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Crossing Numbers of Graphs is the first book devoted to the crossing number, an increasingly popular object of study with surprising connections. The field has matured into a large body of work, which includes identifiable core results and techniques. The book presents a wide variety of ideas and techniques in topological graph theory, discrete geometry, and computer science. The first part of the text deals with traditional crossing number, crossing number values, crossing lemma, related parameters, computational complexity, and algorithms. The second part includes the rich history of alternative crossing numbers, the rectilinear crossing number, the pair crossing number, and the independent odd crossing number.It also includes applications of the crossing number outside topological graph theory.Aimed at graduate students and professionals in both mathematics and computer scienceThe first book of its kind devoted to the topicAuthored by a noted authority in crossing numbers

In 1937 there appeared a paper that was to have a profound influence on the progress of combinatorial enumeration, both in its theoretical and applied aspects. Entitled Kombinatorische Anzahlbest- immungen jUr Gruppen, Graphen und chemische Verbindungen, it was published in Acta Mathematica, Vol. 68, pp. 145 to 254. Its author, George Polya, was already a mathematician of considerable stature, well-known for outstanding work in many branches of mathematics, particularly analysis. The paper in Question was unusual in that it depended almost entirely on a single theorem -- the "Hauptsatz" of Section 4 -- a theorem which gave a method for solving a general type of enumera- tion problem. On the face of it, this is not something that one would expect to run to over 100 pages. Yet the range of the applica- tions of the theorem and of its ramifications was enormous, as Polya clearly showed. In the various sections of his paper he explored many applications to the enumeration of graphs, principally trees, and of chemical isomers, using his theorem to present a comprehen- sive and unified treatment of problems which had previously been solved, if at all, only by ad hoc methods.In the final section he investigated the asymptotic properties of these enumerational results, bringing to bear his formidable insight as an analyst.

To most graph theorists there are two outstanding landmarks in the history of their subject. One is Euler's solution of the Konigsberg Bridges Problem, dated 1736, and the other is the appearance of Denes Konig's textbook in 1936. "From Konigsberg to Konig's book" sings the poetess, "So runs the graphic tale ..." [10]. There were earlier books that took note of graph theory. Veb- len's Analysis Situs, published in 1931, is about general combinato- rial topology. But its first two chapters, on "Linear graphs" and "Two-Dimensional Complexes", are almost exclusively concerned with the territory still explored by graph theorists. Rouse Ball's Mathematical Recreations and Essays told, usually without proofs, of the major graph-theoretical advances ofthe nineteenth century, of the Five Colour Theorem, of Petersen's Theorem on I-factors, and of Cayley's enumerations of trees. It was Rouse Ball's book that kindled my own graph-theoretical enthusiasm. The graph-theoretical papers of Hassler Whitney, published in 1931-1933, would have made an excellent textbook in English had they been collected and published as such.But the honour of presenting Graph Theory to the mathe- matical world as a subject in its own right, with its own textbook, belongs to Denes Konig. Low was the prestige of Graph Theory in the Dirty Thirties. It is still remembered, with resentment now shading into amuse- ment, how one mathematician scorned it as "The slums of Topol- ogy".

This text is a self contained treatment of expander graphs and in particular their explicit construction. Expander graphs are both highly connected but sparse, and besides their interest within combinatorics and graph theory, they also find various applications in computer science and engineering. The reader needs only a background in elementary algebra, analysis and combinatorics; the authors supply the necessary background material from graph theory, number theory, group theory and representation theory. The text can therefore be used as a brief introduction to these subjects as well as an illustration of how such topics are synthesised in modern mathematics.

Students collect, organize, and represent data using frequency, bar, and circle graphs. They use line graphs to describe change over time. They use benchmark fractions and the three measures of central tendency—mode, median, and mean— to describe sets of data.

Spectral Radius of Graphs provides a thorough overview of important results on the spectral radius of adjacency matrix of graphs that have appeared in the literature in the preceding ten years, most of them with proofs, and including some previously unpublished results of the author. The primer begins with a brief classical review, in order to provide the reader with a foundation for the subsequent chapters. Topics covered include spectral decomposition, the Perron-Frobenius theorem, the Rayleigh quotient, the Weyl inequalities, and the Interlacing theorem. From this introduction, the book delves deeper into the properties of the principal eigenvector; a critical subject as many of the results on the spectral radius of graphs rely on the properties of the principal eigenvector for their proofs. A following chapter surveys spectral radius of special graphs, covering multipartite graphs, non-regular graphs, planar graphs, threshold graphs, and others. Finally, the work explores results on the structure of graphs having extreme spectral radius in classes of graphs defined by fixing the value of a particular, integer-valued graph invariant, such as: the diameter, the radius, the domination number, the matching number, the clique number, the independence number, the chromatic number or the sequence of vertex degrees.

Throughout, the text includes the valuable addition of proofs to accompany the majority of presented results. This enables the reader to learn tricks of the trade and easily see if some of the techniques apply to a current research problem, without having to spend time on searching for the original articles. The book also contains a handful of open problems on the topic that might provide initiative for the reader's research.

Dedicated coverage to one of the most prominent graph eigenvalues Proofs and open problems included for further study Overview of classical topics such as spectral decomposition, the Perron-Frobenius theorem, the Rayleigh quotient, the Weyl inequalities, and the Interlacing theorem

This monograph sets out a body of mathematical theory for finite graphs with nodes placed randomly in Euclidean space and edges added to connect points that are close to each other. As an alternative to classical random graph models, these geometric graphs are relevant to the modelling of real-world networks having spatial content, arising in numerous applications such as wireless communications, parallel processing, classification, epidemiology, astronomy, and the internet. Aimed at graduate students and researchers in probability, combinatorics, statistics, and theoretical computer science, it covers topics such as edge and component counts, vertex degrees, cliques, colourings, connectivity, giant component phenomena, vertex ordering and partitioning problems. It also illustrates and extends the application to geometric probability of modern techniques including Stein's method, martingale methods and continuum percolation.

The four-colour theorem is one of the famous problems of mathematics, that frustrated generations of mathematicians from its birth in 1852 to its solution (using substantial assistance from electronic computers) in 1976. The theorem asks whether four colours are sufficient to colour all conceivable maps, in such a way that countries with a common border are coloured with different colours.The book discusses various attempts to solve this problem, and some of the mathematics which developed out of these attempts. Much of this mathematics has developed a life of its own, and forms a fascinating part of the subject now known as graph theory.The book is designed to be self-contained, and develops all the graph-theoretical tools needed as it goes along. It includes all the elementary graph theory that should be included in an introduction to the subject, before concentrating on specific topics relevant to the four-colour problem.Part I covers basic graph theory, Euler's polyhedral formula, and the first published false `proof' of the four-colour theorem. Part II ranges widely through related topics, including map-colouring on surfaces with holes, the famous theorems of Kuratowski, Vizing, and Brooks, the conjectures of Hadwiger and Hajos, and much more besides. In Part III we return to the four-colour theorem, and study in detail the methods which finally cracked the problem.

The four-colour theorem is one of the famous problems of mathematics, that frustrated generations of mathematicians from its birth in 1852 to its solution (using substantial assistance from electronic computers) in 1976. The theorem asks whether four colours are sufficient to colour all conceivable maps, in such a way that countries with a common border are coloured with different colours.The book discusses various attempts to solve this problem, and some of the mathematics which developed out of these attempts. Much of this mathematics has developed a life of its own, and forms a fascinating part of the subject now known as graph theory.The book is designed to be self-contained, and develops all the graph-theoretical tools needed as it goes along. It includes all the elementary graph theory that should be included in an introduction to the subject, before concentrating on specific topics relevant to the four-colour problem.Part I covers basic graph theory, Euler's polyhedral formula, and the first published false `proof' of the four-colour theorem. Part II ranges widely through related topics, including map-colouring on surfaces with holes, the famous theorems of Kuratowski, Vizing, and Brooks, the conjectures of Hadwiger and Hajos, and much more besides. In Part III we return to the four-colour theorem, and study in detail the methods which finally cracked the problem.

Graph theory is a fast-developing discipline and has, over the past few decades, been seen to have important applications, particularly in operations research and in computer science. For this reason there is a need for a reference in the area, comprehensively cataloguing the properties of graphs and accessible to both graph theorists and other researchers. This book contains a wealth of information, including over 10000 diagrams and extensive tables of associated properties. It is the first book to present this information on such a scale, and as such will be an invaluable resource.

This is a book about graph homomorphisms. Graph theory is now an established discipline but the study of graph homomorphisms has only recently begun to gain wide acceptance and interest. The subject gives a useful perspective in areas such as graph reconstruction, products, fractional and circular colourings, and has applications in complexity theory, artificial intelligence, telecommunication, and, most recently, statistical physics.Based on the authors' lecture notes for graduate courses, this book can be used as a textbook for a second course in graph theory at 4th year or master's level and has been used for courses at Simon Fraser University (Vancouver), Charles University (Prague), ETH (Zurich), and UFRJ (Rio de Janeiro). The exercises vary in difficulty. The first few are usually intended to give the reader an opportunity to practice the concepts introduced in the chapter; the later ones explore related concepts, or even introduce new ones. For the harder exercises hints and references are provided.The authors are well known for their research in this area and the book will be invaluable to graduate students and researchers alike.

Graph theory and especially the study of graph homomorphisms has developed exponentially in the last couple of decades. Originally based on the authors' graduate course lecture notes, this is still the only book devoted entirely to graph homomorphisms, bringing together the highlights of the theory and its many applications. It offers a useful perspective on more traditional graph theory topics such as graph reconstruction, various forms of graph colouring, graph products, automorphism groups, chordality, and interval graphs, and has applications in complexity theory, artificial intelligence, telecommunication, and statistical physics. It is intended to be a sampler of this rich subject and focuses on its most interesting results, techniques, and applications, including its algebraic, combinatorial, and algorithmic aspects. The book discusses most of the recent developments in the field, such as the disproof of the Hedetniemi Product Conjecture, the proof of the Feder-Vardi Dichotomy Conjecture of constraint satisfaction problems, as well as the celebrated theory of graph convergence and graphons. The text contains exercises of varying difficulty, intended to support readers' learning and further development.

Generating random networks efficiently and accurately is an important challenge for practical applications, and an interesting question for theoretical study. This book presents and discusses common methods of generating random graphs. It begins with approaches such as Exponential Random Graph Models, where the targeted probability of each network appearing in the ensemble is specified. This section also includes degree-preserving randomisation algorithms, where the aim is to generate networks with the correct number of links at each node, and care must be taken to avoid introducing a bias. Separately, it looks at growth style algorithms (e.g. preferential attachment) which aim to model a real process and then to analyse the resulting ensemble of graphs. It also covers how to generate special types of graphs including modular graphs, graphs with community structure and temporal graphs. The book is aimed at the graduate student or advanced undergraduate. It includes many worked examples and open questions making it suitable for use in teaching. Explicit pseudocode algorithms are included throughout the book to make the ideas straightforward to apply. With larger and larger datasets, it is crucial to have practical and well-understood tools. Being able to test a hypothesis against a properly specified control case is at the heart of the 'scientific method'. Hence, knowledge on how to generate controlled and unbiased random graph ensembles is vital for anybody wishing to apply network science in their research.

Graph theory and especially the study of graph homomorphisms has developed exponentially in the last couple of decades. Originally based on the authors' graduate course lecture notes, this is still the only book devoted entirely to graph homomorphisms, bringing together the highlights of the theory and its many applications. It offers a useful perspective on more traditional graph theory topics such as graph reconstruction, various forms of graph colouring, graph products, automorphism groups, chordality, and interval graphs, and has applications in complexity theory, artificial intelligence, telecommunication, and statistical physics. It is intended to be a sampler of this rich subject and focuses on its most interesting results, techniques, and applications, including its algebraic, combinatorial, and algorithmic aspects. The book discusses most of the recent developments in the field, such as the disproof of the Hedetniemi Product Conjecture, the proof of the Feder-Vardi Dichotomy Conjecture of constraint satisfaction problems, as well as the celebrated theory of graph convergence and graphons. The text contains exercises of varying difficulty, intended to support readers' learning and further development.

Expander families enjoy a wide range of applications in mathematics and computer science, and their study is a fascinating one in its own right. Expander Families and Cayley Graphs: A Beginner's Guide provides an introduction to the mathematical theory underlying these objects.The central notion in the book is that of expansion, which roughly means the quality of a graph as a communications network. Cayley graphs are certain graphs constructed from groups; they play a prominent role in the study of expander families. The isoperimetric constant, the second largest eigenvalue, the diameter, and the Kazhdan constant are four measures of the expansion quality of a Cayley graph. The book carefully develops these concepts, discussing their relationships to one another and to subgroups and quotients as well as their best-case growth rates.Topics include graph spectra (i.e., eigenvalues); a Cheeger-Buser-type inequality for regular graphs; group quotients and graph coverings; subgroups and Schreier generators; the Alon-Boppana theorem on the second largest eigenvalue of a regular graph; Ramanujan graphs; diameter estimates for Cayley graphs; the zig-zag product and its relation to semidirect products of groups; eigenvalues of Cayley graphs; Paley graphs; and Kazhdan constants.The book was written with undergraduate math majors in mind; indeed, several dozen of them field-tested it. The prerequisites are minimal: one course in linear algebra, and one course in group theory. No background in graph theory or representation theory is assumed; the book develops from scatch the required facts from these fields. The authors include not only overviews and quick capsule summaries of key concepts, but also details of potentially confusing lines of reasoning.The book contains ideas for student research projects (for capstone projects, REUs, etc.), exercises (both easy and hard), and extensive notes with references to the literature.

An Introduction to Grids, Graphs, and Networks aims to provide a concise introduction to graphs and networks at a level that is accessible to scientists, engineers, and students. In a practical approach, the book presents only the necessary theoretical concepts from mathematics and considers a variety of physical and conceptual configurations as prototypes or examples. The subject is timely, as the performance of networks is recognized as an important topic in the study of complex systems with applications in energy, material, and information grid transport (epitomized by the internet). The book is written from the practical perspective of an engineer with some background in numerical computation and applied mathematics, and the text is accompanied by numerous schematic illustrations throughout. In the book, Constantine Pozrikidis provides an original synthesis of concepts and terms from three distinct fields-mathematics, physics, and engineering-and a formal application of powerful conceptual apparatuses, like lattice Green's function, to areas where they have rarely been used. It is novel in that it grids, graphs, and networks are connected using concepts from partial differential equations. This original material has profound implications in the study of networks, and will serve as a resource to readers ranging from undergraduates to experienced scientists.

A colourful workbook to help children learn about measuring and simple graphs using the Maths Mastery approach!It's time to master the world of maths with this brand new workbook developed by experts to cover all the essential skills for children in the first stages of their maths journey. Specifically designed for children aged 8-9, this maths exercise book can help your child explore measurement, time, money and simple graphs. Let Maths - No Problem! open your eyes to the wonderful world of numbers! So what are you waiting for? Dive straight in to discover: -Character-led learning with fun and friendly illustrations-Comprehensive and thorough coverage of key concepts-Helpful hints for extra practice-Topics include mass, volume, capacity, height, length, units of measure and unit conversions, money, bar and line graphs. Heard of Maths Mastery but don't know where to begin? Don't worry, DK has got you covered!Maths Mastery is a transformational approach to teaching mathematics that was developed in Singapore and is now used around the world, including in thousands of schools across the UK. Children are taught multiple methods to solve maths problems, enabling them to gain a thorough and adaptable understanding of topics, making it easier for them to apply their knowledge to different scenarios. Each topic is studied in depth to develop a secure understanding of maths concepts. This leads to greater progress in maths learning, as topics are well understood and therefore don't need to be revised in great detail, meaning each new concept builds on prior learning.Based on this widely successful approach, this wonderful workbook is full of top-tips led by appealing characters who offer useful guidance to parents and children alike with the goal of making learning maths both accessible and interesting. This workbook is part of a new series developed in partnership with the award-winning educational publisher Maths - No Problem! The books encourage a deep knowledge of core maths concepts without fostering a reliance on memorisation or rote learning. In fact, Maths - No Problem! is widely used in schools in the UK and its Teacher Hub receives 4,000 daily visits on average during term time!A must-have maths book for children aged 8-9 at a Key Stage 2 level, as well as for adults who wish to become more familiar with maths taught in schools, to better support their children at home.From scales to line graphs and self-check quizzes, this helpful workbook covers it all whilst incorporating an inclusive approach which builds self-confidence so that no child feels they 'can't do maths'.At DK, we believe in the power of discovery. So why not explore the rest of the workbooks in our Maths - No Problem! Master Maths At Home series? Each workbook can be bought individually or as part of a collection, specifically targeted to early learners aged 8-9 at a Key Stage 2 level. Multiply like a magician with Maths - No Problem! Multiplication and Division, Ages 8-9 (Key Stage 2) and have fun finding fractions with Maths - No Problem! Fractions and Decimals, Ages 8-9 (Key Stage 2).It's as easy as 1, 2, 3!

A colourful workbook to help children learn about measuring and graphs using the Maths Mastery approach!It's time to master the world of maths with this brand new workbook developed by experts to cover all the essential skills for children in the first stages of their maths journey. Specifically designed for children aged 9-10, this maths exercise book can help your child learn about different types of graphs and ways ofmeasuring, recording and presenting data.Let Maths - No Problem! open your eyes to the wonderful world of numbers! So what are you waiting for? Dive straight in to discover: -Character-led learning with fun and friendly illustrations-Comprehensive and thorough coverage of key concepts-Helpful hints for extra practice-Topics include measuring and converting mass, volume, capacity and length, time, and graphsHeard of Maths Mastery but don't know where to begin? Don't worry, DK has got you covered!Maths Mastery is a transformational approach to teaching mathematics that was developed in Singapore and is now used around the world, including in thousands of schools across the UK. Children are taught multiple methods to solve maths problems, enabling them to gain a thorough and adaptable understanding of topics, making it easier for them to apply their knowledge to different scenarios. Each topic is studied in depth to develop a secure understanding of maths concepts. This leads to greater progress in maths learning, as topics are well understood and therefore don't need to be revised in great detail, meaning each new concept builds on prior learning.Based on this widely successful approach, this wonderful workbook is full of top-tips led by appealing characters who offer useful guidance to parents and children alike with the goal of making learning maths both accessible and interesting. This workbook is part of a new series developed in partnership with the award-winning educational publisher Maths - No Problem! The books encourage a deep knowledge of core maths concepts without fostering a reliance on memorisation or rote learning. In fact, Maths - No Problem! is widely used in schools in the UK and its Teacher Hub receives 4,000 daily visits on average during term time!A must-have maths book for children aged 9-10 at a Key Stage 2 level, as well as for adults who wish to become more familiar with maths taught in schools, to better support their children at home.From recording information on graphs to calculating capacity, this helpful workbook covers it all whilst incorporating an inclusive approach which builds self-confidence so that no child feels they 'can't do maths'.At DK, we believe in the power of discovery. So why not explore the rest of the workbooks in our Maths - No Problem! Master Maths At Home series? Each workbook can be bought individually or as part of a collection, specifically targeted to early learners aged 9-10, at a Key Stage 2 level. Multiply like a magician with Maths - No Problem! Multiplication and Division, Ages 9-10, (Key Stage 2) and have fun finding fractions and more with Maths - No Problem! Fractions, Decimals and Percentage, Ages 9-10, (Key Stage 2).It's as easy as 1, 2, 3!

A colourful workbook to help children learn about measuring and simple graphs using the Maths Mastery approach!It's time to master the world of maths with this brand new workbook developed by experts to cover all the essential skills for children in the first stages of their maths journey. Specifically designed for children aged 10-11, this maths exercise book can help your child build on their skills with graphs, averages andmeasuring. Let Maths - No Problem! open your eyes to the wonderful world of numbers! So what are you waiting for? Dive straight in to discover: -Character-led learning with fun and friendly illustrations-Comprehensive and thorough coverage of key concepts-Helpful hints for extra practice-Topics include reading and writing pie and line graphs, calculating mean averages, and converting units of length, mass, volume and timeHeard of Maths Mastery but don't know where to begin? Don't worry, DK has got you covered!Maths Mastery is a transformational approach to teaching mathematics that was developed in Singapore and is now used around the world, including in thousands of schools across the UK. Children are taught multiple methods to solve maths problems, enabling them to gain a thorough and adaptable understanding of topics, making it easier for them to apply their knowledge to different scenarios. Each topic is studied in depth to develop a secure understanding of maths concepts. This leads to greater progress in maths learning, as topics are well understood and therefore don't need to be revised in great detail, meaning each new concept builds on prior learning.Based on this widely successful approach, this wonderful workbook is full of top-tips led by appealing characters who offer useful guidance to parents and children alike with the goal of making learning maths both accessible and interesting. This workbook is part of a new series developed in partnership with the award-winning educational publisher Maths - No Problem! The books encourage a deep knowledge of core maths concepts without fostering a reliance on memorisation or rote learning. In fact, Maths - No Problem! is widely used in schools in the UK and its Teacher Hub receives 4,000 daily visits on average during term time!A must-have maths book for children aged 10-11 at a Key Stage 2 level, as well as for adults who wish to become more familiar with maths taught in schools, to better support their children at home.From self-check quizzes to recording data, this helpful workbook covers it all whilst incorporating an inclusive approach which builds self-confidence so that no child feels they 'can't do maths'.At DK, we believe in the power of discovery. So why not explore the rest of the workbooks in our Maths - No Problem! Master Maths At Home series? Each workbook can be bought individually or as part of a collection, specifically targeted to early learners aged 10-11 at a Key Stage 2 level. Know your numbers like a pro with Maths - No Problem! Numbers to 10 Million, Ages 10-11 (Key Stage 2) and have fun finding fractions and more with Maths - No Problem! Fractions, Decimals, Percentage and Ratio, Ages 10-11 (Key Stage 2).It's as easy as 1, 2, 3!

This text is an exploration of automated reasoning and resolution in the expanding field of conceptual structures. Designed not only for computing scientists researching conceptual graphs, but also for anyone interested in exploring the design of knowledge bases, the book explores what are proving to be the fundamental methods for representing semantic relations in knowledge bases. While it provides the first comprehensive treatment of conceptual graph unification and reasoning, the book also addresses fundamental issues of graph matching, automated reasoning, knowledge bases, constraints, ontology and design. With a large number of examples, illustrations, and both formal and informal definitions and discussions, this book should prove useful as a tutorial for the reader new to conceptual graphs, or as a reference book for a senior researcher in artificial intelligence, knowledge representation or automated reasoning.

A revised and expanded advanced-undergraduate/graduate text (first ed., 1978) about optimization algorithms for problems that can be formulated on graphs and networks. This edition provides many new applications and algorithms while maintaining the classic foundations on which contemporary algorithm

This book is a tribute to Paul Erd\H{o}s, the wandering mathematician once described as the "prince of problem solvers and the absolute monarch of problem posers." It examines -- within the context of his unique personality and lifestyle -- the legacy of open problems he left to the world after his death in 1996. Unwilling to succumb to the temptations of money and position, Erd\H{o}s never had a home and never held a job. His "home" was a bag or two containing all his belongings and a record of the collective activities of the mathematical community. His "job" was one at which he excelled: identifying a fundamental roadblock in some particular line of approach and capturing it in a well-chosen, often innocent-looking problem, whose solution would likewise provide insight into the underlying theory. By cataloguing the unsolved problems of Erd\H{o}s in a comprehensive and well-documented volume, the authors hope to continue the work of an unusual and special man who fundamentally influenced the field of mathematics.

Emphasizing the understanding of images and their influences on how they affect our attitudes, beliefs, and actions, this fully updated sixth edition offers consequential ways of looking at images from the perspectives of photographers, critics, theoreticians, historians, curators, and editors. It invites informed conversations about meanings and implications of images, providing multiple and sometimes conflicting answers to questions such as: What are photographs? Should they be called art? Are they ethical? What are their implications for self, society, and the world? From showing how critics verbalize what they see in images and how they persuade us to see similarly, to dealing with what different photographs might mean, the book posits that some interpretations are better than others and explains how to deliberate among competing interpretations. It looks at how the worth of photographs is judged aesthetically and socially, offering samples and practical considerations for both studio critiques for artists and professional criticism for public audiences. This book is a clear and accessible guide for students of art history, photography and criticism, as well as anyone interested in carefully looking at and talking about photographs and their effects on the world in which we live.