Michele D'Adderio, Luca Moci
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.
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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。
Given an oriented graph G, the modular flow polynomial counts the number of nowhere-zero Z_k-flows of G. We give a description of the modular flow polynomial in terms of (open) Ehrhart polynomials of lattice polytopes. Using Ehrhart-Macdonald reciprocity we give a combinatorial interpretation for the values of the modular flow polynomial at negative arguments which answers a question of Beck and Zaslavsky (2006). Our construction extends to Z_l-tensions and we recover Stanley's reciprocity theorem for the chromatic polynomial. Combining the combinatorial reciprocity statements for flows and te…
Extreme coefficients of multiplicity Tutte polynomials
本文推导了重数Tutte多项式的一般系数和六个极端系数的表达式，并作为特例得到了经典Tutte多项式的相应系数表达式。
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 alternating sign with a combinatorial interpretation closely tied to the Dawson partition. Our definition extends in a straightforward way to polymatroids, and in this setting our polynomial has Kálmán's internal and external activity polynomials as its univariate specialisations.
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…
A Proof of a conjecture of Watanabe--Yoshida via Ehrhart Theory
本文通过埃赫拉特理论提供了Watanabe-Yoshida猜想的另一种证明，利用新的显式组合公式和生成函数的Hadamard积。
We introduce a modular (integral) complementary polynomial $κ(G;x,y)$ ($κ_{\mathbbm z}(G;x,y)$) of two variables of a graph $G$ by counting the number of modular (integral) complementary tension-flows (CTF) of $G$ with an orientation $ε$. We study these polynomials by further introducing a cut-Eulerian equivalence relation on orientations and geometric structures such as the complementary open lattice polyhedron $Δ_\textsc{ctf}(G,ε)$, the complementary open 0-1 polytope $Δ^+_\textsc{ctf}(G,ε)$, and the complementary open lattice polytopes $Δ^ρ_\textsc{ctf}(G,ε)$ with respect to orientations $ρ…
Hilbert-Kunz multiplicity of quadrics via Ehrhart theory
本文通过Ehrhart多项式研究二次曲面的Hilbert-Kunz乘数，证明其为特征p的有理函数，并揭示其随维度变化的性质。
Let G be a connected bipartite graph with color classes E and V and root polytope Q. Regarding the hypergraph (V,E) induced by G, we prove that its interior polynomial is equivalent to the Ehrhart polynomial of Q, which in turn is equivalent to the h-vector of any triangulation of Q. It follows that the interior polynomials of (V,E) and its transpose (E,V) agree. When G is a complete bipartite graph, our result recovers a well known hypergeometric identity due to Saalschütz. It also implies that certain extremal coefficients in the Homfly polynomial of a special alternating link can be read of…
In this note we provide a higher-dimensional analogue of Tutte's celebrated theorem on colorings and flows of graphs, by showing that the theory of arithmetic Tutte polynomials and quasi-polynomials encompasses invariants defined for CW complexes by Beck-Breuer-Godkin-Martin and Duval-Klivans-Martin. Furthermore, we answer a question by Bajo-Burdick-Chmutov, concerning the modified Tutte-Krushkal-Renhardy polynomials defined by these authors: to this end, we prove that the product of two arithmetic multiplicity functions on a matroid is again an arithmetic multiplicity function.
Graded Ehrhart Theory of Unimodular Zonotopes
本文通过研究单模棱柱体的分级埃赫拉特理论，证明其与图塔多项式的关系，并揭示其谐波代数的代数性质，回答了Reiner和Rhoades的两个猜想。
Let $M$ be a regular matroid. The Jacobian group ${\rm Jac}(M)$ of $M$ is a finite abelian group whose cardinality is equal to the number of bases of $M$. This group generalizes the definition of the Jacobian group (also known as the critical group or sandpile group) ${\rm Jac}(G)$ of a graph $G$ (in which case bases of the corresponding regular matroid are spanning trees of $G$). There are many explicit combinatorial bijections in the literature between the Jacobian group of a graph ${\rm Jac}(G)$ and spanning trees. However, most of the known bijections use vertices of $G$ in some essential …
We introduce a multiplicity Tutte polynomial M(x,y), with applications to zonotopes and toric arrangements. We prove that M(x,y) satisfies a deletion-restriction recurrence and has positive coefficients. The characteristic polynomial and the Poincare' polynomial of a toric arrangement are shown to be specializations of the associated polynomial M(x,y), likewise the corresponding polynomials for a hyperplane arrangement are specializations of the ordinary Tutte polynomial. Furthermore, M(1,y) is the Hilbert series of the related discrete Dahmen-Micchelli space, while M(x,1) computes the volume …
This paper deals with the location of the complex zeros of the Tutte polynomial for a class of self-dual graphs. For this class of graphs, as the form of the eigenvalues is known, the regions of the complex plane can be focused on the sets where there is only one dominant eigenvalue in particular containing the positive half plane. Thus, in these regions, the analyticity of the pressure can be derived easily. Next, some examples of graphs with their Tutte polynomial having a few number of eigenvalues are given. The cases of the strip of triangles with a double edge, the wheel and the cycle wit…
Recently, Bauer et al. (J Graph Theory 55(4) (2007), 343--358) introduced a graph operator $D(G)$, called the $D$-graph of $G$, which has been useful in investigating the structural aspects of maximal Tutte sets in $G$ with a perfect matching. Among other results, they proved a characterization of maximal Tutte sets in terms of maximal independent sets in the graph $D(G)$ and maximal extreme sets in $G$. This was later extended to graphs without perfect matchings by Busch et al. (Discrete Appl. Math. 155 (2007), 2487--2495). Let $θ$ be a real number and $μ(G,x)$ be the matching polynomial of a…
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