Handbook of the Tutte Polynomial and Related Topics

ISBN-13: 9781482240627 / Angielski / Twarda / 2022 / 784 str.

Joanna A. Ellis-Monaghan; Iain Moffatt

Opis

It has become a well-known fact that most graph polynomials are related to the Tutte Polynomial in some way. In fact, that area of graph polynomials has grown to such an extent that it now has its own subject classification (05C31). This handbook is the first one published on the Tutte Polynomial which is a central, heavily-studied object in the field of combinatorics with applications in a wide range of other fields such as geometry, biology and physics.

Recenzje

"This is a comprehensive reference text on the Tutte polynomial, including its applications and extensions. The book consists of 34 relatively short chapters written by different contributing authors. The individual contributors present the most important theorems in their respective fields and illustrate them with examples. Each chapter ends with a list of open problems. Two brief introductory chapters by the editors-Ellis-Monaghan (Univ. of Amsterdam) and Moffatt (Royal Holloway, University of London)-cover the basic definitions and computational results for Tutte polynomials. The next two-thirds of the book are devoted to applications and extensions, that is, uses and occurrences of Tutte polynomials outside graph theory or matroid theory. Hyperplane arrangements, quantum field theory, network reliability, the sandpile model, and chipfiring games are a few examples of the topics treated. The book concludes with a chapter on the history of the subject written by Graham Farr. It follows from the nature of the volume (i.e., no proofs, no exercises, very broad topical coverage, and more than 50 authors) that classroom use of the book is unlikely. Nonetheless, this work is likely to become the most frequently consulted reference on Tutte polynomials."

Summing Up: Highly recommended. Graduate students and faculty.

-Choice Review

Spis treści

I. Fundamentals. 1. Graph theory. 2. The Tutte Polynomial for Graphs. 3. Essential Properties of the Tutte Polynomial. 4. Matroid theory. 5. Tutte polynomial activities. 6. Tutte Uniqueness and Tutte Equivalence.
II. Computation. 7. Computational Techniques. 8. Computational resources. 9. The Exact Complexity of the Tutte Polynomial. 10. Approximating the Tutte Polynomial. III. Specializations. 11. Foundations of the Chromatic Polynomial. 12. Flows and Colorings. 13. Skein Polynomials and the Tutte Polynomial when x = y. 14. The Interlace Polynomial and the Tutte–Martin Polynomial. IV. Applications. 15. Network Reliability. 16. Codes. 17. The Chip-Firing Game and the Sandpile Model. 18. The Tutte Polynomial and Knot Theory. 19. Quantum Field Theory Connections. 20. The Potts and Random-Cluster Models. 21. Where Tutte and Holant meet: a view from Counting Complexity. 22. Polynomials and Graph Homomorphisms. V. Extensions. 23. Digraph Analogues of the Tutte Polynomial. 24. Multivariable, Parameterized, and Colored Extensions of the Tutte Polynomial. 25. Zeros of the Tutte Polynomial. 26. The U, V and W Polynomials. 27. Valuative invariants on matroid basis polytopes Topological Extensions of the Tutte Polynomial. 28. The Tutte polynomial of Matroid Perspectives. 29. Hyperplane Arrangements and the Finite Field Method. 30. Some Algebraic Structures related to the Tutte Polynomial. 31. The Tutte Polynomial of Oriented Matroids. 32. Valuative Invariants on Matroid Basis Polytopes. 33. Non-matroidal Generalizations. VI. History. 34. The History of Tutte–Whitney Polynomials.