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Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
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Searching... | 30000010064303 | QA166 H364 2004 | Reference Book | Handbook | Searching... |
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Summary
Summary
The Handbook of Graph Theory is the most comprehensive single-source guide to graph theory ever published. Best-selling authors Jonathan Gross and Jay Yellen assembled an outstanding team of experts to contribute overviews of more than 50 of the most significant topics in graph theory-including those related to algorithmic and optimization approaches as well as "pure" graph theory. They then carefully edited the compilation to produce a unified, authoritative work ideal for ready reference.
Designed and edited with non-experts in mind, the Handbook of Graph Theory makes information easy to find and easy to understand. The treatment of each topic includes lists of essential definitions and facts accompanied by examples, tables, remarks, and in some areas, conjectures and open problems. Each section contains a glossary of terms relevant to that topic and an extensive bibliography of references that collectively form an extensive guide to the primary research literature.
The applications of graph theory are fast becoming ubiquitous. Whether your primary area of interest lies in mathematics, computer science, engineering, or operations research, this handbook holds the key to unlocking graph theory's intricacies, applications, and potential.
Reviews 1
Choice Review
To a graph theorist, "graph" means a set (of "vertices") together with a family of some of its two-element subsets ("edges" connecting certain pairs of vertices), or perhaps some small variation thereof. Although considering such unconstrained structures allows the modeling of many phenomena, such generality must work against a unified theory. And so, graph theory breaks into many subdisciplines as additional restrictive hypotheses are posed (e.g., trees, distance-regular graphs); introduces additional structure (e.g., interval graph, graph embedded on a surface); or focuses on special problems, usually of an algorithmic nature (e.g., vertex colorings, edge colorings, Hamiltonian circuits). The recent fashion for compendious, bulky, mathematical "handbooks" serves especially well fields of a focused nature, like homotopy theory, which have produced such detail that even a mere overview now exceeds the means and capacity of the traditional one-author monograph. The case for gathering between two covers all topics falling under the loose rubric of graph theory seems less compelling. Indeed, one sees, e.g., some inevitable overlap between this volume and CRC's Algorithms and Theory of Computation Handbook, edited by Mikhail J. Atallah (CH, Oct'99). Still, the Handbook of Graph Theory makes for pleasant browsing. For that brief interval when it remains up-to-date, it is a fine guide to various literatures, especially for topics like Ramsey theory, where the best results, perpetually subject to small numerical improvements, take work to locate. This good book would make an even better Web site, growing and changing with its subject. Many first-rate mathematicians have contributed, making the exposition's quality high overall. That the tolerance graphs Golumbic (Univ. of Haifa) and Trenk (Wellesley College) study rate just a dozen pages in the Handbook indicates nicely the compression of the Handbook accounts. Tolerance graphs constitute a recent generalization of interval graphs; in an interval graph, each vertex is associated with an interval in the real line; then two vertices are connected, if their corresponding intervals intersect. Interval graphs have a natural application to scheduling resources; they can capture the inability of people to locate themselves in more than one place at a time. In a tolerance graph, each vertex has an associated interval and a numerical tolerance; one may connect two vertices if the size of their overlaps exceeds either tolerance. This models, perhaps, the willingness to miss a little bit of the end of one meeting to get to the next. The concept of tolerance graph raises a number of issues. Absent any available abstract characterization of tolerance graphs, one seeks necessary conditions and sufficient conditions. With a tolerance graph, one may choose intervals and tolerances in some especially nice way. Finally, having a tolerance structure helps with standard algorithmic graph theory questions, such as coloring. Golumbic and Trenk give clear treatment to all these questions and some generalizations beyond. A new subject, their book has no competitors at present. An association scheme, in one view, consists of a certain kind of coloring of some complete graph. One stipulates that the number of ways to complete a given edge of a certain color to a triangle with prescribed colors should depend only on the colors involved, not on the choice of the particular edge. In another formulation, an association scheme consists of certain sets of matrices, and these give rise to so-called Bose-Mesner algebras. Thus, one may study association schemes with (linear) algebraic techniques, and for their applications to experiment design. The mathematical theory of experiment design has a combinatorial component (that decides what data to gather) and an analytical component (that generates statistics once the experiment is done). Obviously, a design that supports the analysis must be chosen, but the distinct intellectual flavors of these two endeavors means that most monographs heavily emphasize one or the other rather than the interaction. Bailey (Univ. of London) makes the connections and justifies this book's place in the library. In the computer age, the mathematically inclined can always find lucrative work in the mathematical sciences. Thus, the population of mathematical amateurs has declined, and anyone who has followed Scientific American knows that this has changed the face of "recreational" mathematics. Where recreational mathematics, usually of a combinatorial or numerical nature, once attracted amateurs while the professional mathematicians stayed busy doing analysis, recreational mathematics now happens mostly on the fringes of the professional mathematical world. Where professional graph theorists will study Hamiltonian circuits in general, hunting for knight tours on chessboards remains sufficiently special to seem just barely frivolous enough to count as recreational mathematics. Watkins (Univ. of Colorado) offers an excellent invitation to serious mathematics, rather than a distraction. ^BSumming Up: All four books: Highly recommended. Lower-division undergraduates through faculty. D. V. Feldman University of New Hampshire
Table of Contents
| Introduction to Graphs |
| Graph Representation |
| Directed Graphs |
| Connectivity and Traversability |
| Colorings and Related Topics |
| Algebraic Graph Theory |
| Topological Graph Theory |
| Analytic Graph Theory |
| Graphical Measurement |
| Graphs in Computer Science |
| Networks and Flows |
