Tutte polynomial
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| Tutte polynomial | |
|---|---|
| Name | Tutte polynomial |
| Field | Combinatorics |
| Introduced | 1950s–1960s |
| Inventor | William Tutte |
Tutte polynomial The Tutte polynomial is a two-variable graph and matroid invariant introduced by William Tutte that encodes numerous combinatorial quantities and bridges disparate areas such as Graph theory, Matroid theory, Statistical mechanics, and Knot theory. It generalizes earlier invariants connected to problems studied by figures like Hassler Whitney, Philip Hall, and Claude Shannon and appears in results related to Brendan McKay, László Lovász, and Paul Seymour. The polynomial's evaluations recover classical results associated with the Four Color Theorem, the Potts model, and the Jones polynomial.
Definition and basic properties
For a finite graph G or a matroid M, the Tutte polynomial T_G(x,y) (or T_M(x,y)) is defined by a deletion–contraction recurrence attributed to William Tutte and earlier ideas of Hassler Whitney; it satisfies T = 1 on a null object, T(G) = T(G \ e) + T(G / e) for ordinary elements that are neither loops nor bridges, and multiplicativity on disjoint unions and direct sums discussed by W. T. Tutte and formalized by James Oxley. Important universal properties connect the Tutte polynomial to invariants studied by Frank Harary, Richard Stanley, and Neil Robertson. Symmetries include duality relations for planar graphs paralleling results from Kenneth Appel and Wolfgang Haken on planar map colorings, and normalization conditions tied to work by Paul Erdős and György Katona.
Equivalent formulations and specializations
Many classical polynomials arise as specializations of the Tutte polynomial: the chromatic polynomial of Eric Temple Bell and George David Birkhoff emerges along one axis, while the flow polynomial related to Claude Shannon and W. T. Tutte appears along another. The spanning-tree enumerator linking to Cayley and Arthur Cayley corresponds to evaluation at (1,1), and the reliability polynomial studied by Claude Shannon and John van Neumann maps to a univariate specialization. Connections to the Martin polynomial of James Martin, the Bollobás–Riordan polynomial associated with Béla Bollobás and Oliver Riordan, and the multivariate Tutte (or Potts) model linked to Fortuin–Kasteleyn highlight equivalences leveraged by M. E. Newman and C. D. Broadbent in percolation theory.
Computation and complexity
Computing the Tutte polynomial is #P-hard at almost all points in the plane, a complexity landscape charted by Leslie Valiant and refined by Jaeger and Matthieu Welsh. Polynomial-time algorithms exist for special graph families studied by Herman Hemmeter and Seymour such as series-parallel graphs, cographs investigated by D. G. Corneil, and graphs of bounded treewidth analyzed by Neil Robertson and Paul Seymour. Exact evaluation methods exploit contraction-deletion recurrences used by William Tutte and dynamic programming approaches tied to Richard Karp and Michael Garey, whereas approximation schemes connect to Markov chain methods developed by Mark Jerrum, Alistair Sinclair, and Eric Vigoda.
Applications in graph theory and matroid theory
The Tutte polynomial encapsulates enumerative invariants central to Graph theory problems addressed by Frank Harary and Paul Erdős, counting spanning trees (linking to Arthur Cayley), forests associated with James W. Moon, colorings in the tradition of George David Birkhoff and Percy Heawood, and nowhere-zero flows connected to William Tutte and Peter Tutte. In Matroid theory, the polynomial unifies rank and nullity enumerations developed by Hassler Whitney and later axiomatized by James Oxley, impacting structural theorems from Paul Seymour and decomposition results examined by Neil Robertson. Network reliability models by Claude Shannon and John von Neumann are interpreted via Tutte evaluations used in studies by E. R. Berlekamp and R. J. Wilson.
Connections to statistical physics and knot invariants
Statistical mechanics frameworks like the Potts model and the random-cluster model, pioneered by Fortuin and Kasteleyn, are encoded by the multivariate Tutte polynomial; work connecting these models to combinatorial invariants involves John Cardy, Michael Fisher, and Leo Kadanoff. In knot theory, the Jones polynomial of links discovered by Vaughan Jones and its relatives relate through specializations and skein-theoretic correspondences explored by Edward Witten and Louis Kauffman, with planar duality paralleling constructions in topological quantum field theory studied at institutions like Institute for Advanced Study and Princeton University.
Examples and notable evaluations
Classical evaluations include: spanning-tree counts at (1,1) traced to Arthur Cayley; the chromatic polynomial at (1−λ,0) recovering results by George David Birkhoff and Heawood; flow polynomial values connected to Claude Shannon and W. T. Tutte; and reliability formulations applied in analyses by John von Neumann. Specific graphs with closed-form Tutte polynomials feature cycles and complete graphs investigated by William Tutte and Frank Harary, series-parallel graphs examined by Paul Seymour, and planar lattices studied by H. N. V. Temperley and E. H. Lieb. Deep evaluations at special points give combinatorial interpretations tied to research by Leslie Valiant, Mark Jerrum, and M. E. J. Newman.