Book Description

An extraordinary variety of disciplines rely on graphs to convey their fundamentals as well as their finer points. With this concise and well-written text, anyone with a firm grasp of general mathematics can follow the development of graph theory and learn to apply its principles in methods both formal and abstract. The work of a distinguished mathematician, this text uses practical examples to illustrate the theory's broad range of applications, from the behavioral sciences, information theory, cybernetics, and other areas, to mathematical disciplines such as set and matrix theory. 1966 ed. 109 b/w illus.
... Read more

Book Description

This book consists of a series of 46 short laboratory investigations using the TI-83 Plus, the TI-83 or the TI-82 graphing calculator. These mini-labs span the topics normally covered in Intermediate and College Algebra courses. They are designed to take between 30 and 60 minutes to complete. Many of the mini-labs emphasize the experimental method, proceeding from collecting data to forming a hypothesis and concluding with a generalization. When the calculator is needed the required keystrokes are shown. Mini-Lab 0 familiarizes the user with the keyboard. Other features of the calculator are introduced as needed to investigate a given problem. ... Read more

Book Description

Improved by more than a dozen new exercises, an augmented section on labeling, the simplification of many proofs, and corrections suggested by classroom users and reviewers, this delightful text on graph theory retains and strengthens the appealing features of the original edition. It is an innovative and stimulating view of mathematics designed to appeal to teachers and students alike. Pearls in Graph Theory is based on twenty years of teaching by the leading researcher in graph theory. Unlike most texts on graph theory, this book is written in an informal style suitable for students in a variety of disciplines, though mathematics majors will find the material of sufficient depth and challenge. Covering major topics and theorems in graph theory, the text provides students with a solid foundation while keeping the material enjoyably accessible and entertaining. This course typically draws 50 to 70 students per year at the University of California, San Diego. The concrete nature of the topics, as well as the broad coverage of the field, allow the book to be used for a survey course at smaller schools with no undergraduate courses in graph theory. The only requirement is some mathematical maturity, about the level attained by a successful calculus student. ... Read more

Reviews (1)

A book on graph theory
Pearls in graph theory begins informally and at an elementary level, suitable for a substantial freshman-sophomore course. After intuitive introductions, concepts and theory are developed with increasing depth, leading into material for a good intermediate-level course. Included also are appropiate open conjetures on, for example the Oberwohlfach problem, magic and antimagic graphs, and colorings of "earth-moon" maps. In addition, for me the final three chapters are a splendid, enticingly elementary yet comprehensive introduction to topological graph theory. Essentially no othr introductory text presents the depth, breadth, and fun of topological graph theory as does this book. ... Read more

Book 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

Book Description

In this substantial revision of a much-quoted monograph first published in 1974, Dr. Biggs aims to express properties of graphs in algebraic terms, then to deduce theorems about them. In the first section, he tackles the applications of linear algebra and matrix theory to the study of graphs; algebraic constructions such as adjacency matrix and the incidence matrix and their applications are discussed in depth. There follows an extensive account of the theory of chromatic polynomials, a subject that has strong links with the "interaction models" studied in theoretical physics, and the theory of knots. The last part deals with symmetry and regularity properties. Here there are important connections with other branches of algebraic combinatorics and group theory.The structure of the volume is unchanged, but the text has been clarified and the notation brought into line with current practice. A large number of "Additional Results" are included at the end of each chapter, thereby covering most of the major advances in the past twenty years.This new and enlarged edition will be essential reading for a wide range of mathematicians, computer scientists and theoretical physicists. ... Read more

Reviews (1)

Well written, intriguing, but somewhat dated
I found this book very clear, concise, and well written. Most proofs seemed effortless on both the author's and reader's part. Moreover, I found the content very interesting, especially the results on the spectrum of a graph, which simply represents the eigenvalues (and their multiplicities) of the adjacency matrix (although recently it has been defined in terms of the normalized incidence matrix). It amazed me just how much mathematics can be put to bear on such a simple notion of vertices and connections between them. The only downside to this book is that algebraic graph theory has moved in many new directions since the first edition (the second edition mostly states some recent results at the end of each chapter), and the interested reader may want to supplement this book or follow up this book with the following: "Spectral Graph Theory", by Fan Chung, "Algebraic Graph Theory", by Godsil et al., and "Modern Graph Theory", by Bollobas, all of which make for graduate-level reading. And to think that some mathematics departments still show little if any desire to support fellow mathematicians who study graph theory! Reading any of these books hopefully might alter some of those entrenched attitudes. ... Read more

Book Description

This book on the theory of graphs provides the reader with a mathematical tool which can be used in the behavioral sciences, in the theory of information, cybernetics, games, transport networks, as well as in set theory and matrix theory. ... Read more

Reviews (1)

Only one of many such books that could be written
What is incredible about this book is not that there are over one hundred pages of unsolved problems posed all or in part by Paul Erdos. The amazing thing is that the word graphs could be replaced by several other mathematical words or phrases and a similar book could be written. Erdos was an expert in so many areas of mathematics and perhaps his greatest ability was in putting forward just the right questions to just the right people. There is very little explanation of the problem proposals, the authors rely on a great deal of the listing of references to fill in the details. Therefore, anyone interested in exploring the problems in greater detail should be prepared to spend some time and effort in tracking down the relevant articles. Fortunately, the authors themselves did a great deal of that, as there are complete references for every problem that appears.
The range of problems is a demonstration of the depth of his understanding of graph theory, and also a demonstration of how little is still unresolved. I put forward no pretense to understanding any more than a few of the problems in this book. However, that did not alter my interest in the problems, as I was able to understand the fundamentals of almost all of them. Reading this book is one of the most educational experiences that I have had in the past year and I encourage all mathematicians at the level of slightest interest in graph theory and above to read it.

Published in Journal of Recreational Mathematics, reprinted with permission. ... Read more

Book Description

Graphical models use graphs to represent and manipulate joint probability distributions. They have their roots in artificial intelligence, statistics, and neural networks. The clean mathematical formalism of the graphical models framework makes it possible to understand a wide variety of network-based approaches to computation, and in particular to understand many neural network algorithms and architectures as instances of a broader probabilistic methodology. It also makes it possible to identify novel features of neural network algorithms and architectures and to extend them to more general graphical models.

This book exemplifies the interplay between the general formal framework of graphical models and the exploration of new algorithms and architectures. The selections range from foundational papers of historical importance to results at the cutting edge of research.

Contributors
H. Attias, C. M. Bishop, B. J. Frey, Z. Ghahramani, D. Heckerman, G. E. Hinton, R. Hofmann, R. A. Jacobs, Michael I. Jordan, H. J. Kappen, A. Krogh, R. Neal, S. K. Riis, F. B. Rodrguez, L. K. Saul, Terrence J. Sejnowski, P. Smyth, M. E. Tipping, V. Tresp, Y. Weiss. ... Read more

Book Description

This book is primarily aimed at graduate students and researchers in graph theory, combinatorics, or discrete mathematics in general. However, all the necessary graph theory is developed from scratch, so the only pre-requisite for reading it is a first course in linear algebra and a small amount of elementary group theory.It should be accessible to motivated upper-level undergraduates. ... Read more

Reviews (1)

an introduction to an interesting subject
--The first part of the book is devoted to quite hard chapters on transitive, arc-transitive graph, homomorphism, etc.

--The second part is about Matrix theory, interlacing, strongly regular graph, two graph, generalized line graph, etc it is the main part of the book.

--The third part is about cut, flows, Knots, etc.

This book can serve as a nice introduction to the subject of Graph theory.

Nevertheless:

--This book lacks some more example, for this see "distance regular graph".

--It is sketchy on chromatic polynomial, planar graph.

--The original book by Norman Biggs is shorter, smarter, nicer ... Read more

Book Description

This book introduces graph theory, a subject with a wide range of applications in real-work situations. This book is designed to be easily accessible to the novice, assuming no more than a good grasp of algebra to understand and relate to the concepts presented. Using many examples, illustrations, and figures, it provides an excellent foundation for the basic knowledge of graphs and their applications.This book includes an introductory chapter that reviews the tools necessary to understand the concepts of graphs, and then goes on to cover such topics as trees and bipartite graphs, distance and connectivity, Eulerian and Hamiltonian graphs, graph coloring, matrices, algorithms, planar graphs, and digraphs and networks.Graph theory has a wide range of applications; this book is useful for those in the fields of anthropology, computer science, chemistry, environmental conservation, fluid dynamics, psychology, sociology, traffic management, telecommunications, and business managers and strategists. ... Read more

Reviews (1)

It's a very good intro text.
Good choice of topics. Easy to read. The concepts are well motivated. The book is full of applications. An undergraduate will be able to follow the proofs. They are clear. I also liked the homework exercises. This is a student-friendly book, as the title suggests. ... Read more

Book Description

Contains a wealth of information previously scattered in research journals, conference proceedings and technical reports. Identifies more than 200 unsolved problems. Every problem is stated in a self-contained, extremely accessible format, followed by comments on its history, related results and literature. The book will stimulate research and help avoid efforts on solving already settled problems. Each chapter concludes with a comprehensive list of references which will lead readers to original sources, important contributions and other surveys. ... Read more

Reviews (1)

A concise catalog of theorems and conjectures.
This is a highly technical book that gathers together in one medium-sized volume (less than 300 pages) hundreds of new and classical theorems and conjectures on every conceivable type of graph coloring problem. It is primarily a mathematical treatment in Theorem-Proof style, although the proofs are left to the references (otherwise the book would have been enormous). The book consists of many short sections, each one often less than a page, formally stating a theorem or a conjecture, then briefly summarizing what is known about it, and where the results have been published. As the authors state in the preface, "We did not intend to write a textbook to be read from beginning to end, but rather a catalog suitable for browsing". It seems they have achieved what was intended; it's quite interesting to simply browse through this book.

To give you an idea of the level of the discussion in the text, here is an excerpt from page 1: After a terse definition of vertex coloring and "chromatic number", the authors state that "The existence of the chromatic number follows from the Well-Ordering Theorem of set theory... However, even if it is not assumed that every set has a well-ordering, but maintaining the property that every set has a cardinality, then the statement 'Any finite or infinite graph has a chromatic number' is equivalent to the Axiom of Choice...". If you are unfamiliar with concepts such as well-ordering or the axiom of choice, such a discussion will be of little value to you. However, if you are familiar with these ideas, you will appreciate how quickly the authors jump into meaty discussions. As another example, the chapter on planar graphs begins with a number of excellent questions: "Does there exist a short proof of the four-color theorem...?", "Is there a short argument to demonstrate that the four-color problem is a finite problem?", and "Is there a short argument that proves the existence of a polynomial algorithm to decide if a given planar graph if 4-colorable?", to mention three. The chapter then proceeds to discuss what is currently know about these problems, and many others.

I suspect that the book would be of great interest not only to mathematicians but also computer scientists, as there are numerous discussions/problems on computational complexity. For example, "Does there exist a function g and a polynomial algorithm that for any given input graph G will find a number s, such that the chromatic number of G satisfies s <= X(G) <= g(s)?" (Here X(G) is the chromatic number of graph G.) The authors state that the question was answered affirmatively by Alon in 1993 if X(G) is replaced by "list-chromatic number". This is typical of the problems cataloged in this book: a terse but formally correct statement of a problem followed by what is currently know, with full citations.

The bibliography at the end of each section is extensive, if not daunting, so there should be little problem looking up all relevant literature concerning a given problem. The authors have also set up an on-line archive for up-to-the-minute research results on these problems. This is an excellent reference for those who are interested in serious research in graph coloring. I would not recommend it to undergraduates in computer science or mathematics, nor to those seeking accessible discussions of classic graph algorithms; this is not an introductory text. ... Read more

Book Description

A comprehensive introduction to the four standard products of graphs and related topics

Addressing the growing usefulness of current methods for recognizing product graphs, this new work presents a much-needed, systematic treatment of the Cartesian, strong, direct, and lexicographic products of graphs as well as graphs isometrically embedded into them. Written by two leading experts in this rapidly evolving area of combinatorics, Product Graphs: Structure and Recognition compiles and consolidates a wealth of information previously scattered throughout the literature, providing researchers in the field with ready access to numerous recent results as well as several new recognition algorithms and proofs. The authors explain all topics from the ground up and make the requisite theory and data structures easily accessible for mathematicians and computer scientists alike.

Coverage includes
* The basic algebraic and combinatorial properties ofproduct graph
* Hypercubes, median graphs, Hamming graphs, triangle-free graphs, and vertex-transitive graphs
* Colorings, automorphisms, homorphisms, domination, and the capacity of products of graphs Sample applications, including novel applications to chemical graph theory Clear connections to other areas of graph theory Figures, exercises, and hundreds of references ... Read more

Reviews (2)

A review of Product Graphs
This book gives an ecellent overview of known results on graph products and related topics. Well worth the $$ and an excellent read! It is also a very usefull tool for anyone that's interested in this area of discrete mathematics.

A review of Product Graphs
The book gives an excellent overview of known result on graph products and related topics. It is a very useful tool for anyone that's interested in graph products. ... Read more

Book Description

Over the past decade, many major advances have been made in the field of graph coloring via the probabilistic method. This monograph provides an accessible and unified treatment of these results, using the Lovasz Local Lemma and Concentration Inequalities developed by Talagrand. This gentle introduction to the probabilistic method will be useful to researchers and graduate students in graph theory, discrete mathematics, theoretical computer science and probabilists. ... Read more

Reviews (1)

Good for every graph theorist
This is definitely for someone who knows graph theory very well
and wants to learn the powerful tool of "probabilistic method".

As far as I know, this is the second book on the probabilistic method (the first one is by Alon and Spencer). It starts with some simple basic notions and gradually takes you to the heart of some deep (and complicated) results in graph theory. Although the technique can be used in different areas of combinatorics and theoretical computer science, almost all examples and problems in the book are related to graph theory (and specially graph coloring). One of the good points about this book is that they usually provide good intuitions for the proofs before going into their details.

If you consider yourself a combinatorist or a theoretical computer scientist and you don't know much about this tool this
book is a good source. ... Read more

Book Description

Mathematica is today’s most advanced technical computing system. It features a rich programming environment, two-and three-dimensional graphics capabilities and hundreds of sophisticated, powerful programming and mathematical functions using state-of-the-art algorithms. Combined with a user-friendly interface, and a complete mathematical typesetting system, Mathematica offers an intuitive, easy-to-hande environment of great power and utility.

The Mathematica Guidebook for Graphics provides a comprehensive step-by-step development of how to use Mathematica to visualize functions and data, manipulate graphics, and optimize their appearance. Two-dimensional graphics, contour plots, plots of surfaces, free-form three-dimensional surfaces, and animations are the core topics. Hundreds of detailed examples and programs show a large variety of visualization techniques, algorithms, methods, and tricks. These tools allow the reader to create virtually any possible graphic, from simple curves to scientific visualizations and artistic images and logos. Mathematica graphics functions are discussed in detail, explained in numerous examples, and put to work in programs that are all contained on the accompanying DVD.

Unique Features:

* Step-by-step introductions to all of Mathematica graphics capabilities

* Comprehensive presentation of two-and three-dimensional graphics primitives and directives, as well as plotting capabilities for functions and data

* Hundreds of unique and innovative scientific visualizations and artistic images

* Website for book with additional materials and updates:

http://www.MathematicaGuideBooks.org

* Accompanying DVD contains all material as an electronic book with complete, executable Mathematica versions 4 and 5 compatible code and programs, rendered color graphics, and animations

Michael Trott is a symbolic computation and computer graphics expert. He holds a Ph.D. in theoretical physics and joined the R&D team at Wolfram Research in 1994, the creators of Mathematica. Since 1998, he has been leading development of the Wolfram Functions Site http://functions.wolfram.com, which currently features more that 80,000 formulas and identities, and thousands of visualizations.

Book Description

Focuses on classical problems in graph theory, including the 5-flow and 4-flow conjectures, the edge-3-coloring conjecture, the 3-flow conjecture, and the cycle double cover conjecture. Highlights the interrelationships between graph coloring, integer flow, cycle covers, and graph minors. Concentrates on graph theoretical methods and results. ... Read more

Book Description

Petri nets are a popular and powerful model for analyzing and modeling concurrent systems, and a rich theory has developed around them. This book focuses on a particular class of Petri nets, free choice Petri nets, which plays a central role in the theory. The text is organized very clearly, with every notion carefully explained and every result proved. The authors give clear exposition of place invariants, siphons, traps and many other important analysis techniques. The book contains classical results of free-choice theory as well as more recent results.The material is organized along the lines of a course, and each chapter contains numerous exercises, making this text ideal for graduate students and research workers alike. ... Read more

Book Description

Children learn to graph as they follow the growth of an orphaned Siberian tiger cub.

A Siberian tiger cub born at the Denver Zoo is orphaned when he is just a few weeks old. At first T. J. refuses to eat his new food, and it requires the full attention of the zoo staff to ensure that he grows into a huge, beautiful, and very healthy tiger.

Through photographs, narrative, and graphs, young readers follow T.J. as he grows from a tiny newborn into a five-hundred-pound adult. A heartwarming story about one tiger’s fight for survival that also introduces a basic math skill.
... Read more

Reviews (3)

Great book for teaching graphing
I am a student teacher in a 3rd grade classroom, and I have been using this to teach graphing. The students love it! I have taught three lessons over several days on pictographs, line graphs, and bar graphs. I have introduced each lesson with a few pages from this book. I highly recommend it! The photographs are wonderful, and it is a great example of using math in everyday life.

What a great book!
If you have a little one, work with children, or just love tigers, this is a must!!!! So many math concepts are taught using nonfiction so children can understand the connection easier. The reader follows the tiger from infancy through adulthood using all sorts of graphic organizers to understand things like number of species alive to different types of species, etc. The photos enable the learner to understand the information in an easy manner. And the photos themselves are rich and beautiful.

I learned quite a bit myself about tigers and so will you!

Makes math fun and interesting
In a format that will surely please elementary school teachers struggling with curriculum frameworks, Tiger Math offers both story and instruction. The tragic tale of a motherless baby Siberian tiger named T.J. will capture reader's sympathy and interest as they read on to discover if will survive. The math part of the book is a natural extension of information. T.J.'s growth and progress is a concern, and is compared to his father's at the same age. Not only are concepts of graphing introduced, but interpreting data in is equally stressed.
Nagda creates a neat transition from picture graphs to bar graphs, and also explains circle graphs and line graphs. Students will learn when to use which type of graph and how to read graphs. Those not interested in the math can simply read the narrative on the right, and ignore the graphs to the left.
The best part of the book is the photos, showing T.J. in a variety of moods. The captions decorating the endpapers are very appropriate and often humorous. The cover photo alone will make the book leap off the shelves, and the tiger striping is very appealing.
An additional page or two on endangered tigers and the habitat and lifestyle of these magnificent creatures would have extended a third use to the book: school reports on animals. The rarity of Siberian tigers is stressed, and a brief note from an environmental point of view could also have been included. ... Read more

Book Description

Graphic modelling is a form of multivariate analysis that uses graphs to represent models. These graphs display the structure of dependencies, both associational and causal, between the variables in the model. This textbook provides an introduction to graphical modelling with emphasis on applications and practicalities rather than on a formal development. It is bsed on the popular software package for graphical modelling, MIM, a freeware version of which can be downloaded from the Internet. Following an introductory chapter which sets the scene and describes some of the basic ideas of graphical modelling, subsequent chapters describe particular families of models, including log-linear models, Gaussian models, and models for mixed discrete and continuous variables. Further chapters cover hypothesis testing and model selection. Chapters 7 and 8 are new to the second edition. Chapter 7 describes the use of directed graphs, chain graphs, and other graphs. Chapter 8 summarizes some recent work on causal inference, relevant when graphical models are given a causal interpretation. This book will provide a useful introduction to this topic for students and researchers. ... Read more

Reviews (1)

this is about directed graphs not graphics
Because graphic methods are very popular in statistics, when you read the title you might think this is a book on the use of graphics in statistics. That is not what the book is about. The directed graph on the cover might be a hint for some. The book deals with the theory of undirected and directed graphs which has applications to causal modeling in statistics and the development of expert systems (which Edwards claim are now more commonly referred to as probabilistic networks).

This subject is being made popular again based on the recent work of Edwards, Pearl and a few others. The book incorporate the approach in many classical statistical problems. This is not commonly seen except in specialized texts on latent variable models.

Edwards discusses implementation of the methods with the freeware MIMS that is available in Denmark and on the web. The book is very well written and applications in MIMS are given throughout the text. Edwards also provides us with an excellent list of references (over 200 with many on causal modeling).

The software LISREL produced by researchers in the US at UCLA for latent variable and path analyses is only briefly mentioned on page 217. The lack of coverage of American and British publications on this topic is the only drawback I see. ... Read more