A convolution formula for the Tutte polynomial

Hopf algebras and Tutte polynomials

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 literature arise from this framework. These polynomials include the classical Tutte polynomial of graphs and matroids, Las Vergnas' Tutte polynomial of the morphism of matroids and his Tutte polynomial for embedded graphs, Bollobas and Riordan's ribbon graph polynomial, the Krushkal polynomial, and the Penrose polynomial. We show that our Tutte polynomials of…

The Tutte Polynomial of a Morphism of Matroids 6. A Multi-Faceted Counting Formula for Hyperplane Regions and Acyclic Orientations

We show that the 4-variable generating function of certain orientation related parameters of an ordered oriented matroid is the evaluation at (x + u, y+v) of its Tutte polynomial. This evaluation contains as special cases the counting of regions in hyperplane arrangements and of acyclic orientations in graphs. Several new 2-variable expansions of the Tutte polynomial of an oriented matroid follow as corollaries. This result hold more generally for oriented matroid perspectives, with specific special cases the counting of bounded regions in hyperplane arrangements or of bipolar acyclic orientat…

f-有界子图的广义Tutte--Berge公式的另一证明

Another Proof of the Generalized Tutte--Berge Formula for $f$-Bounded Subgraphs

本文利用Tutte的f-因子定理为f-有界子图的最大规模min-max关系提供了一个新证明，并说明该结果在f(v)=1时退化为经典的Tutte--Berge匹配公式。

Tutte 划分的不变量与一个 $q$-模拟

Invariants of Tutte Partitions and a $q$-Analogue

本文提出 Tutte 划分构造拟阵与 $q$-拟阵的 Tutte 多项式，证明其包含 Crapo 定义中的划分，并引入 $q$-Tutte-Grothendiek 不变性公理，建立秩多项式与 Tutte 多项式间的卷积关系。

Relative Tutte polynomials of tensor products of colored graphs

The tensor product $(G_1,G_2)$ of a graph $G_1$ and a pointed graph $G_2$ (containing one distinguished edge) is obtained by identifying each edge of $G_1$ with the distinguished edge of a separate copy of $G_2$, and then removing the identified edges. A formula to compute the Tutte polynomial of a tensor product of graphs was originally given by Brylawski. This formula was recently generalized to colored graphs and the generalized Tutte polynomial introduced by Bollobás and Riordan. In this paper we generalize the colored tensor product formula to relative Tutte polynomials of relative graphs…

The Tutte Polynomial of Complex Reflection Groups

This article computes the Tutte polynomial of the hyperplane arrangements associated to the complex reflection groups. The calculations are based on both formulas of De Concini and Procesi for Tutte polynomial and the normaliser of parabolic subgroups in complex reflection groups determined by Krishnasamy and Taylor.

Tutte polynomial of a small-world farey graph

In this paper, we find recursive formulas for the Tutte polynomial of a family of small-world networks: Farey graphs, which are modular and have an exponential degree hierarchy. Then, making use of these formulas, we determine the number of spanning trees, as well as the number of connected spanning subgraphs. Furthermore, we also derive exact expressions for the chromatic polynomial and the reliability polynomial of these graphs.

Ehrhart polynomial and multiplicity Tutte polynomial

We prove that the Ehrhart polynomial of a zonotope is a specialization of the multiplicity Tutte polynomial. We derive some formulae for the volume and the number of integer points of the zonotope.

The $\barγ$-frame for Tutte polynomials of matroids

Specializing the $γ$-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 Tutte polynomials of matroids having rank $r$ and size $n$. Every Tutte polynomial can be expanded as a linear combination with non-negative integer coefficients of elements in this frame. We give explicit formulas for the elements in this frame. These formulas combine to give an expansion of the Tutte polynomial with coefficients obtained by summing numerical invari…

Tutte多项式决定了Ehrhart多项式的线性系数

The Tutte polynomial determines the linear coefficient of the Ehrhart polynomial

本研究证明了Tutte多项式与Ehrhart多项式线性系数的关系公式，刻画了对应极值拟阵并证明了Ferroni的猜想。

多拟阵 Tutte 多项式的递归定义

A recursive definition for the polymatroid Tutte polynomial

本文证明多拟阵 Tutte 多项式的递归公式本身可独立作为其定义，补全了该不变量在拟阵推广下的等价定义体系。

Bollobas-Riordan and relative Tutte polynomials

We establish a relation between the Bollobas-Riordan polynomial of a ribbon graph with the relative Tutte polynomial of a plane graph obtained from the ribbon graph using its projection to the plane in a nontrivial way. Also we give a duality formula for the relative Tutte polynomial of dual plane graphs and an expression of the Kauffman bracket of a virtual link as a specialization of the relative Tutte polynomial.

Tutte多项式的相容集展开

On the compatible sets expansion of the Tutte polynomial

本文回答了Kochol关于拟阵Tutte多项式相容集展开与内部-外部活动公式关系的疑问，将其作为对Las Vergnas拟阵透视三元Tutte多项式展开公式推广的特例，并给出了双射的直接证明。

The Tutte polynomial of some matroids

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 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, engineering, 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 …

Computing the Tutte polynomial of a hyperplane arrangement

We define and study the Tutte polynomial of a hyperplane arrangement. We introduce a method for computing it by solving an enumerative problem in a finite field. For specific arrangements, the computation of Tutte polynomials is then reduced to certain related enumerative questions. As a consequence, we obtain new formulas for the generating functions enumerating alternating trees, labelled trees, semiorders and Dyck paths.

On Tutte polynomial expansion formulas in perspectives of matroids and oriented matroids

We introduce the active partition of the ground set of an oriented matroid perspective (or quotient, or strong map) on a linearly ordered ground set. The reorientations obtained by arbitrarily reorienting parts of the active partition share the same active partition. This yields an equivalence relation for the set of reorientations of an oriented matroid perspective, whose classes are enumerated by coefficients of the Tutte polynomial, and a remarkable partition of the set of reorientations into boolean lattices, from which we get a short direct proof of a 4-variable expansion formula for the …

Arithmetic matroids, Tutte polynomial, and toric arrangements

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 to this setting. Guided by the geometry of generalized toric arrangements, we provide a combinatorial interpretation of the associated arithmetic Tutte polynomial, which can be seen as a generalization of Crapo's formula for the classical Tutte polynomial.

链 Tutte 多项式

Chain Tutte polynomials

本文提出链 Tutte 多项式族，借助通用 Tutte 特征标卷积与赋值不变量框架，填补 Tutte 多项式与 Derksen 的 $\mathcal{G}$-不变量之间的层次空白。

Expectations of Tutte-related functions of random ranked sets with multiplicities

Employing two models, we show that various counting functions of a random variable defined by restriction or contraction of a ranked set with multiplicity (e.g., classical and arithmetic matroids) have expectations given by the corresponding multivariate Tutte polynomial. The first model is based on a generalization of a convolution formula of Kung (2010), extending from matroids to ranked sets with multiplicities. This model enables us to compute the expectations of many familiar polynomials, such as the chromatic, flow and Ehrhart polynomials, generalizing the classical results of Welsh (199…

Recipe theorem for the Tutte polynomial for matroids, renormalization group-like approach

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 of the Hopf algebra of isomorphic classes of matroids, characters which are then related to the Tutte polynomial for matroids. This Hopf algebraic approach also allows to prove, in a new way, a matroid Tutte polynomial convolution formula appearing in W. Kook {\it et. al., J. Comb. Series} {\bf B 76} (1999).

多拟阵Tutte多项式的删除-收缩公式和单调性性质

A deletion-contraction formula and monotonicity properties for the polymatroid Tutte polynomial

本文给出了多拟阵Tutte多项式的删除-收缩公式，并证明了内多项式和外多项式关于包含和子式的系数单调性，但该性质不适用于完整多项式。

Types of embedded graphs and their Tutte polynomials

We take an elementary and systematic approach to the problem of extending the Tutte polynomial to the setting of embedded graphs. Four notions of embedded graphs arise naturally when considering deletion and contraction operations on graphs on surfaces. We give a description of each class in terms of coloured ribbon graphs. We then identify a universal deletion-contraction invariant (i.e., a `Tutte polynomial') for each class. We relate these to graph polynomials in the literature, including the Bollobás--Riordan, Krushkal, and Las Vergnas polynomials, and give state-sum formulations, duality …

半拟阵的Tutte多项式的线性关系

Linear relations for Tutte polynomials of semimatroids

建立半拟阵Tutte多项式系数的线性关系族，引入秩中心集推广半拟阵，并证明相同恒等式成立，应用包括仿射超平面排列和偏斜图的平衡Tutte多项式。

Tutte Polynomial of Ideal Arrangement

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 acyclic orientations. In 2007, Ardila extended the notion of Tutte polynomial to the hyperplane arrangements, and computed the Tutte polynomials of the classical root systems for a certain prime power of the first variable. In this article, we compute Tutte polynomials of ideal arrangements. Those arrangements were introduced in 2006 by Sommers and Tymoczko, and …

Fourientation activities and the Tutte polynomial

A fourientation of a graph $G$ is a choice for each edge of the graph whether to orient that edge in either direction, leave it unoriented, or biorient it. We may naturally view fourientations as a mixture of subgraphs and graph orientations where unoriented and bioriented edges play the role of absent and present subgraph edges, respectively. Building on work of Backman and Hopkins (2015), we show that given a linear order and a reference orientation of the edge set, one can define activities for fourientations of $G$ which allow for a new 12 variable expansion of the Tutte polynomial $T_G$. …

Some variants of the exponential formula, with application to the multivariate Tutte polynomial (alias Potts model)

We prove some variants of the exponential formula and apply them to the multivariate Tutte polynomials (also known as Potts-model partition functions) of graphs. We also prove some further identities for the multivariate Tutte polynomial, which generalize an identity for counting connected graphs found by Riordan, Nijenhuis, Wilf and Kreweras and in more general form by Leroux and Gessel, and an identity for the inversion enumerator of trees found by Mallows, Riordan and Kreweras. Finally, we prove a generalization of Mobius inversion on the partition lattice.

Lattice Points in Orthotopes and a Huge Polynomial Tutte Invariant of Weighted Gain Graphs

A gain graph is a graph whose edges are orientably labelled from a group. A weighted gain graph is a gain graph with vertex weights from an abelian semigroup, where the gain group is lattice ordered and acts on the weight semigroup. For weighted gain graphs we establish basic properties and we present general dichromatic and forest-expansion polynomials that are Tutte invariants (they satisfy Tutte's deletion-contraction and multiplicative identities). Our dichromatic polynomial includes the classical graph one by Tutte, Zaslavsky's two for gain graphs, Noble and Welsh's for graphs with positi…

一个超图Tutte多项式

A Hypergraph Tutte Polynomial

研究引入超图的Tutte多项式$T_{\mathrm{HG}}$及相关的$T_k$，证明$T_{\mathrm{HG}}$有Tutte型性质，$T_k$有通用性定理等，还关联其与其他模型，比较它和$\mathcal{T}_P$，表明两者区分能力不可比。

Bollobás-Riordan和Krushkal多项式的张量积公式

Tensor product formulas for the Bollobás-Riordan and Krushkal polynomials

本文定义了嵌入伪曲面图的张量积，推广并统一了已知的多项式张量积公式，为Bollobás-Riordan和Krushkal多项式提供了Brylawski风格的公式。

The Active Bijection 2.a - Decomposition of activities for matroid bases, and Tutte polynomial of a matroid in terms of beta invariants of minors

We introduce and study filtrations of a matroid on a linearly ordered ground set, which are particular sequences of nested sets. A given basis can be decomposed into a uniquely defined sequence of bases of minors, such that these bases have an internal/external activity equal to 1/0 or 0/1 (in the sense of Tutte polynomial activities). This decomposition, which we call the active filtration/partition of the basis, refines the known partition of the ground set into internal and external elements with respect to a given basis. It can be built by a certain closure operator, which we call the acti…

Evaluations of topological Tutte polynomials

We find new properties of the topological transition polynomial of embedded graphs, $Q(G)$. We use these properties to explain the striking similarities between certain evaluations of Bollobás and Riordan's ribbon graph polynomial, $R(G)$, and the topological Penrose polynomial, $P(G)$. The general framework provided by $Q(G)$ also leads to several other combinatorial interpretations these polynomials. In particular, we express $P(G)$, $R(G)$, and the Tutte polynomial, $T(G)$, as sums of chromatic polynomials of graphs derived from $G$; show that these polynomials count $k$-valuations of media…

Renormalization group-like proof of the universality of the Tutte polynomial for matroids

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 are solutions of some differential equations which are of the same type as the differential equations used to describe the renormalization group flow in quantum field theory. This approach allows us to also prove, in a different way, a matroid Tutte polynomial convolution formula published by Kook, Reiner and Stanton (1999). This FPSAC contribution is an ex…

K-theoretic Tutte polynomials of morphisms of matroids

We generalize the Tutte polynomial of a matroid to a morphism of matroids via the K-theory of flag varieties. We introduce two different generalizations, and demonstrate that each has its own merits, where the trade-off is between the ease of combinatorics and geometry. One generalization recovers the Las Vergnas Tutte polynomial of a morphism of matroids, which admits a corank-nullity formula and a deletion-contraction recursion. The other generalization does not, but better reflects the geometry of flag varieties.

多拟阵的张量积与转移多项式的Brylawski型公式

Tensor products of multimatroids and a Brylawski-type formula for the transition polynomial

本文针对多拟阵给出转移多项式的张量积公式，证明Brylawski公式及其拓扑类似形式为其特例，还得到多拟阵二元和与星积的转移多项式公式。

作为双代数态射的 Fortuin–Kasteleyn 多项式及其在 Tutte 多项式中的应用

The Fortuin-Kasteleyn polynomial as a bialgebra morphism and applications to the Tutte polynomial

本文证明 Fortuin–Kasteleyn 多项式是图双代数上的态射，并借助全无环部分定向、覆盖结构及所有定向映射，给出其反极元公式与 Tutte 多项式的组合解释。

Tutte Polynomials of Tensor Products of Signed Graphs and their Applications in Knot Theory

It is well-known that the Jones polynomial of an alternating knot is closely related to the Tutte polynomial of a special graph obtained from a regular projection of the knot. Relying on the results of Bollobás and Riordan, we introduce a generalization of Kauffman's Tutte polynomial of signed graphs for which describing the effect of taking a signed tensor product of signed graphs is very simple. We show that this Tutte polynomial of a signed tensor product of signed graphs may be expressed in terms of the Tutte polynomials of the original signed graphs by using a simple substitution rule. Ou…

Tutte polynomials and random-cluster models in Bernoulli cell complexes

This paper studies Bernoulli cell complexes from the perspective of persistent homology, Tutte polynomials, and random-cluster models. Following the previous work [9], we first show the asymptotic order of the expected lifetime sum of the persistent homology for the Bernoulli cell complex process on the $\ell$-cubical lattice. Then, an explicit formula of the expected lifetime sum using the Tutte polynomial is derived. Furthermore, we study a higher dimensional generalization of the random-cluster model derived from the Edwards-Sokal type coupling, and show some basic results such as the posit…

Tutte relations, TQFT, and planarity of cubic graphs

It has been known since the work of Tutte that the value of the chromatic polynomial of planar triangulations at $(3+\sqrt{5})/2$ has a number of remarkable properties. We investigate to what extent Tutte's relations characterize planar graphs. A version of the Tutte linear relation for the flow polynomial at $(3-\sqrt{5})/2$ is shown to give a planarity criterion for $3$-connected cubic graphs. A conjecture is formulated that the golden identity for the flow polynomial characterizes planarity of cubic graphs as well. In addition, Tutte's upper bound on the chromatic polynomial of planar trian…

The Complexity of Computing the Sign of the Tutte Polynomial

We study the complexity of computing the sign of the Tutte polynomial of a graph. As there are only three possible outcomes (positive, negative, and zero), this seems at first sight more like a decision problem than a counting problem. Surprisingly, however, there are large regions of the parameter space for which computing the sign of the Tutte polynomial is actually #P-hard. As a trivial consequence, approximating the polynomial is also #P-hard in this case. Thus, approximately evaluating the Tutte polynomial in these regions is as hard as exactly counting the satisfying assignments to a CNF…

Exponential Time Complexity of the Permanent and the Tutte Polynomial

We show conditional lower bounds for well-studied #P-hard problems: (a) The number of satisfying assignments of a 2-CNF formula with n variables cannot be counted in time exp(o(n)), and the same is true for computing the number of all independent sets in an n-vertex graph. (b) The permanent of an n x n matrix with entries 0 and 1 cannot be computed in time exp(o(n)). (c) The Tutte polynomial of an n-vertex multigraph cannot be computed in time exp(o(n)) at most evaluation points (x,y) in the case of multigraphs, and it cannot be computed in time exp(o(n/polylog n)) in the case of simple graphs…

关于简单有根图的Greedoid Tutte多项式

On the Greedoid Tutte Polynomial for simple rooted graphs

本文针对Gordon和McMahon提出的关于简单有根图greedoid Tutte多项式x-adic赋值等于根邻接叶点数的猜想，给出了完整证明，推进了图论与greedoid理论中多项式不变量的研究。

单变量Tutte多项式固定一个变量的系数

Coefficients of univariate Tutte polynomials with one variable fixed

本文引入两个对偶参数，给出了拟阵单变量Tutte多项式高次项系数的公式，并证明了当t≤1时这些系数的单峰性。

Matroid variant of Matiyasevich formula and its application

In 1977, Yu. V. Matiyasevich proposed a formula expressing the chromatic polynomial of an arbitrary graph as a linear combination of flow polynomials of subgraphs of the original graph. In this paper, we prove that this representation is a particular case of one (easily verifiable) formula, namely, the representation of the characteristic polynomial of an arbitrary matroid as a linear combination of characteristic polynomials of dual matroids. As an application, we represent the flow polynomial of a complete graph with $n$ vertices as the sum of elementary terms with respect to all partitions …

A determinant formula for the Jones polynomial of pretzel knots

This paper presents an algorithm to construct a weighted adjacency matrix of a plane bipartite graph obtained from a pretzel knot diagram. The determinant of this matrix after evaluation is shown to be the Jones polynomial of the pretzel knot by way of perfect matchings (or dimers) of this graph. The weights are Tutte's activity letters that arise because the Jones polynomial is a specialization of the signed version of the Tutte polynomial. The relationship is formalized between the familiar spanning tree setting for the Tait graph and the perfect matchings of the plane bipartite graph above.…

Tutte chromatic identities from the Temperley-Lieb algebra

This paper introduces a conceptual framework, in the context of quantum topology and the algebras underlying it, for analyzing relations obeyed by the chromatic polynomial χ(Q) of planar graphs. Using it we give new proofs and substantially extend a number of classical results concerning the combinatorics of the chromatic polynomial. In particular, we show that Tutte's golden identity is a consequence of level-rank duality for SO(N) topological quantum field theories and Birman-Murakami-Wenzl algebras. This identity is a remarkable feature of the chromatic polynomial relating χ({ϕ+2}) for any …

有理链的Jones多项式新公式

Generalized formulas for the Jones polynomial of a rational link

本文推导了有理链的标准图表示下Jones多项式和Kauffman括号多项式的新公式，扩展了Qazaqzeh等人关于Tait图的 Tutte多项式结果，并引入有限自动机计算交叉符号以确定有理链标准图的扭度。

A most general edge elimination graph polynomial

We look for graph polynomials which satisfy recurrence relations on three kinds of edge elimination: edge deletion, edge contraction and deletion of edges together with their end points. Like in the case of deletion and contraction only (W. Tutte, 1954), it turns out that there is a most general polynomial satisfying such recurrence relations, which we call $ξ(G,x,y,z)$. We show that the new polynomial simultaneously generalizes the Tutte polynomial, the matching polynomial, and the recent generalization of the chromatic polynomial proposed by K.Dohmen, A.Pönitz and P.Tittman (2003), including…

A polynomial invariant and duality for triangulations

The Tutte polynomial is a classical invariant, important in combinatorics and statistical mechanics. An essential feature of the Tutte polynomial is the duality for planar graphs G, $T_G(X,Y)\; =\; {T}_{G^*}(Y,X)$ where $G^*$ denotes the dual graph. We examine this property from the perspective of manifold topology, formulating polynomial invariants for higher-dimensional simplicial complexes. Polynomial duality for triangulations of a sphere follows as a consequence of Alexander duality. The main goal of this paper is to introduce and begin the study of a more general 4-variable polynomial fo…

Macdonald cumulants, $G$-inversion polynomials and $G$-parking functions

We prove a combinatorial formula for Macdonald cumulants which generalizes the celebrated formula of Haglund for Macdonald polynomials. We provide several applications of our formula. Firstly, it gives a new, constructive proof of a strong factorization property of Macdonald polynomials proven recently by the author of this paper. Moreover it proves that Macdonald cumulants are $q,t$--positive in the monomial and in the fundamental quasisymmetric bases. Furthermore, we use our formula to prove the recent higher-order Macdonald positivity conjecture for the coefficients of the Schur polynomials…

On the number of Hamiltonian cycles and polynomial invariants of graphs

Firstly, for a general graph, we find a recursion formula on the number of Hamiltonian cycles and one on cycles. By this result, we give some new polynomial invariants. Secondly, we give a condition to tell whether a polynomial defined by recursion on edges is a invariant. Then, we give an generaliztion of the Tutte polynomial. Finally, We have a try on distinguishing different graphs by using these polynomials.

Extensions of Barrier Sets to Nonzero Roots of the Matching Polynomials

In matching theory, barrier sets (also known as Tutte sets) have been studied extensively due to its connection to maximum matchings in a graph. In this paper, we first define $θ$-barrier sets. Our definition of a $θ$-barrier set is slightly different from that of a barrier set. However we show that $θ$-barrier sets and barrier sets have similar properties. In particular, we prove a generalized Berge's Formula and give a characterization for the set of all $θ$-special vertices in a graph.

半拟阵的特征多项式及其与拟阵、超平面构形和图着色的联系

Characteristic polynomials of semimatroids and their connections to matroids, hyperplane arrangements and graph colorings

该研究探究半拟阵特征多项式的性质，推广惠特尼断链定理与罗塔-赫伦-韦尔奇猜想，给出相关卷积恒等式，并引入指派拟阵建立半拟阵、超平面构形与图着色的联系。

A proof of Catalan's Convolution formula

We give a new proof of the $k$-fold convolution of the Catalan numbers. This is done by enumerating a certain class of polygonal dissections called $k$-in-$n$ dissections. Furthermore, we give a formula for the average number of cycles in a triangulation.

A simple combinatorial proof of Shapiro's Catalan convolution

Shapiro proved an elegant convolution formula involving Catalan numbers of even index. This paper gives a simple combinatorial proof of his formula. In addition, we show that it is equivalent with the alternating convolution formula of central binomial coefficients.