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This book features a unified derivation of the mathematical theory of the three classical types of invariant random matrix ensembles--orthogonal, unitary, and symplectic. The authors follow the approach of Tracy and Widom, but the exposition here contains a substantial amount of additional material, in particular, facts from functional analysis and the theory of Pfaffians. The main result in the book is a proof of universality for orthogonal and symplectic ensembles corresponding to generalized Gaussian type weights following the authors' prior work. New, quantitative error estimates are derived. The book is based in part on a graduate course given by the first author at the Courant Institute in fall 2005. Subsequently, the second author gave a modified version of this course at the University of Rochester in spring 2007. Anyone with some background in complex analysis, probability theory, and linear algebra and an interest in the mathematical foundations of random matrix theory will benefit from studying this valuable reference. ... Read more

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Continuing to provide a carefully written, thorough introduction, Graphs & Digraphs, Fifth Edition expertly describes the concepts, theorems, history, and applications of graph theory. Nearly 50 percent longer than its bestselling predecessor, this edition reorganizes the material and presents many new topics.

New to the Fifth Edition

• New or expanded coverage of graph minors, perfect graphs, chromatic polynomials, nowhere-zero flows, flows in networks, degree sequences, toughness, list colorings, and list edge colorings
• New examples, figures, and applications to illustrate concepts and theorems
• Expanded historical discussions of well-known mathematicians and problems
• More than 300 new exercises, along with hints and solutions to odd-numbered exercises at the back of the book
• Reorganization of sections into subsections to make the material easier to read
• Bolded definitions of terms, making them easier to locate

Despite a field that has evolved over the years, this student-friendly, classroom-tested text remains the consummate introduction to graph theory. It explores the subject’s fascinating history and presents a host of interesting problems and diverse applications.

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Revised throughout

Includes new chapters on the network simplex algorithm and a section on the five color theorem

Recent developments are discussed

Customer Reviews (3)

THE graph algorithm book
This is the definitive guide to graph algorithms.Every algorithm is well documented with proofs and complexity estimates.A general knowledge of graph theory is presupposed.This is a very good thing, since then neither paper or time needs to be vasted on elementaries.There are heaps of introductory books around.
Some areas are rather complex.But that's the fun part, right?

Comprehensive book on graphs and networks
This is a comprehensive book on graphs and networks. It is concise and terse with few excessive prose. It is not for a beginner who is learning graphs and networks for the first time. A beginner would do well to study other elementary books such as Graphs and Applications by Robin Wilson. That being said, readers with some knowledge of the subject will profit immensely by studying this well-written book.

NOT a good book for beginners
It may be a good book for experts in Graph Theory but definitely not for beginners.

The content in this book is like a record of discussion in an expert meeting. It doesn't contain enough description which a text book should have. To summarize the cons:

* It always supposes you know every basic points without reviewing it.
* It's lazy to give a figure to explain a complex theory or proof.
* Also, proofs are simple in obscure text and really unclear.
* It gives thedefinitions in the context without using bold text. You need to read very line to find an important definition.
* It leaves too many things for reader to do.

However, covering everything of graph field in the contents may be a good point to mention about this book, which is attractive to buyers. ... Read more

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This monograph provides and explains the mathematics behind geometric graph theory, which studies the properties of a graph that consists of nodes placed in Euclidean space so that edges can be added to connect points that are close to one another. For example, a collection of trees scattered in a forest and the disease that is passed between them, a set of nests of animals or birds on a region and the communication between them or communication between communications stations or nerve cells. Aimed at graduate students and researchers in probability, statistics, combinatorics and graph theory including computer scientists, it covers topics such as: technical tools, edge and component counts, vertex degrees, clique and chromatic number, and connectivity. Applications of this theory are used in the study of neural networks, spread of disease, astrophysics and spatial statistics. ... Read more

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The first chapter of this book provides a brief treatment ofthe basics of the subject. The other chapters deal with the variousdecompositions of non-negative matrices, Birkhoff type theorems, thestudy of the powers of non-negative matrices, applications of matrixmethods to other combinatorial problems, and applications ofcombinatorial methods to matrix problems and linear algebra problems.

The coverage of prerequisites has been kept to a minimum.Nevertheless, the book is basically self-contained (an Appendixprovides the necessary background in linear algebra, graph theory andcombinatorics). There are many exercises, all of which are accompaniedby sketched solutions. Audience: The book is suitable for a graduate course as well asbeing an excellent reference and a valuable resource formathematicians working in the area of combinatorial matrix theory. ... Read more

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Applications of Graph Theory and Topology in Inorganic Cluster and Coordination Chemistry is a text-reference that provides inorganic chemists with a rudimentary knowledge of topology, graph theory, and related mathematical disciplines. The book emphasizes the application of these topics to metal clusters and coordination compounds.The book's initial chapters present background information in topology, graph theory, and group theory, explaining how these topics relate to the properties of atomic orbitals and are applied to coordination polyhedra. Subsequent chapters apply these ideas to the structure and chemical bonding in diverse types of inorganic compounds, including boron cages, metal clusters, solid state materials, metal oxide derivatives, superconductors, icosahedral phases, and carbon cages (fullerenes). The book's final chapter introduces the application of topology and graph theory for studying the dynamics of rearrangements in coordination and cluster polyhedra. ... Read more

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The only text available on graph theory at the freshman/sophomore level, it covers properties of graphs, presents numerous algorithms, and describes actual applications to chemistry, genetics, music, linguistics, control theory and the social sciences. Illustrated. ... Read more

Customer Reviews (1)

Great starter for even self-study
Graph theory and analysis is becoming increasingly popular and useful, and not just in pure mathematics. Indeed, a significant amount of computer network theory connects with some of the models and mathematical modeling presented here.

What is especially surprising in this text is the ease with which one can self-study a significant amount of the material. There are, of course, numerous drawings to clarify concepts.Plan on making a bunch of your own sketches too with this text. There are enough answers in the back to verify your understanding of the concepts.

If you have seen a chapter or so on graph theory in another introductory textbook on discrete mathematics (or computer mathematics) this can be a great continuation of that study. By being fairly accessible to students immediately after precalculus (or strong algebra/geomtery) the text understandably does not go into the deepest elements of graph theory for pure mathematics studies. So this book is probably ideally suited to students who need a fair amount of mathematics but whom are not necessarily math majors. Computer and social science students will most likely benefit from the modeling presented here (different models for each of those fields but there are enough examples here to assure a strong background for either field).

A used copy of this text (due to cost) is a great idea to supplement any discrete math textbook to explore the token chapter or two on graph theory. ... Read more

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Graphs are extremely useful in modelling systems in physical sciences and engineering problems, because of their intuitive diagrammatic nature. This text gives a reasonably deep account of material closely related to engineering applications. Topics like directed-graph solutions of linear equations, topological analysis of linear systems, state equations, rectangle dissection and layouts, and minimal cost flows are included. A major theme of the book is electrical network theory. This book is basically intended as a reference text for researchers, and requires a certain level of mathematical maturity. However the text may equally well be used for graduate level courses on network topology and linear systems and circuits. Some of the later chapters are suitable as topics for advanced seminars. A special feature of the book is that references to other published literature are included for almost all the results presented, making the book handy for those wishing to continue with a study of special topics. ... Read more

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Sperner's theorem stimulated the development of a fast-growing theory dealing with external problems on finite sets and, more generally, on finite partially ordered sets. This book presents Sperner theory from a unified point of view, bringing combinatorial techniques together with methods from programming, linear algebra, Lie-algebra representations and eigenvalue methods, probability theory, and enumerative combinatorics. ... Read more

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The graph isomorphism problem belongs to the part of Complexity Theory that focuses on the structure of complexity classes involved in the classification of computational problems and in the relations among them. It consists in deciding whether two given graphs are isomorphic, i.e. whether there is a bijective mapping from the nodes of one graph to the nodes of the second graph such that the edge connections are respected. It is a problem of considerable practical as wen as theoretical importance that is, as of now, unresolved in the sense that no efficient algorithm for it has yet been found. Given this fact, it is natural to ask whether such an algorithm exists at an or whether the problem is intractable. -Be book focuses on this issue and presents several recent results that provide a better understanding of the relative position of the graph isomorphism problem in the class NP as well as in other complexity classes. It also uses the problem to illustrate important concepts in structural complexity, providing a look into the more general theory.'The book is basically self-contained; the only prerequisite for reading it is some elementary knowledge from Complexity Theory and Probability Theory. Its level of presentation makes it eminently suitable for a seminar or graduate course devoted to the problem, or as a rich source of examples for a standard graduate course in Complexity Theory. ... Read more

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Combinatorial games are games of pure strategy involving two players, with perfect information and no element of chance. Starting from the very basics of gameplay and strategy, the authors cover a wide range of topics, from game algebra and surreal numbers to special classes of games. Classic techniques are introduced and applied in novel ways to analyze both old and new games, several appearing for the first time in this book. This book makes an excellent guide for undergraduates or for self-study by the enterprising reader, with a generous collection of exercises and problems scattered throughout the book. ... Read more

Customer Reviews (1)

understanding many types of games
The cover illustration is quite well done. It hints at the range of games considered in the book. Inside, the discourse is highly mathematical. Not a trivial read, but suited to a reader who has already taken a course in discrete maths.

The book explains how to classify games by various criteria. So there could be impartial games and non-impartial games, for example. Another viewpoint is that some games have move order being vital in determining the outcome. Think chess or go.

The authors have also generously supplied many problems in each chapter. Including solutions. But what could be of interest to some readers is the description of the Combinatorial Game Suite. An open source Java program that lets you get at many built in games. While the Java programmer can extend it to incorporate other games written in Java. ... Read more

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Here is an accessible, algorithmically oriented guide to some of the most interesting techniques of complexity theory. The book shows that simple algorithms are at the heart of complexity theory. The book is organized by technique rather than by topic. Each chapter focuses on one technique: what it is, and what results and applications it yields.

Customer Reviews (1)

Good idea, but badly executed
The aim of this book is to introduce complexity theory using a more technique-oriented approach, which is not seen in other complexity theory textbooks. Techniques covered are:
- self-reducibility
- one-way function
- tournament divide and conquer
- isolation technique (aka. isolation lemma)
- witness reduction
- polynomial interpolation
- nonsolvable group (used in Barrington's theorem to show that width-five branching program can simulate any Boolean formula and thus the complexity class NC1)
- random restriction technique
- polynomial technique
This seems like an awesome approach and I can't agree with the authors more about the organization of techniques. This was the reason why I bought the book.

Unfortunately, the results are badly presented in a lot of places. Proofs are extremely tedious with unnecessary details and notations. This overly formal approach prevents readers from seeing the big strategy or ideas behind the proofs. For example, Barrington's theorem has a very elegant and readable 1-page proof (cf. Boppana-Sipser survey "The Complexity of Finite Functions", 1989). But in this book they decided to include an ugly 6 page proof!

The students will learn more from "Computational Complexity" by Arora and Barak, where only the most elegant and readable versions of proofs are selected. Papadimitriou does not cover recent results, but is also a solid textbook with full of excellent exercises. ... Read more

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Analyzing the behavior of complex networks is an important element in the design of new man-made structures such as communication systems and biologically engineered molecules. Because any complex network can be represented by a graph, and therefore in turn by a matrix, graph theory has become a powerful tool in the investigation of network performance. This self-contained book provides a concise introduction to the theory of graph spectra and its applications to the study of complex networks. Covering a range of types of graphs and topics important to the analysis of complex systems, this guide provides the mathematical foundation needed to understand and apply spectral insight to real-world systems. In particular, the general properties of both the adjacency and Laplacian spectrum of graphs are derived and applied to complex networks. An ideal resource for researchers and students in communications networking as well as in physics and mathematics. ... Read more

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Professor Hodges emphasizes definability and methods of construction, and introduces the reader to advanced topics such as stability.He also provides the reader with much historical information and a full bibliography, enhancing the book's use as a reference. ... Read more

Customer Reviews (4)

Not for self study
I study model theory as a hobby and this book was recommended to me by the model theorist Carol Wood who said it contains a lot of gossip which I learned to interpert as chatty. I bought the book around 1995 and since then developed a friendship and correspondence with the author. My copy of the book is well worn and the binding has been mended with duct tape several times.I persued a career in industry rather than a professorship in math so I mainly use math books for self study and this is where this book is weakest. Too many abstractions, too many incomplete proofs, too many errors, too few examples, too much left to the exercises( Morley's category Theorem!), etc. So the reader beware if left on his own. As best I can tell Hodge's view about this is although a text may be trying the reader will learn through his suffering which is what readers of the works of the master logician Sharon Shelah teaches you.On the other hand reading what a master like Hodges has to say is inspriational! His insights about the similarities between game theory and model theory and the role of almagation diagrams being at the hearbeat of model theory are worth the price of the book.In summary if I were to teach a course in model theory I would recomment to the brightest students to pick up this book as a reference and then would go on to teach a sixty year course to cover its contents.

Hodges' Model Theory
Vast in scope, full of careful attention to detail; proofs are generally clear and well-turned. Style is generally more accessible than that of a typical graduate textbook. Contains generous bibliography. This properly areference and not a textbook, but would serve well the interested novice.

Encyclopedic but readable
The style is light, even glib, but still demanding. The prerequisites are not stated, but I would say undergraduate logic, graduate algebra, and a good deal of general mathematical experience. Motivation and context are very good and examples are frequent and clearly illustrative. What makes it challenging is that many details are glossed-over without the reader being warned about what knowledge is being assumed. His proofs also make many off-hand references rather than include every detail. Another thing I think it could use is more illustration of procedures; all of the examples seem to be of cases rather than methods. Still, explanations of specific points are often much clearer than in Marker.

Excellent text, very clearly written
This is an excellent text on the model theory of first-order languages. Acondensed version has also been published as: "A Shorter ModelTheory."

This book is a classic. Its influence should be comparableto Keisler & Chang's famous book as *the* standard account of therelationship between the language of f.o. logic and mathematicalstructures. (N.B. Chang & Keisler's last edition is still usefulalongside this book since the contents are not identical!)

The text isclear and fluent, as one has come to expect from this remarkable author.The book includes a detailed bibliography and suggestions for furtherreading. The binding and typesetting are beautifully done as well.

Theshort-comings are: (1) the large number of typos including an"incorrect lemma" (but with corrections available on the author'shomepage), (2) the almost exclusive concentration on (untyped) first-orderlanguages and (3) avoidance of certain speacialized topics.

All in all,this is an excellent book. ... Read more

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Primarily of interest to researchers interested in graph theory from mathematics, combinatorics, computer science and telecommunications. The book will also appeal to students of graph theory. ... Read more

Customer Reviews (1)

For advanced undergraduate students and above
This is not a friendly book in any sense of the word.I didn't find it as cohesive as I imagined it to be.It contains a somewhat random smattering of important original papers about perfect graphs.I used this book as a source for my thesis, and I expected to get more out of it than I did.I ended up only using history from it because it was too advanced for me.I am a senior undergraduate math major with only 1/3 of a semester's experience in graph theory, in an easycombinatorics class.I'm sure someone with more experience will find this book much more useful. ... Read more

Product Description
This book is designed to describe fundamentalalgorithmic techniques for constructing drawings of graphs. Suitable as a book or reference manual, its chapters offer anaccurate, accessible reflection of the rapidly expanding field of graphdrawing. ... Read more

Customer Reviews (7)

The best book I can find on the topic, but not really that good
I'm giving this book 4 stars only because it is dramatically better than the other books that are available on this topic.If there were better competition I would give it 3 stars since it has a range of flaws.The major flaw has been mentioned by other reviewers; it gives only terse and highly mathematical descriptions of the algorithms and the effort to convert anything in this book into an actual algorithm is huge.However on the plus side, the book is mostly well written and is actually much *less* mathematical than almost every other book on the topic.Most other planar graph drawing books just lay down some formulas and assume implementation is obvious (very far from true in this topic).However, this book does at least give a nod to the algorithm side and lays out a general framework for an implementation of most of the important layout types.The book also gives a strong foundation for almost the entire field; probably because the authors represent some of the strongest contributors to this field.So the bottom line is that, as of today you almost have to buy this book if you want to work on this type of software since no other book comes close to explaining as much and does it as well as this book.

Graph drawing, 1st edition
Worth reading, if a little dated now. These guys all know their stuff, as one would expect from such distinguished writers in the field. Can't wait for the second edition.

Not the best book ever...
This graph drawing book is, according to my lecturer, one of the few books on this subject. There is a different book too, written by some japanese authors. The drawback of the latter book is that it is too technical sometimes, while this book discusses intuitively understandable algorithms.
But, there are also some major drawbacks concerning this book. Not all chapters are equally good: some are horrible to read while others are very understandable.
Furthermore, the book is not printed anymore, so you just purchase a black-and-white copy of the original book with some fancy cover. Since the book is just copied, some pictures do not look like the way they should and one picture isn't shown at all. This results in some difficulties understanding the pictures and discussed material. I purchased one of the cheaper variants of the book, maybe the more expensive ones are printed versions.
Also, the book is absolutely not free of mistakes!
To conclude: the book is probably OK for understanding the basics of graph drawing, considering what's for sale. But beware of the drawbacks of this book and use it with care.

Who is this book for?
I am a mathematician/computer scientist quite interested in the subject matter, but the book is almost useless, since it mostly discusses ad hoc methods, and avoids proving any of the actual theorems in the subject. Unfortunate, since there is certainly room for a good book on the subject.

Not useful for me, maybe for other people
To me the book is not useful, because I need to draw graphs in which the distance between two connected vertices is fixed. The book doesn't mention any method to handle graphs with such a restriction, although the chapter on force-directed methods inspired me to use something similar. If you draw graphs without that restriction, the book might be useful to you - that's why I'm careful and give it 4 stars.

I disagree with Viv. R who said it doesn't contain pseudo code, because the book contains quite some of it, though not in every chapter. But even lack of pseudo code doesn't bother me, because for an experienced mathematical programmer that should be no problem. ... Read more

Product Description
Graph theory is an important branch of contemporary combinatorial mathematics.By describing recent results in algebraic graph theory and demonstrating how linear algebra can be used to tackle graph-theoretical problems, the authors provide new techniques for specialists in graph theory. The book explains how the spectral theory of finite graphs can be strengthened by exploiting properties of the eigenspaces of adjacency matrices associated with a graph. The extension of spectral techniques proceeds at three levels: using eigenvectors associated with an arbitrary labeling of graph vertices, using geometrical invariants of eigenspaces such as graph angles and main angles, and introducing certain kinds of canonical eigenvectors by means of star partitions and star bases.Current research on these topics is part of a wider effort to forge closer links between algebra and combinatorics. Problems of graph reconstruction and identification are used to illustrate the importance of graph angles and star partitions in relation to graph structure. Specialists in graph theory will welcome this treatment of important new research. ... Read more

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This book provides a pedagogical and comprehensive introduction to graph theory and its applications. It contains all the standard basic material and develops significant topics and applications, such as: colorings and the timetabling problem, matchings and the optimal assignment problem, and Hamiltonian cycles and the traveling salesman problem, to name but a few. Exercises at various levels are given at the end of each chapter, and a final chapter presents a few general problems with hints for solutions, thus providing the reader with the opportunity to test and refine their knowledge on the subject. An appendix outlines the basis of computational complexity theory, in particular the definition of NP-completeness, which is essential for algorithmic applications. ... Read more