The Tutte polynomial

@article{Crapo1969TheTP,
  title={The Tutte polynomial},
  author={Henry Crapo},
  journal={aequationes mathematicae},
  year={1969},
  volume={3},
  pages={211-229}
}
  • H. Crapo
  • Published 1 October 1969
  • Mathematics
  • aequationes mathematicae
$q$-Matroids are defined on complemented modular support lattices. Minors of length 2 are of four types as in a "classical" matroid. Tutte polynomials $\tau(x,y)$ of matroids are calculated either by recursion over deletion/contraction of single elements, by an enumeration of bases with respect to internal/external activities, or by substitution $x \to (x-1),\; y \to (y-1)$ in their rank generating functions $\rho(x,y)$. The $q$-analogue of the passage from a Tutte polynomial to its… 
The Tutte q-Polynomial
$q$-Matroids are defined on complemented modular support lattices. Minors of length 2 are of four types as in a "classical" matroid. Tutte polynomials $\tau(x,y)$ of matroids are calculated either by
Interpretations of the Tutte and characteristic polynomials of matroids
  • M. Kochol
  • Mathematics
    Journal of Algebraic Combinatorics
  • 2019
We study interpretations of the Tutte and characteristic polynomials of matroids. If M is a matroid with rank function r whose ground set E is given with a linear ordering <, then $$X\subseteq E$$ X
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A corollary gives the computational complexity of various enumeration problems for planar graphs.
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Given a linear space $L$ in affine space $\mathbb{A}^n$, we study its closure $\widetilde{L}$ in the product of projective lines $(\mathbb{P}^1)^n$. We show that the degree, multigraded Betti
Tutte Polynomial Activities.
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A New Graph Polynomial and Generalized Tutte-Grothendieck Invariant from Quantum Circuits
TLDR
The main theorems represent Q_G as a quasi-specialization of the rank-generating polynomial \(S_G(x,y) of Oxley and Whittle, J Comb Theory Ser B 59:210–244, 1993, 1993) and show that \(Q_G\) is likewise a generalized Tutte–Grothendieck invariant.
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References

SHOWING 1-10 OF 17 REFERENCES
A contribution to the theory of chromatic polynomials
Summary Two polynomials θ(G, n) and ϕ(G, n) connected with the colourings of a graph G or of associated maps are discussed. A result believed to be new is proved for the lesser-known polynomial ϕ(G,
Single-element extensions of matroids
Extensions of matroids to se ts containing one additional ele ment are c ha rac te ri zed in te rm s or mod ular cuts of the latti ce or closed s ubse ts. An equivalent charac teri za ti o n is give
On The Foundations of Combinatorial Theory: Combinatorial Geometries
It has been clear within the last ten years that combinatorial geometry, together with its order-theoretic counterpart, the geometric lattice, can serve to catalyze the whole field of combinatorial
Möbius Inversion in Lattices
In the development of computational techniques for combinatorial theory, attention has lately centered on ROTA’S theory of Mobius inversion [6]. The main theorem of ROTA’S paper, concerning the
A Ring in Graph Theory
We call a point set in a complex K a O-cell if it contains just one point of K, and a I-cell if it is an open arc. A set L of O-cells and I-cells of K is called a linear graph on K
On the Abstract Properties of Linear Dependence
Let C1 , C2 ,· · · ,Cm be the columns of a matrix M. Any subset of these columns is either linearly independent or linearly dependent; the subsets thus fall into two classes. These classes are not
Lectures on matroids
Lattice Differentials
  • and the Theory of Combinatorial Independence, Research Reports, Northeastern University
  • 1964
On the Theory of Combinatorial Independence
  • Doctoral Thesis,
  • 1964
...
1
2
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