Harmonic Tutte polynomials of matroids (original) (raw)

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.

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 quasisymmetric function for matroids

European Journal of Combinatorics, 2009

A new isomorphism invariant of matroids is introduced, in the form of a quasisymmetric function. This invariant • defines a Hopf morphism from the Hopf algebra of matroids to the quasisymmetric functions, which is surjective if one uses rational coefficients, • is a multivariate generating function for integer weight vectors that give minimum total weight to a unique base of the matroid, • is equivalent, via the Hopf antipode, to a generating function for integer weight vectors which keeps track of how many bases minimize the total weight, • behaves simply under matroid duality, • has a simple expansion in terms of P -partition enumerators, and • is a valuation on decompositions of matroid base polytopes. This last property leads to an interesting application: it can sometimes be used to prove that a matroid base polytope has no decompositions into smaller matroid base polytopes. Existence of such decompositions is a subtle issue arising in work of Lafforgue, where lack of such a decomposition implies the matroid has only a finite number of realizations up to scalings of vectors and overall change-of-basis.

Weighted Tutte-Grothendieck polynomials of graphs

arXiv (Cornell University), 2022

In this paper, we introduce the notion of weighted chromatic polynomials of a graph associated to a weight function f of a certain degree, and discuss some of its properties. As a generalization of this concept, we define the weighted Tutte-Grothendieck polynomials of graphs. When f is harmonic, we notice that there is a correspondence between the weighted Tutte-Grothendieck polynomials of graphs and the weighted Tutte polynomials of matroids. Moreover, we present some constructions of the weighted Tutte-Grothendieck invariants for graphs as well as the weighted Tutte invariants for matroids. Finally, we give a remark on the categorification of the weighted chromatic polynomials of graphs and weighted Tutte polynomials of matroids.

Cycle index, weight enumerator, and Tutte polynomial

2007

With every linear code is associated a permutation group whose cycle index is the weight enumerator of the code (up to a trivial normalisation). There is a class of permutation groups (the IBIS groups) which includes the groups obtained from codes as above. With every IBIS group is associated a matroid; in the case of a group from a code, the matroid differs only trivially from that which arises directly from the code. In this case, the Tutte polynomial of the code specialises to the weight enumerator (by Greene's Theorem), and hence also to the cycle index. However, in another subclass of IBIS groups, the base-transitive groups, the Tutte polynomial can be derived from the cycle index but not vice versa.

The Arithmetic Tutte polynomial of two matrices associated to Trees

Special Matrices, 2018

Arithmetic matroids arising from a list A of integral vectors in Zn are of recent interest and the arithmetic Tutte polynomial MA(x, y) of A is a fundamental invariant with deep connections to several areas. In this work, we consider two lists of vectors coming from the rows of matrices associated to a tree T. Let T = (V, E) be a tree with |V| = n and let LTbe the q-analogue of its Laplacian L in the variable q. Assign q = r for r ∈ ℤ with r/= 0, ±1 and treat the n rows of LTafter this assignment as a list containing elements of ℤn. We give a formula for the arithmetic Tutte polynomial MLT(x, y) of this list and show that it depends only on n, r and is independent of the structure of T. An analogous result holds for another polynomial matrix associated to T: EDT, the n × n exponential distance matrix of T. More generally, we give formulae for the multivariate arithmetic Tutte polynomials associated to the list of row vectors of these two matriceswhich shows that even the multivariat...