# The Tutte polynomial

@article{Crapo1969TheTP, title={The Tutte polynomial}, author={Henry Crapo}, journal={aequationes mathematicae}, year={1969}, volume={3}, pages={211-229} }

$q$-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…

## 224 Citations

The Tutte q-Polynomial

- Mathematics
- 2017

$q$-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

- MathematicsJournal 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…

The Computational Complexity of Tutte Invariants for Planar Graphs

- MathematicsSIAM J. Comput.
- 2005

A corollary gives the computational complexity of various enumeration problems for planar graphs.

Matroids and quotients of spheres

- Mathematics
- 2002

Abstract. For any linear quotient of a sphere, $X=S^{n-1}/\Gamma,$ where $\Gamma$ is an elementary abelian p–group, there is a corresponding ${\mathbb F}_p$ representable matroid $M_X$ which only…

The Tutte expansion revisited

- Mathematics
- 2016

The Tutte polynomial of a connected graph was originally defined by Tutte as a sum over all spanning trees of monomials depending on a fixed linear order on the set of edges. Tuttle proved that while…

Closures of Linear Spaces

- Mathematics
- 2013

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.

- Mathematics
- 2019

Unlike Whitney's definition of the corank-nullity generating function $T(G;x+1,y+1)$, Tutte's definition of his now eponymous polynomial $T(G;x,y)$ requires a total order on the edges of which the…

The Tutte decomposition

- Mathematics
- 1986

“Simple ideas are often the most powerful.” This adage is best exemplified in matroid theory by the method of Tutte (-Grothendieck) decomposition. This method has its origin in the following…

A New Graph Polynomial and Generalized Tutte-Grothendieck Invariant from Quantum Circuits

- MathematicsACSS
- 2020

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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