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The Tutte Polynomial Part I: General Theory

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Part of the C.I.M.E. Summer Schools book series (CIME,volume 83)

Abstract

Matroid theory (sometimes viewed as the theory of combinatorial geometries or geometric lattices) is reasonably young as a mathematical theory (its traditional birthday is given as 1935 with the appearance of [159]) but has steadily developed over the years and shown accelerated growth recently due, in large part, to two applications. The first is in the field of algorithms. To coin an oversimplification: “when a good algorithm is known, a matroid structure is probably hidden away somewhere.” In any event, many of the standard good algorithms (such as the greedy algorithm) and many important ones whose complexities are currently being scrutinized (e.g., existence of a Hamiltonian path) can be thought of as matroid algorithms. In the accompanying lecture notes of Professor Welsh the connections between matroids and algorithms are presented.

Another important application of matroids is the theory of the Tutte polynomial

$$ {\text{t}}\left( {{\text{M;}}\,{\text{x, y}}} \right) = \sum {{\text{a}}_{{\text{ij}}} \left( {{\text{x - 1}}} \right)^{\text{i}} \left( {{\text{y - 1}}} \right)^{\text{j}} } $$

Keywords

  • Boolean Algebra
  • Combinatorial Geometry
  • Tutte Polynomial
  • Chromatic Polynomial
  • Geometric Lattice

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Brylawski, T. (2010). The Tutte Polynomial Part I: General Theory. In: Barlotti, A. (eds) Matroid Theory and its Applications. C.I.M.E. Summer Schools, vol 83. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11110-5_3

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