Related papers: A generalization of the Tutte polynomials
Specializing the $\gamma$-basis for the vector space $\mathcal{G}(n,r)$ spanned by the set of symbols on bit sequences with $r$ $1$'s and $n-r$ $0$'s, we obtain a frame or spanning set for the vector space $\mathcal{T}(n,r)$ spanned by…
We introduce an arithmetic version of the multivariate Tutte polynomial, and (for representable arithmetic matroids) a quasi-polynomial that interpolates between the two. A generalized Fortuin-Kasteleyn representation with applications to…
By considering Tutte polynomials of Hopf algebras, we show how a Tutte polynomial can be canonically associated with combinatorial objects that have some notions of deletion and contraction. We show that several graph polynomials from the…
This is a chapter destined for the book "Handbook of the Tutte Polynomial". The chapter is a composite. The first part is a brief introduction to Orlik-Solomon algebras. The second part sketches the theory of evaluative functions on matroid…
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 is originally a bivariate polynomial enumerating the colorings of a graph and of its dual graph. But it reveals more of the internal structure of the graph like its number of forests, of spanning subgraphs, and of…
The Tutte polynomial of a graph, or equivalently the $q$-state Potts model partition function, is a two-variable polynomial graph invariant of considerable importance in both combinatorics and statistical physics. The computation of this…
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…
Morphisms of matroids are combinatorial abstractions of linear maps and graph homomorphisms. We introduce the notion of basis for morphisms of matroids, and show that its generating function is strongly log-concave. As a consequence, we…
We introduce the notion of an arithmetic matroid, whose main example is given by a list of elements of a finitely generated abelian group. In particular we study the representability of its dual, providing an extension of the Gale duality…
This paper surveys a comprehensive, although not exhaustive, sampling of graph polynomials with the goal of providing a brief overview of a variety of techniques defining a graph polynomial and then for decoding the combinatorial…
Brylawski proved the coefficients of the Tutte polynomial of a matroid satisfy a set of linear relations. We extend these relations to a generalization of the Tutte polynomial that includes greedoids and antimatroids. This leads to families…
We prove that the Tutte polynomial of a coloopless paving matroid is convex along the portions of the line segments x+y=p lying in the positive quadrant. Every coloopless paving matroids is in the class of matroids which contain two…
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…
Using a quantum field theory renormalization group-like differential equation, we give a new proof of the recipe theorem for the Tutte polynomial for matroids. The solution of such an equation is in fact given by some appropriate characters…
In this note we introduce a family of polynomials on a matroid derived from chain Tutte polynomials which generalize the classic and ubiquitous characteristic polynomial. We show that the coefficients of these polynomials alternate and…
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…
A catalogue of all non-isomorphic simple connected regular matroids ${\cal M}$ of cardinality $n \leq 15$ is provided on the net. These matroids are given as binary matrix matroids and are sieved from the large pool of all non-isomorphic…
In this work, we study matroids over a domain and several classical combinatorial and algebraic invariants related. We define their Grothendieck-Tutte polynomial $T_{\mathcal{M}}(x,y)$, extending the definition given by Fink and Moci in…
In this paper we give a new proof of the universality of the Tutte polynomial for matroids. This proof uses appropriate characters of Hopf algebra of matroids, algebra introduced by Schmitt (1994). We show that these Hopf algebra characters…