相关论文: A catalogue of small regular matroids and their Tu…
The Tutte polynomial is a fundamental invariant of graphs and matroids. In this article, we define a generalization of the Tutte polynomial to oriented graphs and regular oriented matroids. To any regular oriented matroid $N$, we associate…
The Tutte polynomial of a graph or a matroid, named after W. T. Tutte, has the important universal property that essentially any multiplicative graph or network invariant with a deletion and contraction reduction must be an evaluation of…
$q$-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…
We provide a full classification of all families of matroids that are closed under duality and minors, and for which the Tutte polynomial is a universal valuative invariant. There are four inclusion-wise maximal families, two of which are…
Many important enumerative invariants of a matroid can be obtained from its Tutte polynomial, and many more are determined by two stronger invariants, the $\mathcal{G}$-invariant and the configuration of the matroid. We show that the same…
We give a general multiplication-convolution identity for the multivariate and bivariate rank generating polynomial of a matroid. The bivariate rank generating polynomial is transformable to and from the Tutte polynomial by simple algebraic…
The Tutte polynomial is a well-studied invariant of graphs and matroids. We first extend the Tutte polynomial from graphs to hypergraphs, and more generally from matroids to polymatroids, as a two-variable polynomial. Our definition is…
The configuration of a matroid $M$ is the abstract lattice of cyclic flats (flats that are unions of circuits) where we record the size and rank of each cyclic flat, but not the set. One can compute the Tutte polynomial of $M$, and stronger…
We introduce the minor-closed, dual-closed class of multi-path matroids. We give a polynomial-time algorithm for computing the Tutte polynomial of a multi-path matroid, we describe their basis activities, and we prove some basic structural…
In the paper [Proceedings of the Japan Academy, Ser. A Mathematical Sciences, 95(10) 111-113], the authors introduce the concept of the Tutte polynomials of genus $g$ and announce that each matroid $M$ can be reconstructed from its Tutte…
We introduce a binary matroid M(IAS(G)) associated with a looped simple graph G. M(IAS(G)) classifies G up to local equivalence, and determines the delta-matroid and isotropic system associated with G. Moreover, a parametrized form of its…
We show how the Tutte polynomial of a matroid $M$ can be computed from its condensed configuration, which is a statistic of its lattice of cyclic flats. The results imply that the Tutte polynomial of $M$ is already determined by the…
We study algebraic properties of the Tutte polynomial of a matroid and its generalizations to other combinatorially defined bivariate polynomial invariants. Merino, de Mier and Noy showed that the Tutte polynomial of a connected matroid is…
The multivariate arithmetic Tutte polynomial of arithmetic matroids is a generalization of the multivariate Tutte polynomial of matroids. In this note, we give the convolution formulas for the multivariate arithmetic Tutte polynomial of the…
The multivariate Tutte polynomial $\hat Z_M$ of a matroid $M$ is a generalization of the standard two-variable version, obtained by assigning a separate variable $v_e$ to each element $e$ of the ground set $E$. It encodes the full structure…
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 these monomials do depend on the linear…
The Tutte polynomial for matroids is not directly applicable to polymatroids. For instance, deletion-contraction properties do not hold. We construct a polynomial for polymatroids which behaves similarly to the Tutte polynomial of a…
We recover the Tutte polynomial of a matroid, up to change of coordinates, from an Ehrhart-style polynomial counting lattice points in the Minkowski sum of its base polytope and scalings of simplices. Our polynomial has coefficients of…
The catenary data of a matroid $M$ of rank $r$ on $n$ elements is the vector $(\nu(M;a_0,a_1,\ldots,a_r))$, indexed by compositions $(a_0,a_1,\ldots,a_r)$, where $a_0 \geq 0$,\, $a_i > 0$ for $i \geq 1$, and $a_0+ a_1 + \cdots + a_r = n$,…
We introduce and investigate multivariate Tutte polynomials, dichromatic polynomials, subset-corank polynomials, size-corank polynomials, and rank generating polynomials of semimatroids, which generalize the corresponding polynomial…