation learning methods with a focus on graph neural networks. The book providers researchers and practitioners with an understanding of the fundamental issues as well as a launch point for discussing the latest trends in the science. The authors emphasize several frontier aspects of graph neural networks and utilize graph data to describe pairwise relations for real-world data from many different domains, including social science, chemistry, and biology. Several frontiers of graph neural networks are introduced, which enable readers to acquire the needed techniques of advances in graph neural networks via theoretical models and real-world applications. 

This book treats the elements of discrete mathematics that have important applications in computer science, thus providing the necessary tools for the reader to come to a competent mathematical judgement of modern developments in the age of information. Almost all assertions are shown with full proofs. Exercises are provided, with solutions presented in full detail.

This book includes chapters which discuss effective and efficient approaches in dealing with various aspects of social media analysis by using machine learning techniques from clustering to deep learning. A variety of theoretical aspects, application domains and case studies are covered to highlight how it is affordable to maximize the benefit of various applications from postings on social media platforms. Social media platforms have significantly influenced and reshaped various social aspects. They have set new means of communication and interaction between people, turning the whole world into a small village where people with internet connect can easily communicate without feeling any barriers. This has attracted the attention of researchers who have developed techniques and tools capable of studying various aspects of posts on social media platforms with main concentration on Twitter. This book addresses challenging applications in this dynamic domain where it is not possible to continue applying conventional techniques in studying social media postings. The content of this book helps the reader in developing own perspective about how to benefit from machine learning techniques in dealing with social media postings and how social media postings may directly influence various applications.

This book is intended to provide graduate students and researchers in graph theory with an overview of the elementary methods of graph Ramsey theory. It is especially targeted towards graduate students in extremal graph theory, graph Ramsey theory, and related fields, as the included contents allow the text to be used in seminars. It is structured in thirteen chapters which are application-focused and largely independent, enabling readers to target specific topics and information to focus their study. The first chapter includes a true beginner's overview of elementary examples in graph Ramsey theory mainly using combinatorial methods. The following chapters progress through topics including the probabilistic methods, algebraic construction, regularity method, but that's not all. Many related interesting topics are also included in this book, such as the disproof for a conjecture of Borsuk on geometry, intersecting hypergraphs, Turan numbers and communication channels, etc.

This proceedings volume gathers selected, revised papers presented at the 51st Southeastern International Conference on Combinatorics, Graph Theory and Computing (SEICCGTC 2020), held at Florida Atlantic University in Boca Raton, USA, on March 9-13, 2020. The SEICCGTC is broadly considered to be a trendsetter for other conferences around the world - many of the ideas and themes first discussed at it have subsequently been explored at other conferences and symposia.The conference has been held annually since 1970, in Baton Rouge, Louisiana and Boca Raton, Florida. Over the years, it has grown to become the major annual conference in its fields, and plays a major role in disseminating results and in fostering collaborative work.This volume is intended for the community of pure and applied mathematicians, in academia, industry and government, working in combinatorics and graph theory, as well as related areas of computer science and the interactions among these fields.

One of the most preeminent ways of applying mathematics in real-world scenario modeling involves graph theory. A graph can be undirected or directed depending on whether the pairwise relationships among objects are symmetric or not. Nevertheless, in many real-world situations, representing a set of complex relational objects as directed or undirected is not su¢ cient. Weighted graphs o§er a framework that helps to over come certain conceptual limitations. We show using the concept of an isomorphism that weighted graphs have a natural connection to fuzzy graphs. As we show in the book, this allows results to be carried back and forth between weighted graphs and fuzzy graphs. This idea is in keeping with the important paper by Klement and Mesiar that shows that many families of fuzzy sets are lattice isomorphic to each other. We also outline the important work of Head and Weinberger that show how results from ordinary mathematics can be carried over to fuzzy mathematics. We focus on the concepts connectivity, degree sequences and saturation, and intervals and gates in weighted graphs.

This book is designed to meet the requirement of undergraduate and postgraduate students pursuing computer science, information technology, mathematical science, and physical science course. No formal prerequisites are needed to understand the text matter except a very reasonable background in college algebra. The text contains in-depth coverage of all major topics proposed by professional institutions and universities for a discrete mathematics course. It emphasizes on problem-solving techniques, pattern recognition, conjecturing, induction, applications of varying nature, proof technique, algorithmic development, algorithm correctness, and numeric computations. A sufficient amount of theory is included for those who enjoy the beauty in development of the subject and a wealth of applications as well as for those who enjoy the power of problem-solving techniques.Biographical sketches of nearly 25 mathematicians and computer scientists who have played a significant role in the development of the field are threaded into the text to provide a human dimension and attach a human face to major discoveries. Each section of the book contains a generous selection of carefully tailored examples to classify and illuminate various concepts and facts. Theorems are backbone of mathematics. Consequently, this book contains the various proof techniques, explained and illustrated in details. Most of the concepts, definitions, and theorems in the book are illustrated with appropriate examples. Proofs shed additional light on the topic and enable students to sharpen thin problem-solving skills. Each chapter ends with a summary of important vocabulary, formulae, properties developed in the chapter, and list of selected references for further exploration and enrichment.

The present book is based on the curriculum of undergraduate and postgraduate courses of universities in India and abroad. Every effort is made to present the various topics in the theory of graphs in a logical manner with adequate historical background and include suitable figures to illustrate concepts and results ideally. The formidable exercises, neither easy nor straightforward, are bold faced and highlighted. The theory portion of each chapter is studied thoroughly as it helps solve many of the problems with comparative ease. Selected material from this book is used for a semester course on graph theory, while the entire book serves for a whole session course.

This book provides a foundation for the mathematics of blockchain. It is inspired by a general analysis and synthesis of the current knowledge of blockchain technology and starts by laying a foundation for the mathematics of blockchain. The aim is for research in the area of blockchain to lead their study from the construction highlighted in this book.First, the basis of a blockchain is set: block, transaction contents, block header, Merkle tree, nonce, Proof-of-Work. Connections with elliptic curves and cryptographic signatures are made.Second, the book ties this with a Graph and Matrix Theories approach and models the peer-to-peer relationship through the Bitcoin Network.Third, it is proposed further modelling, notably around halving, optimal storing, or diffusion of information, which are consequences of the mathematical foundation. The notion of Entropy of Privacy and the Particles model are introduced.Finally, the mathematical statements therein are proven and essential reminders can be found before each section, so the content can be accessible from a graduate level.

The main focus of this book is a pair of cooperative control problems: consensus and cooperative output regulation. Its emphasis is on complex multi-agent systems characterized by strong nonlinearity, large uncertainty, heterogeneity, external disturbances and jointly connected switching communication topologies. The cooperative output regulation problem is a generalization of the classical output regulation problem to multi-agent systems and it offers a general framework for handling a variety of cooperative control problems such as consensus, formation, tracking and disturbance rejection. The book strikes a balance between rigorous mathematical proof and engineering practicality. Every design method is systematically presented together with illustrative examples and all the designs are validated by computer simulation. The methods presented are applied to several practical problems, among them the leader-following consensus problem of multiple Euler-Lagrange systems, attitude synchronization of multiple rigid-body systems, and power regulation of microgrids. The book gives a detailed exposition of two approaches to the design of distributed control laws for complex multi-agent systems-the distributed-observer and distributed-internal-model approaches. Mastering both will enhance a reader's ability to deal with a variety of complex real-world problems. Cooperative Control of Multi-agent Systems can be used as a textbook for graduate students in engineering, sciences, and mathematics, and can also serve as a reference book for practitioners and theorists in both industry and academia. Some knowledge of the fundamentals of linear algebra, calculus, and linear systems are needed to gain maximum benefit from this book. Advances in Industrial Control reports and encourages the transfer of technology in control engineering. The rapid development of control technology has an impact on all areas of the control discipline. The series offers an opportunity for researchers to present an extended exposition of new work in all aspects of industrial control.

The first half of the book walks the reader through methods of counting, both direct elementary methods and the more advanced method of generating functions. Then, in the second half of the book, the reader learns how to apply these methods to fascinating objects, such as graphs, designs, random variables, partially ordered sets, and algorithms. In short, the first half emphasizes depth by discussing counting methods at length; the second half aims for breadth, by showing how numerous the applications of our methods are.New to this fifth edition of A Walk Through Combinatorics is the addition of Instant Check exercises - more than a hundred in total - which are located at the end of most subsections. As was the case for all previous editions, the exercises sometimes contain new material that was not discussed in the text, allowing instructors to spend more time on a given topic if they wish to do so. With a thorough introduction into enumeration and graph theory, as well as a chapter on permutation patterns (not often covered in other textbooks), this book is well suited for any undergraduate introductory combinatorics class.

Im täglichen Leben sind wir zunehmend von Codes umgeben, die mathematisch konstruiert werden. Sie sind teils leicht erkennbar (Strichcode, ISBN, IBAN, QR) und teils eher verborgen (GPS, WLAN, CD, DVD). In diesem Buch werden solche Codes vorgestellt. Es wird dargelegt, wie sie aufgebaut sind, wie sie funktionieren und welche Mathematik zu ihrer Entwicklung und Anwendung notwendig ist. Die Lesenden lernen, eigenhändig Codes zu erstellen, Fehler zu erkennen und zu korrigieren:EAN, ISBN und deren Barcodedarstellung sowie die internationale Bankkontonummer IBAN werden erarbeitet.Kleine QR-Codes werden mit den vorgestellten Methoden (Paritätsprüfung, Linearcode, Polynomcode, zyklischer Code und Reed-Solomon Code) anschaulich realisiert.An der Herstellung einer Mini-CD mit einem CIRC-Code über einem kleinen Körper werden wesentliche Konstruktionsprinzipien von neuen Codes aus bestehenden Codes, wie z.B. Kürzen, Erweitern, Spreizen (Interleaving) und gekreuztes Spreizen (Cross-Interleaving) veranschaulicht.Das Verstehen von Mathematik wird durch diese selbstständige Erstellung und Verwendung didaktisch maßgeschneiderter Codes wesentlich gefördert.Ein besonderer Fokus des Buchs liegt auf elementaren Methoden des Rechnens mit ganzen Zahlen und Polynomen. Für diese benötigt man nur den Satz von der Division mit Rest als zentrale Aussage ¿ daher können große Abschnitte bereits mit Lernenden der Sekundarstufe II erarbeitet und die Grundlagen wesentlicher Teile der Codierungstheorie von den Lernenden mathematisch korrekt erfasst werden. Für Ausführungen, zu deren Verständnis Kenntnisse notwendig sind, die über die Mathematik der Sekundarstufe II hinausgehen, liegt ein ausführlicher Anhang vor (Vektorräume, Matrizen, Rechnen in endlichen Körpern).

This monograph is designed to be an in-depth introduction to domination in graphs. It focuses on three core concepts: domination, total domination, and independent domination. It contains major results on these foundational domination numbers, including a wide variety of in-depth proofs of selected results providing the reader with a toolbox of proof techniques used in domination theory. Additionally, the book is intended as an invaluable reference resource for a variety of readerships, namely, established researchers in the field of domination who want an updated, comprehensive coverage of domination theory; next, researchers in graph theory who wish to become acquainted with newer topics in domination, along with major developments in the field and some of the proof techniques used; and, graduate students with interests in graph theory, who might find the theory and many real-world applications of domination of interest for masters and doctoral thesis topics. The focusedcoverage also provides a good basis for seminars in domination theory or domination algorithms and complexity.The authors set out to provide the community with an updated and comprehensive treatment on the major topics in domination in graphs. And by Jove, they¿ve done it! In recent years, the authors have curated and published two contributed volumes: Topics in Domination in Graphs, © 2020 and Structures of Domination in Graphs, © 2021. This book rounds out the coverage entirely. The reader is assumed to be acquainted with the basic concepts of graph theory and has had some exposure to graph theory at an introductory level. As graph theory terminology sometimes varies, a glossary of terms and notation is provided at the end of the book.

Dieses Lehrbuch bietet eine kompakte Einführung in die Grundlagen der Graphentheorie und die Methoden der Netzwerkanalyse. Zahlreiche praktische Beispiele und Übungsaufgaben mit Lösungsvorschlägen helfen Leser:innen dabei, die theoretischen Konzepte besser zu verstehen und anzuwenden. Dabei werden unterschiedliche Technologien und Programmiersprachen verwendet, um ein breites Spektrum an Anwendungen abzudecken. Darüber hinaus beleuchten spezielle Kapitel die Methodik mit Blick auf die Planung und Durchführung eigener Netzwerkanalyseprojekte sowie ethische und datenschutzrechtliche Aspekte. So liefert das Buch nicht nur einen theoretischen Überblick, sondern auch praktische Tipps und Anleitungen für die Untersuchung eigener netzwerkanalytischer Fragestellungen. Dieses Buch eignet sich nicht nur als Nachschlagewerk für Studierende und Dozierende vielfältiger Fachdisziplinen mit curricularem Bezug zum Thema, sondern auch als Ergänzung des Repertoires von Praktiker:innen im Bereich Data Science mit Interesse an der Untersuchung von Netzwerken. Ob als theoretischer Einstieg oder als praktischer Ratgeber - dieses Buch leistet einen Beitrag für die Untersuchung und Analyse von Netzwerken und bietet eine Grundlage für weiterführende Studien und Projekte.

This book contains contributions in the area of Network Science, presented at the 14th International Conference on Complex Networks (CompleNet), 24-28 April, 2023 in Aveiro, Portugal. CompleNet is an international conference on complex networks that brings together researchers and practitioners from diverse disciplines¿from sociology, biology, physics, and computer science¿who share a passion to better understand the interdependencies within and across systems. CompleNet is a venue to discuss ideas and findings about all types networks, from biological, to technological, to informational and social. It is this interdisciplinary nature of complex networks that CompleNet aims to explore and celebrate.The audience of the work are professionals and academics working in Network Science, a highly-multidisciplinary field.

This book examines some issues involving climate change, human trafficking, and other serious world challenges made worse by climate change. Climate change increases the risk of natural disasters and thus creates poverty and can cause situations of conflict and instability. Displacement can occur giving traffickers an opportunity to exploit affected people. In the fuzzy graph theory part of the book, the relatively new concepts of fuzzy soft semigraphs and graph structures are used to study human trafficking, as well as its time intuitionistic fuzzy sets that have been introduced to model forest fires. The notion of legal and illegal incidence strength is used to analyze immigration to the USA. The examination of return refugees to their origin countries is undertaken. The neighborhood connectivity index is determined for trafficking in various regions in the world. The cycle connectivity measure for the directed graph of the flow from South Americato the USA is calculated. It is determined that there is a need for improvement in government response by countries. Outside the area of fuzzy graph theory, a new approach to examine climate change is introduced. Social network theory is used to study feedback processes that effect climate forcing. Tipping points in climate change are considered. The relationship between terrorism and climate change is examined. Ethical issues concerning the obligation of business organizations to reduce carbon emissions are also considered. Nonstandard analysis is a possible new area that could be used by scholars of mathematics of uncertainty. A foundation is laid to aid the researcher in the understanding of nonstandard analysis. In order to accomplish this, a discussion of some basic concepts from first-order logic is presented as some concepts of mathematics of uncertainty. An application to the theory of relativity is presented.

Neste livro, são realizados exercícios sobre os seguintes tópicos matemáticos:cálculo combinatório e cálculo de probabilidadeprobabilidade condicional e teorema de Bayescálculo de parâmetros estatísticos: média, mediana, moda e variânciaTambém são apresentadas dicas teóricas iniciais para que a execução dos exercícios seja compreendida

This book gives a comprehensive view of graph theory in informational retrieval (IR) and natural language processing(NLP). This book provides number of graph techniques for IR and NLP applications with examples. It also provides understanding of graph theory basics, graph algorithms and networks using graph. The book is divided into three parts and contains nine chapters. The first part gives graph theory basics and graph networks, and the second part provides basics of IR with graph-based information retrieval. The third part covers IR and NLP recent and emerging applications with case studies using graph theory. This book is unique in its way as it provides a strong foundation to a beginner in applying mathematical structure graph for IR and NLP applications. All technical details that include tools and technologies used for graph algorithms and implementation in Information Retrieval and Natural Language Processing with its future scope are explained in a clear and organized format.

Dieses Buch ist die erste fachübergreifende Einführung in die Kombinatorik in deutscher Sprache und umfasst Gesetzmäßigkeiten, Rechenregeln, Anwendungen sowie 50 Übungsaufgaben mit Lösungen. Es zeigt, wie dieses oft unterschätzte Fachgebiet zahlreiche Entwicklungen in Linguistik und Lyrik, Mathematik und Informatik, Chemie, Musik, Psychologie, Soziologie und Ökonomie beeinflusste.

This book highlights cutting-edge research in the field of network science, offering scientists, researchers, students, and practitioners a unique update on the latest advances in theory and a multitude of applications. It presents the peer-reviewed proceedings of the X International Conference on Complex Networks and their Applications (COMPLEX NETWORKS 2021). The carefully selected papers cover a wide range of theoretical topics such as network models and measures; community structure, network dynamics; diffusion, epidemics and spreading processes; resilience and control as well as all the main network applications, including social and political networks; networks in finance and economics; biological and neuroscience networks, and technological networks.

This interdisciplinary volume collects contributions from experts in their respective fields with as common theme diagrams.Diagrams play a fundamental role in the mathematical visualization and philosophical analysis of forms in space. Some of the most interesting and profound recent developments in contemporary sciences, whether in topology, geometry, dynamic systems theory, quantum field theory or string theory, have been made possible by the introduction of new types of diagrams, which, in addition to their essential role in the discovery of new classes of spaces and phenomena, have contributed to enriching and clarifying the meaning of the operations, structures and properties that are at the heart of these spaces and phenomena.The volume gives a closer look at the scope and the nature of diagrams as constituents of mathematical and physical thought, their function in contemporary artistic work, and appraise, in particular, the actual importance of the diagrams of knots, of braids, of fields, of interaction, of strings in topology and geometry, in quantum physics and in cosmology, but also in theory of perception, in plastic arts and in philosophy. The editors carefully curated this volume to be an inspiration to students and researchers in philosophy, phenomenology, mathematics and the sciences, as well as artists, musicians and the general interested audience.

Das Kompendium prasentiert die fur ein Mathematik- oder Informatikstudium benotigten Grundlagen der diskreten Mathematik in kompakter Form. Alle wichtigen Themen wie Mengen, Relationen und Funktionen, Logik, Graphen, abstrakte und lineare Algebra sowie diskrete Wahrscheinlichkeitstheorie werden abgedeckt. Damit erweist sich das Buch als nutzlicher Begleiter fur viele Vorlesungen im Studium.