This book provides an extensive set of tools for applying fuzzy mathematics and graph theory to real-life problems. Balancing the basics and latest developments in fuzzy graph theory, this book starts with existing fundamental theories such as connectivity, isomorphism, products of fuzzy graphs, and different types of paths and arcs in fuzzy graphs to focus on advanced concepts such as planarity in fuzzy graphs, fuzzy competition graphs, fuzzy threshold graphs, fuzzy tolerance graphs, fuzzy trees, coloring in fuzzy graphs, bipolar fuzzy graphs, intuitionistic fuzzy graphs, m-polar fuzzy graphs, applications of fuzzy graphs, and more. Each chapter includes a number of key representative applications of the discussed concept. An authoritative, self-contained, and inspiring read on the theory and modern applications of fuzzy graphs, this book is of value to advanced undergraduate and graduate students of mathematics, engineering, and computer science, as well as researchers interested in new developments in fuzzy logic and applied mathematics.

There is a sympathy of ideas among the fields of knot theory, infinite discrete group theory, and the topology of 3-manifolds. This book contains fifteen papers in which new results are proved in all three of these fields. These papers are dedicated to the memory of Ralph H. Fox, one of the world's leading topologists, by colleagues, former students, and friends. In knot theory, papers have been contributed by Goldsmith, Levine, Lomonaco, Perko, Trotter, and Whitten. Of these several are devoted to the study of branched covering spaces over knots and links, while others utilize the braid groups of Artin. Cossey and Smythe, Stallings, and Strasser address themselves to group theory. In his contribution Stallings describes the calculation of the groups In/In+1 where I is the augmentation ideal in a group ring RG. As a consequence, one has for each k an example of a k-generator l-relator group with no free homomorphs. In the third part, papers by Birman, Cappell, Milnor, Montesinos, Papakyriakopoulos, and Shalen comprise the treatment of 3-manifolds. Milnor gives, besides important new results, an exposition of certain aspects of our current knowledge regarding the 3- dimensional Brieskorn manifolds.

This volume comprises 16 contributions that present advanced topics in graph domination, featuring open problems, modern techniques, and recent results. The focus is on primary dominating sets such as paired domination, connected domination, restrained domination, dominating functions, Roman domination, and power domination. Additionally, surveys include known results with a sample of proof techniques for each parameter. Of extra benefit to the reader, the first chapter includes a glossary of commonly used terms; the second chapter provides an overview of models of domination from which the parameters are defined. The book is intended to provide a reference for established researchers in the fields of domination and graph theory and graduate students who wish to gain knowledge of the topics covered as well as an overview of the major accomplishments in the field and proof techniques used.

Oriented matroids are a very natural mathematical concept which presents itself in many different guises and which has connections and applications to many different areas. These include discrete and computational geometry, combinatorics, convexity, topology, algebraic geometry, operations research, computer science and theoretical chemistry. This is the second edition of the first comprehensive, accessible account of the subject. It is intended for a diverse audience: graduate students who wish to learn the subject from scratch; researchers in the various fields of application who want to concentrate on certain aspects of the theory; specialists who need a thorough reference work; and others at academic points in between. A list of exercises and open problems ends each chapter. For the second edition, the authors have expanded the bibliography greatly to ensure that it remains comprehensive and up-to-date, and they have also added an appendix surveying research since the work was first published.

Impending technological advances will widen an adversary's attack plane over the next decade. Visualizing what the future will hold, and what new threat vectors could emerge, is a task that traditional planning mechanisms struggle to accomplish given the wide range of potential issues. Understanding and preparing for the future operating environment is the basis of an analytical method known as Threatcasting. It is a method that gives researchers a structured way to envision and plan for risks ten years in the future. Threatcasting uses input from social science, technical research, cultural history, economics, trends, expert interviews, and even a little science fiction to recognize future threats and design potential futures. During this human-centric process, participants brainstorm what actions can be taken to identify, track, disrupt, mitigate, and recover from the possible threats. Specifically, groups explore how to transform the future they desire into reality while avoiding an undesired future. The Threatcasting method also exposes what events could happen that indicate the progression toward an increasingly possible threat landscape. This book begins with an overview of the Threatcasting method with examples and case studies to enhance the academic foundation. Along with end-of-chapter exercises to enhance the reader's understanding of the concepts, there is also a full project where the reader can conduct a mock Threatcasting on the topic of "the next biological public health crisis." The second half of the book is designed as a practitioner's handbook. It has three separate chapters (based on the general size of the Threatcasting group) that walk the reader through how to apply the knowledge from Part I to conduct an actual Threatcasting activity. This book will be useful for a wide audience (from student to practitioner) and will hopefully promote new dialogues across communities and novel developments in the area.

This book presents original peer-reviewed contributions from the London Mathematical Society (LMS) Midlands Regional Meeting and Workshop on 'Galois Covers, Grothendieck-Teichmuller Theory and Dessinsd'Enfants', which took place at the University of Leicester, UK, from 4 to 7 June, 2018. Within the theme of the workshop, the collected articles cover a broad range of topics and explore exciting new links between algebraic geometry, representation theory, group theory, number theory and algebraic topology. The book combines research and overview articles by prominent international researchers and provides a valuable resource for researchers and students alike.

Graph-Theoretical Matrices in Chemistry presents a systematic survey of graph-theoretical matrices and highlights their potential uses. This comprehensive volume is an updated, extended version of a former bestseller featuring a series of mathematical chemistry monographs. In this edition, nearly 200 graph-theoretical matrices are included. This second edition is organized like the previous one-after an introduction, graph-theoretical matrices are presented in five chapters: The Adjacency Matrix and Related Matrices, Incidence Matrices, The Distance Matrix and Related Matrices, Special Matrices, and Graphical Matrices. Each of these chapters is followed by a list of references. Among the matrices presented several are novel and some are known only to a few. The properties and potential usefulness of many of the presented graph-theoretical matrices in chemistry have yet to be investigated. Most of the graph-theoretical matrices presented have been used as sources of molecular descriptors usually referred to as topological indices. They are particularly concerned with a special class of graphs that represents chemical structures involving molecules. Due to its multidisciplinary scope, this book will appeal to a broad audience ranging from chemistry and mathematics to pharmacology.

The Nature of Complex Networks provides a systematic introduction to the statistical mechanics of complex networks and the different theoretical achievements in the field that are now finding strands in common. The book presents a wide range of networks and the processes taking place on them, including recently developed directions, methods, and techniques. It assumes a statistical mechanics view of random networks based on the concept of statistical ensembles but also features the approaches and methods of modern random graph theory and their overlaps with statistical physics. This book will appeal to graduate students and researchers in the fields of statistical physics, complex systems, graph theory, applied mathematics, and theoretical epidemiology.

This volume collects together research and survey papers written by invited speakers of the conference celebrating the 70th birthday of Laszlo Lovasz. The topics covered include classical subjects such as extremal graph theory, coding theory, design theory, applications of linear algebra and combinatorial optimization, as well as recent trends such as extensions of graph limits, online or statistical versions of classical combinatorial problems, and new methods of derandomization. Laszlo Lovasz is one of the pioneers in the interplay between discrete and continuous mathematics, and is a master at establishing unexpected connections, "building bridges" between seemingly distant fields. His invariably elegant and powerful ideas have produced new subfields in many areas, and his outstanding scientific work has defined and shaped many research directions in the last 50 years. The 14 contributions presented in this volume, all of which are connected to Laszlo Lovasz's areas of research, offer an excellent overview of the state of the art of combinatorics and related topics and will be of interest to experienced specialists as well as young researchers.

This book collects the extended abstracts of the accepted contributions to EuroComb21. A similar book is published at every edition of EuroComb (every two years since 2001) collecting the most recent advances in combinatorics, graph theory, and related areas. It has a wide audience in the areas, and the papers are used and referenced broadly.

This book bridges the gaps between logic, mathematics and computer science by delving into the theory of well-quasi orders, also known as wqos. This highly active branch of combinatorics is deeply rooted in and between many fields of mathematics and logic, including proof theory, commutative algebra, braid groups, graph theory, analytic combinatorics, theory of relations, reverse mathematics and subrecursive hierarchies. As a unifying concept for slick finiteness or termination proofs, wqos have been rediscovered in diverse contexts, and proven to be extremely useful in computer science. The book introduces readers to the many facets of, and recent developments in, wqos through chapters contributed by scholars from various fields. As such, it offers a valuable asset for logicians, mathematicians and computer scientists, as well as scholars and students.

When you picture human-data interactions (HDI), what comes to mind? The datafication of modern life, along with open data initiatives advocating for transparency and access to current and historical datasets, has fundamentally transformed when, where, and how people encounter data. People now rely on data to make decisions, understand current events, and interpret the world. We frequently employ graphs, maps, and other spatialized forms to aid data interpretation, yet the familiarity of these displays causes us to forget that even basic representations are complex, challenging inscriptions and are not neutral; they are based on representational choices that impact how and what they communicate. This book draws on frameworks from the learning sciences, visualization, and human-computer interaction to explore embodied HDI. This exciting sub-field of interaction design is based on the premise that every day we produce and have access to quintillions of bytes of data, the exploration and analysis of which are no longer confined within the walls of research laboratories. This volume examines how humans interact with these data in informal (not work or school) environments, paritcularly in museums. The first half of the book provides an overview of the multi-disciplinary, theoretical foundations of HDI (in particular, embodied cognition, conceptual metaphor theory, embodied interaction, and embodied learning) and reviews socio-technical theories relevant for designing HDI installations to support informal learning. The second half of the book describes strategies for engaging museum visitors with interactive data visualizations, presents methodologies that can inform the design of hand gestures and body movements for embodied installations, and discusses how HDI can facilitate people's sensemaking about data. This cross-disciplinary book is intended as a resource for students and early-career researchers in human-computer interaction and the learning sciences, as well as for more senior researchers and museum practitioners who want to quickly familiarize themselves with HDI.

This book gives the state-of-the-art on transversals in linear uniform hypergraphs. The notion of transversal is fundamental to hypergraph theory and has been studied extensively. Very few articles have discussed bounds on the transversal number for linear hypergraphs, even though these bounds are integral components in many applications. This book is one of the first to give strong non-trivial bounds on the transversal number for linear hypergraphs. The discussion may lead to further study of those problems which have not been solved completely, and may also inspire the readers to raise new questions and research directions. The book is written with two readerships in mind. The first is the graduate student who may wish to work on open problems in the area or is interested in exploring the field of transversals in hypergraphs. This exposition will go far to familiarize the student with the subject, the research techniques, and the major accomplishments in the field. The photographs included allow the reader to associate faces with several researchers who made important discoveries and contributions to the subject. The second audience is the established researcher in hypergraph theory who will benefit from having easy access to known results and latest developments in the field of transversals in linear hypergraphs.

This book focusses on techniques for automating the procedure of creating external labelings, also known as callout labelings. In this labeling type, the features within an illustration are connected by thin leader lines (called leaders) with their labels, which are placed in the empty space surrounding the image. In general, textual labels describing graphical features in maps, technical illustrations (such as assembly instructions or cutaway illustrations), or anatomy drawings are an important aspect of visualization that convey information on the objects of the visualization and help the reader understand what is being displayed. Most labeling techniques can be classified into two main categories depending on the "distance" of the labels to their associated features. Internal labels are placed inside or in the direct neighborhood of features, while external labels, which form the topic of this book, are placed in the margins outside the illustration, where they do not occlude the illustration itself. Both approaches form well-studied topics in diverse areas of computer science with several important milestones. The goal of this book is twofold. The first is to serve as an entry point for the interested reader who wants to get familiar with the basic concepts of external labeling, as it introduces a unified and extensible taxonomy of labeling models suitable for a wide range of applications. The second is to serve as a point of reference for more experienced people in the field, as it brings forth a comprehensive overview of a wide range of approaches to produce external labelings that are efficient either in terms of different algorithmic optimization criteria or in terms of their usability in specific application domains. The book mostly concentrates on algorithmic aspects of external labeling, but it also presents various visual aspects that affect the aesthetic quality and usability of external labeling.

This volume highlights the mathematical research presented at the 2019 Association for Women in Mathematics (AWM) Research Symposium held at Rice University, April 6-7, 2019. The symposium showcased research from women across the mathematical sciences working in academia, government, and industry, as well as featured women across the career spectrum: undergraduates, graduate students, postdocs, and professionals. The book is divided into eight parts, opening with a plenary talk and followed by a combination of research paper contributions and survey papers in the different areas of mathematics represented at the symposium: algebraic combinatorics and graph theory algebraic biology commutative algebra analysis, probability, and PDEs topology applied mathematics mathematics education

The emergence of multilayer networks as a concept from the field of complex systems provides many new opportunities for the visualization of network complexity, and has also raised many new exciting challenges. The multilayer network model recognizes that the complexity of relationships between entities in real-world systems is better embraced as several interdependent subsystems (or layers) rather than a simple graph approach. Despite only recently being formalized and defined, this model can be applied to problems in the domains of life sciences, sociology, digital humanities, and more. Within the domain of network visualization there already are many existing systems, which visualize data sets having many characteristics of multilayer networks, and many techniques, which are applicable to their visualization. In this Synthesis Lecture, we provide an overview and structured analysis of contemporary multilayer network visualization. This is not only for researchers in visualization, but also for those who aim to visualize multilayer networks in the domain of complex systems, as well as those solving problems within application domains. We have explored the visualization literature to survey visualization techniques suitable for multilayer network visualization, as well as tools, tasks, and analytic techniques from within application domains. We also identify the research opportunities and examine outstanding challenges for multilayer network visualization along with potential solutions and future research directions for addressing them.

Die Theorie der regularen Graphen (The Theory of Regular Graphs), written by the Danish Mathematician Julius Petersen in 1891, is often considered the first strictly theoretical paper dealing with graphs. In the 130 years since then, regular graphs have been a common and popular area of study. While regular graphs are typically considered to be graphs whose vertices all have the same degree, a more general interpretation is that of graphs possessing some common characteristic throughout their structure. During the past several decades, however, there has been some increased interest in investigating graphs possessing a property that is, in a sense, opposite to regularity. It is this topic with which this book deals, giving rise to a study of what might be called irregularity in graphs. Here, various irregularity concepts dealing with several topics in graph theory are described, such as degrees of vertices, graph labelings, weightings, colorings, graph structures, Eulerian and Hamiltonian properties, graph decompositions, and Ramsey-type problems.

This book provides a detailed description of network science concepts applied to power systems and electricity markets, offering an appropriate blend of theoretical background and practical applications for operation and power system planning. It discusses an approach to understanding power systems from a network science perspective using the direct recognition of the interconnectivity provided by the transmission system. Further, it explores the network properties in detail and characterizes them as a tool for online and offline applications for power system operation. The book includes an in-depth explanation of electricity markets problems that can be addressed from a graph theory perspective. It is intended for advanced undergraduate and graduate students in the fields of electric energy systems, operations research, management science and economics. Practitioners in the electric energy sector also benefit from the concepts and techniques presented here.

This book focuses on the latest developments in behaviormetrics and data science, covering a wide range of topics in data analysis and related areas of data science, including analysis of complex data, analysis of qualitative data, methods for high-dimensional data, dimensionality reduction, visualization of such data, multivariate statistical methods, analysis of asymmetric relational data, and various applications to real data. In addition to theoretical and methodological results, it also shows how to apply the proposed methods to a variety of problems, for example in consumer behavior, decision making, marketing data, and social network structures. Moreover, it discuses methodological aspects and applications in a wide range of areas, such as behaviormetrics; behavioral science; psychology; and marketing, management and social sciences. Combining methodological advances with real-world applications collected from a variety of research fields, the book is a valuable resource for researchers and practitioners, as well as for applied statisticians and data analysts.

This concise monograph present the complete history of the domination game and its variants up to the most recent developments and will stimulate research on closely related topics, establishing a key reference for future developments. The crux of the discussion surrounds new methods and ideas that were developed within the theory, led by the imagination strategy, the Continuation Principle, and the discharging method of Bujtas, to prove results about domination game invariants. A toolbox of proof techniques is provided for the reader to obtain results on the domination game and its variants. Powerful proof methods such as the imagination strategy are presented. The Continuation Principle is developed, which provides a much-used monotonicity property of the game domination number. In addition, the reader is exposed to the discharging method of Bujtas. The power of this method was shown by improving the known upper bound, in terms of a graph's order, on the (ordinary) domination number of graphs with minimum degree between 5 and 50. The book is intended primarily for students in graph theory as well as established graph theorists and it can be enjoyed by anyone with a modicum of mathematical maturity. The authors include exact results for several families of graphs, present what is known about the domination game played on subgraphs and trees, and provide the reader with the computational complexity aspects of domination games. Versions of the games which involve only the "slow" player yield the Grundy domination numbers, which connect the topic of the book with some concepts from linear algebra such as zero-forcing sets and minimum rank. More than a dozen other related games on graphs and hypergraphs are presented in the book. In all these games there are problems waiting to be solved, so the area is rich for further research. The domination game belongs to the growing family of competitive optimization graph games. The game is played by two competitors who take turns adding a vertex to a set of chosen vertices. They collaboratively produce a special structure in the underlying host graph, namely a dominating set. The two players have complementary goals: one seeks to minimize the size of the chosen set while the other player tries to make it as large as possible. The game is not one that is either won or lost. Instead, if both players employ an optimal strategy that is consistent with their goals, the cardinality of the chosen set is a graphical invariant, called the game domination number of the graph. To demonstrate that this is indeed a graphical invariant, the game tree of a domination game played on a graph is presented for the first time in the literature.

This volume provides accessible and self-contained research problems designed for undergraduate student projects, and simultaneously promotes the development of sustainable undergraduate research programs. The chapters in this work span a variety of topical areas of pure and applied mathematics and mathematics education. Each chapter gives a self-contained introduction on a research topic with an emphasis on the specific tools and knowledge needed to create and maintain fruitful research programs for undergraduates. Some of the topics discussed include:* Disease modeling* Tropical curves and surfaces* Numerical semigroups* Mathematics EducationThis volume will primarily appeal to undergraduate students interested in pursuing research projects and faculty members seeking to mentor them. It may also aid students and faculty participating in independent studies and capstone projects.

Discrete mathematics has been rising in prominence in the past fifty years, both as a tool with practical applications and as a source of new and interesting mathematics. The topics in discrete mathematics have become so well developed that it is easy to forget that common threads connect the different areas, and it is through discovering and using these connections that progress is often made. For over fifty years, Ron Graham has been able to illuminate some of these connections and has helped to bring the field of discrete mathematics to where it is today. To celebrate his contribution, this volume brings together many of the best researchers working in discrete mathematics, including Fan Chung, Erik D. Demaine, Persi Diaconis, Peter Frankl, Alfred W. Hales, Jeffrey C. Lagarias, Allen Knutson, Janos Pach, Carl Pomerance, N. J. A. Sloane, and of course, Ron Graham himself.

This book covers pattern recognition techniques applied to various areas of biomedicine, including disease diagnosis and prognosis, and several problems of classification, with a special focus on-but not limited to-pattern recognition modeling of biomedical signals and images. Multidisciplinary by definition, the book's topic blends computing, mathematics and other technical sciences towards the development of computational tools and methodologies that can be applied to pattern recognition processes. In this work, the efficacy of such methods and techniques for processing medical information is analyzed and compared, and auxiliary criteria for determining the correct diagnosis and treatment strategies are recommended and applied. Researchers in applied mathematics, the computer sciences, engineering and related fields with a focus on medical applications will benefit from this book, as well as professionals with a special interest in state-of-the-art pattern recognition techniques as applied to biomedicine.

This book comprehensively covers the important efforts in improving the quality of images in visual cryptography (VC), with a focus on cases with gray scale images. It not only covers schemes in traditional VC and extended VC for binary secret images, but also the latest development in the analysis-by-synthesis approach. This book distinguishes itself from the existing literature in three ways. First, it not only reviews traditional VC for binary secret images, but also covers recent efforts in improving visual quality for gray scale secret images. Second, not only traditional quality measures are reviewed, but also measures that were not used for measuring perceptual quality of decrypted secret images, such as Radially Averaged Power Spectrum Density (RAPSD) and residual variance, are employed for evaluating and guiding the design of VC algorithms. Third, unlike most VC books following a mathematical formal style, this book tries to make a balance between engineering intuition and mathematical reasoning. All the targeted problems and corresponding solutions are fully motivated by practical applications and evaluated by experimental tests, while important security issues are presented as mathematical proof. Furthermore, important algorithms are summarized as pseudocodes, thus enabling the readers to reproduce the results in the book. Therefore, this book serves as a tutorial for readers with an engineering background as well as for experts in related areas to understand the basics and research frontiers in visual cryptography.

This proceedings presents the result of the 8th International Conference in Network Analysis, held at the Higher School of Economics, Moscow, in May 2018. The conference brought together scientists, engineers, and researchers from academia, industry, and government. Contributions in this book focus on the development of network algorithms for data mining and its applications. Researchers and students in mathematics, economics, statistics, computer science, and engineering find this collection a valuable resource filled with the latest research in network analysis. Computational aspects and applications of large-scale networks in market models, neural networks, social networks, power transmission grids, maximum clique problem, telecommunication networks, and complexity graphs are included with new tools for efficient network analysis of large-scale networks. Machine learning techniques in network settings including community detection, clustering, and biclustering algorithms are presented with applications to social network analysis.