Handbook of the Tutte Polynomial and Related Topics

Joanna A. Ellis-Monaghan is a professor of discrete mathematics at the Korteweg - de Vries Instituut voor Wiskunde at the Universiteit van Amsterdam. Her research focuses on algebraic combinatorics, especially graph polynomials, as well as applications of combinatorics to DNA self-assembly, statistical mechanics, computer chip design, and bioinformatics. She also has an interest in mathematical pedagogy. She has published over 50 papers in these areas.Iain Moffatt is a professor of mathematics in Royal Holloway, University of London. His main research interests lie in the interactions between topology and combinatorics. He is especially interested in graph polynomials, topological graph theory, matroid theory, and knot theory. He has written more than 40 papers in these areas and is also the author of the book An Introduction to Quantum and Vassiliev Knot invariants. Ellis-Monaghan and Moffatt have authored several papers on the Tutte polynomial and related graph polynomials together as well as the book Graphs on surfaces: Dualities, Polynomials, and Knots.

• 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.

"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