This book gives an overview of research on graphs associated with commutative rings. The study of the connections between algebraic structures and certain graphs, especially finite groups and their Cayley graphs, is a classical subject which has attracted a lot of interest. More recently, attention has focused on graphs constructed from commutative rings, a field of study which has generated an extensive amount of research over the last three decades. The aim of this text is to consolidate this large body of work into a single volume, with the intention of encouraging interdisciplinary research between algebraists and graph theorists, using the tools of one subject to solve the problems of the other. The topics covered include the graphical and topological properties of zero-divisor graphs, total graphs and their transformations, and other graphs associated with rings. The book will be of interest to researchers in commutative algebra and graph theory and anyone interested in learning about the connections between these two subjects.

This textbook offers an accessible introduction to combinatorics, infused with Solomon Golomb's insights and illustrative examples. Core concepts in combinatorics are presented with an engaging narrative that suits undergraduate study at any level. Featuring early coverage of the Principle of Inclusion-Exclusion and a unified treatment of permutations later on, the structure emphasizes the cohesive development of ideas. Combined with the conversational style, this approach is especially well suited to independent study. Falling naturally into three parts, the book begins with a flexible Chapter Zero that can be used to cover essential background topics, or as a standalone problem-solving course. The following three chapters cover core topics in combinatorics, such as combinations, generating functions, and permutations. The final three chapters present additional topics, such as Fibonacci numbers, finite groups, and combinatorial structures. Numerous illuminating examples are included throughout, along with exercises of all levels. Three appendices include additional exercises, examples, and solutions to a selection of problems. Solomon Golomb's Course on Undergraduate Combinatorics is ideal for introducing mathematics students to combinatorics at any stage in their program. There are no formal prerequisites, but readers will benefit from mathematical curiosity and a willingness to engage in the book's many entertaining challenges.

This book is based on a capstone course that the author taught to upper division undergraduate students with the goal to explain and visualize the connections between different areas of mathematics and the way different subject matters flow from one another. In teaching his readers a variety of problem solving techniques as well, the author succeeds in enhancing the readers' hands on knowledge of mathematics and provides glimpses into the world of research and discovery. The connections between different techniques and areas of mathematics are emphasized throughout and constitute one of the most important lessons this book attempts to impart. This book is interesting and accessible to anyone with a basic knowledge of high school mathematics and a curiosity about research mathematics. The author is a professor at the University of Missouri and has maintained a keen interest in teaching at different levels since his undergraduate days at the University of Chicago. He has run numerous summer programs in mathematics for local high school students and undergraduate students at his university.The author gets much of his research inspiration from his teaching activities and looks forward to exploring this wonderful and rewarding symbiosis for years to come.

This textbook treats graph colouring as an algorithmic problem, with a strong emphasis on practical applications. The author describes and analyses some of the best-known algorithms for colouring graphs, focusing on whether these heuristics can provide optimal solutions in some cases; how they perform on graphs where the chromatic number is unknown; and whether they can produce better solutions than other algorithms for certain types of graphs, and why. The introductory chapters explain graph colouring, complexity theory, bounds and constructive algorithms. The author then shows how advanced, graph colouring techniques can be applied to classic real-world operational research problems such as designing seating plans, sports scheduling, and university timetabling. He includes many examples, suggestions for further reading, and historical notes, and the book is supplemented by an online suite of downloadable code. The book is of value to researchers, graduate students, and practitioners in the areas of operations research, theoretical computer science, optimization, and computational intelligence. The reader should have elementary knowledge of sets, matrices, and enumerative combinatorics.

Graph algorithms is a well-established subject in mathematics and computer science. Beyond classical application fields, such as approximation, combinatorial optimization, graphics, and operations research, graph algorithms have recently attracted increased attention from computational molecular biology and computational chemistry. Centered around the fundamental issue of graph isomorphism, this text goes beyond classical graph problems of shortest paths, spanning trees, flows in networks, and matchings in bipartite graphs. Advanced algorithmic results and techniques of practical relevance are presented in a coherent and consolidated way. This book introduces graph algorithms on an intuitive basis followed by a detailed exposition in a literate programming style, with correctness proofs as well as worst-case analyses. Furthermore, full C++ implementations of all algorithms presented are given using the LEDA library of efficient data structures and algorithms.

The main aim of the book is to give a review of all relevant information regarding a well-known and important problem of Feedback Arc Set (FAS). This review naturally also includes a history of the problem, as well as specific algorithms. To this point such a work does not exist: There are sources where one can find incomplete and perhaps untrustworthy information. With this book, information about FAS can be found easily in one place: formulation, description, theoretical background, applications, algorithms etc. Such a compendium will be of help to people involved in research, but also to people that want to quickly acquaint themselves with the problem and need reliable information. Thus research, professional work and learning can proceed in a more streamlined and faster way.

This new edition to the classic book by ggplot2 creator Hadley Wickham highlights compatibility with knitr and RStudio. ggplot2 is a data visualization package for R that helps users create data graphics, including those that are multi-layered, with ease. With ggplot2, it's easy to: produce handsome, publication-quality plots with automatic legends created from the plot specification superimpose multiple layers (points, lines, maps, tiles, box plots) from different data sources with automatically adjusted common scales add customizable smoothers that use powerful modeling capabilities of R, such as loess, linear models, generalized additive models, and robust regression save any ggplot2 plot (or part thereof) for later modification or reuse create custom themes that capture in-house or journal style requirements and that can easily be applied to multiple plots approach a graph from a visual perspective, thinking about how each component of the data is represented on the final plot This book will be useful to everyone who has struggled with displaying data in an informative and attractive way. Some basic knowledge of R is necessary (e.g., importing data into R). ggplot2 is a mini-language specifically tailored for producing graphics, and you'll learn everything you need in the book. After reading this book you'll be able to produce graphics customized precisely for your problems, and you'll find it easy to get graphics out of your head and on to the screen or page.

This book is a result of a workshop, the 8th of the successful TopoInVis workshop series, held in 2019 in Nykoeping, Sweden. The workshop regularly gathers some of the world's leading experts in this field. Thereby, it provides a forum for discussions on the latest advances in the field with a focus on finding practical solutions to open problems in topological data analysis for visualization. The contributions provide introductory and novel research articles including new concepts for the analysis of multivariate and time-dependent data, robust computational approaches for the extraction and approximations of topological structures with theoretical guarantees, and applications of topological scalar and vector field analysis for visualization. The applications span a wide range of scientific areas comprising climate science, material sciences, fluid dynamics, and astronomy. In addition, community efforts with respect to joint software development are reported and discussed.

Combinatorial Methods with Computer Applications provides in-depth coverage of recurrences, generating functions, partitions, and permutations, along with some of the most interesting graph and network topics, design constructions, and finite geometries. Requiring only a foundation in discrete mathematics, it can serve as the textbook in a combinatorial methods course or in a combined graph theory and combinatorics course. After an introduction to combinatorics, the book explores six systematic approaches within a comprehensive framework: sequences, solving recurrences, evaluating summation expressions, binomial coefficients, partitions and permutations, and integer methods. The author then focuses on graph theory, covering topics such as trees, isomorphism, automorphism, planarity, coloring, and network flows. The final chapters discuss automorphism groups in algebraic counting methods and describe combinatorial designs, including Latin squares, block designs, projective planes, and affine planes. In addition, the appendix supplies background material on relations, functions, algebraic systems, finite fields, and vector spaces. Paving the way for students to understand and perform combinatorial calculations, this accessible text presents the discrete methods necessary for applications to algorithmic analysis, performance evaluation, and statistics as well as for the solution of combinatorial problems in engineering and the social sciences.

This book provides an overview of many interesting properties of natural numbers, demonstrating their applications in areas such as cryptography, geometry, astronomy, mechanics, computer science, and recreational mathematics. In particular, it presents the main ideas of error-detecting and error-correcting codes, digital signatures, hashing functions, generators of pseudorandom numbers, and the RSA method based on large prime numbers. A diverse array of topics is covered, from the properties and applications of prime numbers, some surprising connections between number theory and graph theory, pseudoprimes, Fibonacci and Lucas numbers, and the construction of Magic and Latin squares, to the mathematics behind Prague's astronomical clock. Introducing a general mathematical audience to some of the basic ideas and algebraic methods connected with various types of natural numbers, the book will provide invaluable reading for amateurs and professionals alike.

This book focuses on linear time eigenvalue location algorithms for graphs. This subject relates to spectral graph theory, a field that combines tools and concepts of linear algebra and combinatorics, with applications ranging from image processing and data analysis to molecular descriptors and random walks. It has attracted a lot of attention and has since emerged as an area on its own. Studies in spectral graph theory seek to determine properties of a graph through matrices associated with it. It turns out that eigenvalues and eigenvectors have surprisingly many connections with the structure of a graph. This book approaches this subject under the perspective of eigenvalue location algorithms. These are algorithms that, given a symmetric graph matrix M and a real interval I, return the number of eigenvalues of M that lie in I. Since the algorithms described here are typically very fast, they allow one to quickly approximate the value of any eigenvalue, which is a basic step in most applications of spectral graph theory. Moreover, these algorithms are convenient theoretical tools for proving bounds on eigenvalues and their multiplicities, which was quite useful to solve longstanding open problems in the area. This book brings these algorithms together, revealing how similar they are in spirit, and presents some of their main applications. This work can be of special interest to graduate students and researchers in spectral graph theory, and to any mathematician who wishes to know more about eigenvalues associated with graphs. It can also serve as a compact textbook for short courses on the topic.

This textbook offers an accessible introduction to combinatorics, infused with Solomon Golomb's insights and illustrative examples. Core concepts in combinatorics are presented with an engaging narrative that suits undergraduate study at any level. Featuring early coverage of the Principle of Inclusion-Exclusion and a unified treatment of permutations later on, the structure emphasizes the cohesive development of ideas. Combined with the conversational style, this approach is especially well suited to independent study. Falling naturally into three parts, the book begins with a flexible Chapter Zero that can be used to cover essential background topics, or as a standalone problem-solving course. The following three chapters cover core topics in combinatorics, such as combinations, generating functions, and permutations. The final three chapters present additional topics, such as Fibonacci numbers, finite groups, and combinatorial structures. Numerous illuminating examples are included throughout, along with exercises of all levels. Three appendices include additional exercises, examples, and solutions to a selection of problems. Solomon Golomb's Course on Undergraduate Combinatorics is ideal for introducing mathematics students to combinatorics at any stage in their program. There are no formal prerequisites, but readers will benefit from mathematical curiosity and a willingness to engage in the book's many entertaining challenges.

This book focuses on the application of virtual reality (VR) technology in mining machinery. It gives a detailed introduction to the application of VR technology in virtual assembly, virtual planning, and virtual monitoring. Based on the theory of digital twin, VR technology and collaborative control technology are applied to coal mining machinery equipment, which lays a foundation for the digitalization and intellectualization of coal machinery equipment and broadens the application scope of virtual reality technology in the mechanical engineering field. Through the application of VR technology in coal machinery equipment, this book provides new methods and ideas for teaching activities, scientific research activities, and actual production with rich illustrations, related table introduction, unique research ideas, and other unique contents. This book could be a useful reference for researchers in mining machinery, simulation and modeling, computer-aided engineering (CAD and CAE) and design, visualization, mechanical engineering, and other disciplines.

The Workshop for Women in Graph Theory and Applications was held at the Institute for Mathematics and Its Applications (University of Minnesota, Minneapolis) on August 19-23, 2019. During this five-day workshop, 42 participants performed collaborative research, in six teams, each focused on open problems in different areas of graph theory and its applications. The research work of each team was led by two experts in the corresponding area, who prior to the workshop, carefully selected relevant and meaningful open problems that would yield high-quality research and results of strong impact. As a result, all six teams have made significant contributions to several open problems in their respective areas. The workshop led to the creation of the Women in Graph Theory and Applications Research Collaboration Network, which provided the framework to continue collaborating and to produce this volume. This book contains six chapters, each of them on one of the different areas of research at the Workshop for Women in Graph Theory and Applications, and written by participants of each team.

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 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.

The inverse scattering problem is central to many areas of science and technology such as radar, sonar, medical imaging, geophysical exploration and nondestructive testing. This book is devoted to the mathematical and numerical analysis of the inverse scattering problem for acoustic and electromagnetic waves. In this fourth edition, a number of significant additions have been made including a new chapter on transmission eigenvalues and a new section on the impedance boundary condition where particular attention has been made to the generalized impedance boundary condition and to nonlocal impedance boundary conditions. Brief discussions on the generalized linear sampling method, the method of recursive linearization, anisotropic media and the use of target signatures in inverse scattering theory have also been added.

This book surveys the mathematical and computational properties of finite sets of points in the plane, covering recent breakthroughs on important problems in discrete geometry, and listing many open problems. It unifies these mathematical and computational views using forbidden configurations, which are patterns that cannot appear in sets with a given property, and explores the implications of this unified view. Written with minimal prerequisites and featuring plenty of figures, this engaging book will be of interest to undergraduate students and researchers in mathematics and computer science. Most topics are introduced with a related puzzle or brain-teaser. The topics range from abstract issues of collinearity, convexity, and general position to more applied areas including robust statistical estimation and network visualization, with connections to related areas of mathematics including number theory, graph theory, and the theory of permutation patterns. Pseudocode is included for many algorithms that compute properties of point sets.

Features recent advances and new applications in graph edge coloring

Reviewing recent advances in the Edge Coloring Problem, "Graph Edge Coloring: Vizing's Theorem and Goldberg's Conjecture" provides an overview of the current state of the science, explaining the interconnections among the results obtained from important graph theory studies. The authors introduce many new improved proofs of known results to identify and point to possible solutions for open problems in edge coloring.

The book begins with an introduction to graph theory and the concept of edge coloring. Subsequent chapters explore important topics such as:

Use of Tashkinov trees to obtain an asymptotic positive solution to Goldberg's conjecture

Application of Vizing fans to obtain both known and new results

Kierstead paths as an alternative to Vizing fans

Classification problem of simple graphs

Generalized edge coloring in which a color may appear more than once at a vertex

This book also features first-time English translations of two groundbreaking papers written by Vadim Vizing on an estimate of the chromatic class of a p-graph and the critical graphs within a given chromatic class.

Written by leading experts who have reinvigorated research in the field, "Graph Edge Coloring" is an excellent book for mathematics, optimization, and computer science courses at the graduate level. The book also serves as a valuable reference for researchers interested in discrete mathematics, graph theory, operations research, theoretical computer science, and combinatorial optimization.

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.

Scientific visualization has always been an integral part of discovery, starting first with simplified drawings of the pre-Enlightenment and progressing to present day. Mathematical formalism often supersedes visual methods, but their use is at the core of the mental process. As historical examples, a spatial description of flow led to electromagnetic theory, and without visualization of crystals, structural chemistry would not exist. With the advent of computer graphics technology, visualization has become a driving force in modern computing. A Concise Introduction to Scientific Visualization - Past, Present, and Future serves as a primer to visualization without assuming prior knowledge. It discusses both the history of visualization in scientific endeavour, and how scientific visualization is currently shaping the progress of science as a multi-disciplinary domain.

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.

Since the early eighteenth century, the theory of networks and graphs has matured into an indispensable tool for describing countless real-world phenomena. However, the study of large-scale features of a network often requires unrealistic limits, such as taking the network size to infinity or assuming a continuum. These asymptotic and analytic approaches can significantly diverge from real or simulated networks when applied at the finite scales of real-world applications. This book offers an approach to overcoming these limitations by introducing operator graph theory, an exact, non-asymptotic set of tools combining graph theory with operator calculus. The book is intended for mathematicians, physicists, and other scientists interested in discrete finite systems and their graph-theoretical description, and in delineating the abstract algebraic structures that characterise such systems. All the necessary background on graph theory and operator calculus is included for readers to understand the potential applications of operator graph theory.

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.

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.