The Tutte polynomial of matroids constructed by a family of splitting operations (original) (raw)

A new formula for an evaluation of the Tutte polynomial of a matroid

Discrete Mathematics, 2005

Given a matroid M and its Tutte polynomial T M (x, y), T M (0, 1) is an invariant of M with various interesting combinatorial and topological interpretations. Being a Tutte-Grothendieck invariant, T M (0, 1) may be computed via deletion-contraction recursions. In this note we derive a new recursion formula for this invariant that involves contractions of M through the circuits containing a fixed element of M.

The Tutte polynomial of some matroids

2014

The Tutte polynomial of a graph or a matroid, named after W. T. Tutte, has the important universal property that essentially any mul-tiplicative graph or network invariant with a deletion and contraction reduction must be an evaluation of it. The deletion and contraction operations are natural reductions for many network models arising from a wide range of problems at the heart of computer science, engi-neering, optimization, physics, and biology. Even though the invariant is #P-hard to compute in general, there are many occasions when we face the task of computing the Tutte polynomial for some families of graphs or matroids. In this work we compile known formulas for the Tutte polynomial of some families of graphs and matroids. Also, we give brief explanations of the techniques that were use to find the for-mulas. Hopefully, this will be useful for researchers in Combinatorics and elsewhere.

Tutte polynomials for oriented matroids

arXiv (Cornell University), 2022

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 a polynomial invariant AN (q, y, z), which we call the A-polynomial. The A-polynomial has the following interesting properties among many others: • a specialization of AN gives the Tutte polynomial of the underlying unoriented matroid N , • when the oriented matroid N corresponds to an unoriented matroid (that is, when the elements of the ground set come in pairs with opposite orientations), the invariant AN is equivalent to the Tutte polynomial of this unoriented matroid (up to a change of variables), • the invariant AN detects, among other things, whether N is acyclic and whether N is totally cyclic. We explore various properties and specializations of the A-polynomial. We show that some of the known properties of the Tutte polynomial of matroids can be extended to the A-polynomial of regular oriented matroids. For instance, we show that a specialization of AN counts all the acyclic orientations obtained by reorienting some elements of N , according to the number of reoriented elements. Let us mention that in a previous article we had defined an invariant of oriented graphs that we called the B-polynomial, which is also a generalization of the Tutte polynomial. However, the B-polynomial of an oriented graph N is not equivalent to its A-polynomial, and the B-polynomial cannot be extended to an invariant of regular oriented matroids.

A note on some inequalities for the Tutte polynomial of a matroid

2009

We prove that if a matroid M contains two disjoint bases (or, dually, if its ground set is the union of two bases), then T M (a, a) ≤ max{T M (2a, 0), T M (0, 2a)} for a ≥ 2. This resembles the conjecture that appears in C. Merino and D.J.A. Welsh, Forests, colourings and acyclic orientations of the square lattice, Annals of Combinatorics 3 (1999) pp. 417-429: If G is a 2-connected graph with no loops, then T G (1, 1) ≤ max{T G (2, 0), T (0, 2)}. We conjecture that T M (1, 1) ≤ max{T M (2, 0), T M (0, 2)} for matroids which contains two disjoint bases or its ground set is the union of two bases. We also prove the latter for some families of graphs and matroids.

On dual of the split-off matroids

2019

Azadi et al. [Generalization of Splitting off Operation to Binary Matroids, Electronic Notes in Discrete Math, 15 (2003), 186–188] have generalized the splitting off (or in short split-off) operation on graphs to binary matroids. The dual of a split-off matroid is not always equal to the split-off of dual of the original matroid. In this paper, for a given matroid M and two elements x and y from E(M), we first characterize the cobases of the split-off matroid Mxy in terms of the cobases of the matroid M . Then, by using the set of cobases of Mxy and the set of bases (Azadi characterized this set) of (M)xy, we characterize those binary matroids for which (Mxy) ∗ = (M)xy. Indeed, for a binary matroid M on a set E with x, y ∈ E, we prove that (Mxy) = (M)xy if and only if M = N ⊕N ′ where N is an arbitrary binary matroid and N ′ is U0,2 or U2,2 such that x, y ∈ E(N ′). Key–Words: Binary matroid, Uniform matroid, Direct sum, Split-off operation

Harmonic Tutte polynomials of matroids II

Cornell University - arXiv, 2022

In this work, we introduce the harmonic generalization of the m-tuple weight enumerators of codes over finite Frobenius rings. A harmonic version of the MacWilliams-type identity for m-tuple weight enumerators of codes over finite Frobenius ring is also given. Moreover, we define the demi-matroid analogue of well-known polynomials from matroid theory, namely Tutte polynomials and coboundary polynomials, and associate them with a harmonic function. We also prove the Greene-type identity relating these polynomials to the harmonic m-tuple weight enumerators of codes over finite Frobenius rings. As an application of this Greene-type identity, we provide a simple combinatorial proof of the MacWilliams-type identity for harmonic m-tuple weight enumerators over finite Frobenius rings. Finally, we provide the structure of the relative invariant spaces containing the harmonic mtuple weight enumerators of self-dual codes over finite fields.

Harmonic Tutte polynomials of matroids

Designs, Codes and Cryptography

In the present paper, we introduce the concept of harmonic Tutte polynomials and discuss some of its properties. In particular, we generalize Greene's theorem to the case between harmonic Tutte polynomials and harmonic weight enumerators.