The rapidly expanding area of algebraic graph theory uses two different branches of algebra to explore various aspects of graph theory: linear algebra (for spectral theory) and group theory (for studying graph symmetry). These areas have links with other areas of mathematics, such as logic and harmonic analysis, and are increasingly being used in such areas as computer networks where symmetry is an important feature. Other books cover portions of this material, but this book is unusual in covering both of these aspects and there are no other books with such a wide scope. Peter J. Cameron, internationally recognized for his substantial contributions to the area, served as academic consultant for this volume, and the result is ten expository chapters written by acknowledged international experts in the field. Their well-written contributions have been carefully edited to enhance readability and to standardize the chapter structure, terminology and notation throughout the book. To help the reader, there is an extensive introductory chapter that covers the basic background material in graph theory, linear algebra and group theory. Each chapter concludes with an extensive list of references.

This monograph discusses decision making methods under bipolar fuzzy graphical models with the aim of overcoming the lack of mathematical approach towards bipolar information-positive and negative. It investigates the properties of bipolar fuzzy graphs, their distance functions, and concept of their isomorphism. It presents certain notions, including irregular bipolar fuzzy graphs, domination in bipolar fuzzy graphs, bipolar fuzzy circuits, energy in bipolar fuzzy graphs, bipolar single-valued neutrosophic competition graphs, and bipolar neutrosophic graph structures. This book also presents the applications of mentioned concepts to real-world problems in areas of product manufacturing, international relations, psychology, global terrorism and more, making it valuable for researchers, computer scientists, social scientists and alike.

Rosemary Bailey covers in this study the mathematics of association schemes--an area lying between pure mathematics and statistics that relates to the optimal design of scientific experiments. The book is accessible to mathematicians as well as statisticians. Arising from a graduate course taught by the author, it appeals to students as well as researchers as a valuable reference work from which to learn about the statistical/combinatorial aspects of their work.

This book collects original peer-reviewed contributions to the conferences organised by the international research network "Minimal surfaces: Integrable Systems and Visualization" financed by the Leverhulme Trust. The conferences took place in Cork, Granada, Munich and Leicester between 2016 and 2019. Within the theme of the network, the presented articles cover a broad range of topics and explore exciting links between problems related to the mean curvature of surfaces in homogeneous 3-manifolds, like minimal surfaces, CMC surfaces and mean curvature flows, integrable systems and visualisation. Combining research and overview articles by prominent international researchers, the book offers a valuable resource for both researchers and students who are interested in this research area.

This book constitutes the proceedings of the 27th International Conference on Computing and Combinatorics, COCOON 2021, held in Tainan, Taiwan, in October 2021. Due to the COVID-19 pandemic, COCOON 2021 was organized as a hybrid conference. The 56 papers presented in this volume were carefully reviewed and selected from 131 submissions. The papers are divided into the following topical sub-headings: algorithms, approximation algorithms, automata, computational geometry, fault tolerant computing and fault diagnosis, graph algorithms, graph theory and applications, network and algorithms, online algorithm and stream algorithms, parameterized complexity and algorithms, and recreational games.

This book is the first general and extensive review on the algorithmics and mathematical results of beyond planar graphs. Most real-world data sets are relational and can be modelled as graphs consisting of vertices and edges. Planar graphs are fundamental for both graph theory and graph algorithms and are extensively studied. Structural properties and fundamental algorithms for planar graphs have been discovered. However, most real-world graphs, such as social networks and biological networks, are non-planar. To analyze and visualize such real-world networks, it is necessary to solve fundamental mathematical and algorithmic research questions on sparse non-planar graphs, called beyond planar graphs.This book is based on the National Institute of Informatics (NII) Shonan Meeting on algorithmics on beyond planar graphs held in Japan in November, 2016. The book consists of 13 chapters that represent recent advances in various areas of beyond planar graph research. The main aims and objectives of this book include 1) to timely provide a state-of-the-art survey and a bibliography on beyond planar graphs; 2) to set the research agenda on beyond planar graphs by identifying fundamental research questions and new research directions; and 3) to foster cross-disciplinary research collaboration between computer science (graph drawing and computational geometry) and mathematics (graph theory and combinatorics). New algorithms for beyond planar graphs will be in high demand by practitioners in various application domains to solve complex visualization problems. This book therefore will be a valuable resource for researchers in graph theory, algorithms, and theoretical computer science, and will stimulate further deep scientific investigations into many areas of beyond planar graphs.

This book describes a set of novel statistical algorithms designed to infer functional connectivity of large-scale neural assemblies. The algorithms are developed with the aim of maximizing computational accuracy and efficiency, while faithfully reconstructing both the inhibitory and excitatory functional links. The book reports on statistical methods to compute the most significant functional connectivity graph, and shows how to use graph theory to extract the topological features of the computed network. A particular feature is that the methods used and extended at the purpose of this work are reported in a fairly completed, yet concise manner, together with the necessary mathematical fundamentals and explanations to understand their application. Furthermore, all these methods have been embedded in the user-friendly open source software named SpiCoDyn, which is also introduced here. All in all, this book provides researchers and graduate students in bioengineering, neurophysiology and computer science, with a set of simplified and reduced models for studying functional connectivity in in silico biological neuronal networks, thus overcoming the complexity of brain circuits.

Steven Finch provides 136 essays, each devoted to a mathematical constant or a class of constants, from the well known to the highly exotic. This book is helpful both to readers seeking information about a specific constant, and to readers who desire a panoramic view of all constants coming from a particular field, for example, combinatorial enumeration or geometric optimization. Unsolved problems appear virtually everywhere as well. This work represents a scholarly attempt to bring together all significant mathematical constants in one place.

Introduction to Chemical Graph Theory is a concise introduction to the main topics and techniques in chemical graph theory, specifically the theory of topological indices. These include distance-based, degree-based, and counting-based indices. The book covers some of the most commonly used mathematical approaches in the subject. It is also written with the knowledge that chemical graph theory has many connections to different branches of graph theory (such as extremal graph theory, spectral graph theory). The authors wrote the book in an appealing way that attracts people to chemical graph theory. In doing so, the book is an excellent playground and general reference text on the subject, especially for young mathematicians with a special interest in graph theory. Key Features: A concise introduction to topological indices of graph theory Appealing to specialists and non-specialists alike Provides many techniques from current research About the Authors: Stephan Wagner grew up in Graz (Austria), where he also received his PhD from Graz University of Technology in 2006. Shortly afterwards, he moved to South Africa, where he started his career at Stellenbosch University as a lecturer in January 2007. His research interests lie mostly in combinatorics and related areas, including connections to other scientific fields such as physics, chemistry and computer science. Hua Wang received his PhD from University of South Carolina in 2005. He held a Visiting Research Assistant Professor position at University of Florida before joining Georgia Southern University in 2008. His research interests include combinatorics and graph theory, elementary number theory, and related problems

Analytic Combinatorics: A Multidimensional Approach is written in a reader-friendly fashion to better facilitate the understanding of the subject. Naturally, it is a firm introduction to the concept of analytic combinatorics and is a valuable tool to help readers better understand the structure and large-scale behavior of discrete objects. Primarily, the textbook is a gateway to the interactions between complex analysis and combinatorics. The study will lead readers through connections to number theory, algebraic geometry, probability and formal language theory. The textbook starts by discussing objects that can be enumerated using generating functions, such as tree classes and lattice walks. It also introduces multivariate generating functions including the topics of the kernel method, and diagonal constructions. The second part explains methods of counting these objects, which involves deep mathematics coming from outside combinatorics, such as complex analysis and geometry. Features Written with combinatorics-centric exposition to illustrate advanced analytic techniques Each chapter includes problems, exercises, and reviews of the material discussed in them Includes a comprehensive glossary, as well as lists of figures and symbols About the author Marni Mishna is a professor of mathematics at Simon Fraser University in British Columbia. Her research investigates interactions between discrete structures and many diverse areas such as representation theory, functional equation theory, and algebraic geometry. Her specialty is the development of analytic tools to study the large-scale behavior of discrete objects.

This textbook, suitable for an early undergraduate up to a graduate course, provides an overview of many basic principles and techniques needed for modern data analysis. In particular, this book was designed and written as preparation for students planning to take rigorous Machine Learning and Data Mining courses. It introduces key conceptual tools necessary for data analysis, including concentration of measure and PAC bounds, cross validation, gradient descent, and principal component analysis. It also surveys basic techniques in supervised (regression and classification) and unsupervised learning (dimensionality reduction and clustering) through an accessible, simplified presentation. Students are recommended to have some background in calculus, probability, and linear algebra. Some familiarity with programming and algorithms is useful to understand advanced topics on computational techniques.

This book is a state-of-the-art look at combinatorial games, that is, games not involving chance or hidden information. It contains a fascinating collection of articles by some of the top names in the field, such as Elwyn Berlekamp and John Conway, plus other researchers in mathematics and computer science, together with some top game players. The articles run the gamut from new theoretical approaches (infinite games, generalizations of game values, 2-player cellular automata, Alpha-Beta pruning under partial orders) to the very latest in some of the hottest games (Amazons, Chomp, Dot-and-Boxes, Go, Chess, Hex). Many of these advances reflect the interplay of the computer science and the mathematics. The book ends with an updated bibliography by A. Fraenkel and an updated and annotated list of combinatorial game theory problems by R. K. Guy.

This book A Guide to Graph Algorithms offers high-quality content in the research area of graph algorithms and explores the latest developments in graph algorithmics. The reader will gain a comprehensive understanding of how to use algorithms to explore graphs. It is a collection of texts that have proved to be trend setters and good examples of that. The book aims at providing the reader with a deep understanding of the structural properties of graphs that are useful for the design of efficient algorithms. These algorithms have applications in finite state machine modelling, social network theory, biology, and mathematics. The book contains many exercises, some up at present-day research-level. The exercises encourage the reader to discover new techniques by putting things in a clear perspective. A study of this book will provide the reader with many powerful tools to model and tackle problems in real-world scenarios.

This book highlights new and original contributions on Graph Theory and Combinatorial Optimization both from the theoretical point of view and from applications in all fields. The book chapters describe models and methods based on graphs, structural properties, discrete optimization, network optimization, mixed-integer programming, heuristics, meta-heuristics, math-heuristics, and exact methods as well as applications. The book collects selected contributions from the CTW2020 international conference (18th Cologne-Twente Workshop on Graphs and Combinatorial Optimization), held online on September 14-16, 2020. The conference was organized by IASI-CNR with the contribution of University of Roma Tre, University Roma Tor Vergata, and CNRS-LIX and with the support of AIRO. It is addressed to researchers, PhD students, and practitioners in the fields of Graph Theory, Discrete Mathematics, Combinatorial Optimization, and Operations Research.

This book presents a printed testimony for the fact that George Andrews, one of the world's leading experts in partitions and q-series for the last several decades, has passed the milestone age of 80. To honor George Andrews on this occasion, the conference "Combinatory Analysis 2018" was organized at the Pennsylvania State University from June 21 to 24, 2018. This volume comprises the original articles from the Special Issue "Combinatory Analysis 2018 - In Honor of George Andrews' 80th Birthday" resulting from the conference and published in Annals of Combinatorics. In addition to the 37 articles of the Andrews 80 Special Issue, the book includes two new papers. These research contributions explore new grounds and present new achievements, research trends, and problems in the area. The volume is complemented by three special personal contributions: "The Worlds of George Andrews, a daughter's take" by Amy Alznauer, "My association and collaboration with George Andrews" by Krishna Alladi, and "Ramanujan, his Lost Notebook, its importance" by Bruce Berndt. Another aspect which gives this Andrews volume a truly unique character is the "Photos" collection. In addition to pictures taken at "Combinatory Analysis 2018", the editors selected a variety of photos, many of them not available elsewhere: "Andrews in Austria", "Andrews in China", "Andrews in Florida", "Andrews in Illinois", and "Andrews in India". This volume will be of interest to researchers, PhD students, and interested practitioners working in the area of Combinatory Analysis, q-Series, and related fields.

Combinatorics on words has arisen independently within several branches of mathematics, for instance, number theory, group theory and probability, and appears frequently in problems related to theoretical computer science. The first unified treatment of the area was given in Lothaire's Combinatorics on Words. Since its publication, the area has developed and the authors now aim to present several more topics as well as giving deeper insights into subjects that were discussed in the previous volume. An introductory chapter provides the reader with all the necessary background material. There are numerous examples, full proofs whenever possible and a notes section discussing further developments in the area. This book is both a comprehensive introduction to the subject and a valuable reference source for researchers.

This book describes active illumination techniques in computer vision. We can classify computer vision techniques into two classes: passive and active techniques. Passive techniques observe the scene statically and analyse it as is. Active techniques give the scene some actions and try to facilitate the analysis. In particular, active illumination techniques project specific light, for which the characteristics are known beforehand, to a target scene to enable stable and accurate analysis of the scene. Traditional passive techniques have a fundamental limitation. The external world surrounding us is three-dimensional; the image projected on a retina or an imaging device is two-dimensional. That is, reduction of one dimension has occurred. Active illumination techniques compensate for the dimensional reduction by actively controlling the illumination. The demand for reliable vision sensors is rapidly increasing in many application areas, such as robotics and medical image analysis. This book explains this new endeavour to explore the augmentation of reduced dimensions in computer vision. This book consists of three parts: basic concepts, techniques, and applications. The first part explains the basic concepts for understanding active illumination techniques. In particular, the basic concepts of optics are explained so that researchers and engineers outside the field can understand the later chapters. The second part explains currently available active illumination techniques, covering many techniques developed by the authors. The final part shows how such active illumination techniques can be applied to various domains, describing the issue to be overcome by active illumination techniques and the advantages of using these techniques. This book is primarily aimed at 4th year undergraduate and 1st year graduate students, and will also help engineers from fields beyond computer vision to use active illumination techniques. Additionally, the book is suitable as course material for technical seminars.

Use the Pharo interactive development environment to significantly reduce the cost of creating interactive visualizations. This book shows how Pharo leverages visualization development against traditional frameworks and toolkits. Agile Visualization with Pharo focuses on the Roassal visualization engine and first presents the basic and necessary tools to visualize data, including an introduction to the Pharo programming language. Once you've grasped the basics, you'll learn all about the development environment offered by Roassal. The book provides numerous ready-to-use examples. You'll work on several applications, including visualizing the training phase of reinforcement learning (a powerful machine learning algorithm) and generating software visualizations from GitHub. This book covers aspects that are relevant for engineers and academics to successfully design and implement interactive visualizations. What You Will Learn Implement agile data visualization using the Pharo programming language Chart, plot, and curve using Grapher Build and draw graphs using Mondrian Implement reinforcement learning (Q-Learning, from scratch) and use visualizations to monitor learning and state exploration Use GitHub Action to generate software visualizations (UML class diagram, test coverage) at each commit Who This Book Is For Programmers with some prior exposure to data visualization and computer vision who may be new to the Pharo programming language. This book is also for those with some Pharo experience looking to apply it to data visualization.

This brief investigates the asymptotic behavior of some PDEs on networks. The structures considered consist of finitely interconnected flexible elements such as strings and beams (or combinations thereof), distributed along a planar network. Such study is motivated by the need for engineers to eliminate vibrations in some dynamical structures consisting of elastic bodies, coupled in the form of chain or graph such as pipelines and bridges. There are other complicated examples in the automotive industry, aircraft and space vehicles, containing rather than strings and beams, plates and shells. These multi-body structures are often complicated, and the mathematical models describing their evolution are quite complex. For the sake of simplicity, this volume considers only 1-d networks.

This book reports on the development and assessment of a novel framework for studying neural interactions (the connectome) and their dynamics (the chronnectome). Using EEG recordings taken during an auditory oddball task performed by 48 patients with schizophrenia and 87 healthy controls, and applying local and network measures, changes in brain activation from pre-stimulus to cognitive response were assessed, and significant differences were observed between the patients and controls. This book investigates the source of the network abnormalities and presents new evidence for the disconnection hypothesis and the aberrant salience hypothesis with regard to schizophrenia. Moreover, it puts forward a novel approach to combining local regularity measures and graph measures in order to characterize schizophrenia brain dynamics, and presents interesting findings on the regularity of brain patterns in healthy control subjects versus patients with schizophrenia. Besides providing new evidence for the disconnection hypothesis, it offers a source of inspiration for future research directions in the field.

This book offers a holistic framework to study behavior and evolutionary dynamics in large-scale, decentralized, and heterogeneous crowd networks. In the emerging crowd cyber-ecosystems, millions of deeply connected individuals, smart devices, government agencies, and enterprises actively interact with each other and influence each other's decisions. It is crucial to understand such intelligent entities' behaviors and to study their strategic interactions in order to provide important guidelines on the design of reliable networks capable of predicting and preventing detrimental events with negative impacts on our society and economy. This book reviews the fundamental methodologies to study user interactions and evolutionary dynamics in crowd networks and discusses recent advances in this emerging interdisciplinary research field. Using information diffusion over social networks as an example, it presents a thorough investigation of the impact of user behavior on the network evolution process and demonstrates how this can help improve network performance. Intended for graduate students and researchers from various disciplines, including but not limited to, data science, networking, signal processing, complex systems, and economics, the book encourages researchers in related research fields to explore the many untouched areas in this domain, and ultimately to design crowd networks with efficient, effective, and reliable services.

This textbook teaches readers how to turn geometry into an image on a computer screen. This exciting journey begins in the schools of the ancient Greek philosophers, and describes the major events that changed people's perception of geometry. The readers will learn how to see geometry and colors beyond simple mathematical formulas and how to represent geometric shapes, transformations and motions by digital sampling of various mathematical functions.Special multiplatform visualization software developed by the author will allow readers to explore the exciting world of visual immersive mathematics, and the book software repository will provide a starting point for their own sophisticated visualization applications. Making Images with Mathematics serves as a self-contained text for a one-semester computer graphics and visualization course for computer science and engineering students, as well as a reference manual for researchers and developers.

This collection of peer-reviewed workshop papers provides comprehensive coverage of cutting-edge research into topological approaches to data analysis and visualization. It encompasses the full range of new algorithms and insights, including fast homology computation, comparative analysis of simplification techniques, and key applications in materials and medical science. The book also addresses core research challenges such as the representation of large and complex datasets, and integrating numerical methods with robust combinatorial algorithms. In keeping with the focus of the TopoInVis 2017 Workshop, the contributions reflect the latest advances in finding experimental solutions to open problems in the sector. They provide an essential snapshot of state-of-the-art research, helping researchers to keep abreast of the latest developments and providing a basis for future work. Gathering papers by some of the world's leading experts on topological techniques, the book represents a valuable contribution to a field of growing importance, with applications in disciplines ranging from engineering to medicine.

Designed for the nonspecialist, this classic text by a world expert is an invaluable reference tool for those interested in a basic understanding of the subject. Exercises, notes and exhaustive references follow each chapter, making it outstanding as both a text and reference for students and researchers in graph theory and its applications. The reader will delight to discover that the topics in this book are coherently unified and include some of the deepest and most beautiful developments in graph theory.

Mahler measure, a height function for polynomials, is the central theme of this book. It has many interesting properties, obtained by algebraic, analytic and combinatorial methods. It is the subject of several longstanding unsolved questions, such as Lehmer's Problem (1933) and Boyd's Conjecture (1981). This book contains a wide range of results on Mahler measure. Some of the results are very recent, such as Dimitrov's proof of the Schinzel-Zassenhaus Conjecture. Other known results are included with new, streamlined proofs. Robinson's Conjectures (1965) for cyclotomic integers, and their associated Cassels height function, are also discussed, for the first time in a book.One way to study algebraic integers is to associate them with combinatorial objects, such as integer matrices. In some of these combinatorial settings the analogues of several notorious open problems have been solved, and the book sets out this recent work. Many Mahler measure results are proved for restricted sets of polynomials, such as for totally real polynomials, and reciprocal polynomials of integer symmetric as well as symmetrizable matrices. For reference, the book includes appendices providing necessary background from algebraic number theory, graph theory, and other prerequisites, along with tables of one- and two-variable integer polynomials with small Mahler measure. All theorems are well motivated and presented in an accessible way. Numerous exercises at various levels are given, including some for computer programming. A wide range of stimulating open problems is also included. At the end of each chapter there is a glossary of newly introduced concepts and definitions. Around the Unit Circle is written in a friendly, lucid, enjoyable style, without sacrificing mathematical rigour. It is intended for lecture courses at the graduate level, and will also be a valuable reference for researchers interested in Mahler measure. Essentially self-contained, this textbook should also be accessible to well-prepared upper-level undergraduates.