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 $ρ$. The polynomial $κ(G;x,y)$ ($κ_{\mathbbm z}(G;x,y)$) is a common generalization of the modular (integral) tension polynomial $τ(G,x)$ ($τ_\mathbbm{z}(G,x)$) and the modular (integral) flow polynomial $ϕ(G,y)$ ($ϕ_\mathbbm{z}(G,y)$), and can be decomposed into a sum of product Ehrhart polynomials of complementary open 0-1 polytopes $Δ^+_\textsc{ctf}(G,ρ)$. There are dual complementary polynomials $\barκ(G;x,y)$ and $\barκ_{\mathbbm z}(G;x,y)$, dual to $κ$ and $κ_{\mathbbm z}$ respectively, in the sense that the lattice-point counting to the Ehrhart polynomials is taken inside a topological sum of the dilated closed polytopes $\barΔ^+_\textsc{ctf}(G,ρ)$. It turns out that the polynomial $\barκ(G;x,y)$ is Whitney's rank generating polynomial $R_G(x,y)$, which gives rise to a combinatorial interpretation on the values of the Tutte polynomial $T_G(x,y)$ at positive integers. In particular, some special values of $κ_\mathbbm{z}$ and $\barκ_\mathbbm{z}$ ($κ$ and $\barκ$) count the number of certain special kinds (of equivalence classes) of orientations.
Unlike Whitney's definition of the corank-nullity generating function $T(G;x+1,y+1)$, Tutte's definition of his now eponymous polynomial $T(G;x,y)$ requires a total order on the edges of which the polynomial is a posteriori independent. Tutte presented his definition in terms of internal and external activities of maximal spanning forests. Although Tutte's original definition may appear somewhat ad hoc upon first inspection, subsequent work by various researchers has demonstrated that activity is a deep combinatorial concept. In this survey, we provide an introduction to activities for graphs …
Plethysm is a fundamental operation in symmetric function theory, derived directly from its connection with representation theory. However, it does not admit a simple combinatorial interpretation, and finding coefficients of Schur function plethysms is a major open question. In this paper, we introduce a graph-theoretic interpretation for any plethysm based on the chromatic symmetric function. We use this interpretation to give simple proofs of new and previously known plethystic identities, as well as chromatic symmetric function identities.
Using Tutte's combinatorial definition of a map we define a $Δ$-matroid purely combinatorially and show that it is identical to Bouchet's topological definition.
On Some Quadratic Algebras I $\frac{1}{2}$: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials
We associate two modules, the $G$-parking critical module and the toppling critical module, to an undirected connected graph $G$. The $G$-parking critical module and the toppling critical module are canonical modules (with suitable twists) of quotient rings of the well-studied $G$-parking function ideal and the toppling ideal, respectively. For each critical module, we establish a Tutte-like short exact sequence relating the modules associated to $G$, an edge contraction $G/e$ and an edge deletion $G \setminus e$ ($e$ is a non-bridge). We obtain purely combinatorial consequences of Tutte short…
The Bollobás-Riordan polynomial [Math. Ann. 323, 81 (2002)] is a universal polynomial invariant for ribbon graphs. We find an extension of this polynomial for a particular family of combinatorial objects, called rank 3 weakly-colored stranded graphs. Stranded graphs arise in the study of tensor models for quantum gravity in physics, and generalize graphs and ribbon graphs. We present a seven-variable polynomial invariant of these graphs, which obeys a contraction/deletion recursion relation similar to that of the Tutte and Bollobás-Riordan polynomials. However, it is defined on a much broader …
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 2016, and we show that such polynomial has the classical deletion-contraction property. Moreover, we study the set of independencies for a realizable matroid over a domain, generalizing the definition of \emph{poset of torsions} $Gr(\mathcal{M})$ given by the second author in 2017. This is a union of identical simplicial posets as for (quasi-)arithmetic matroid…
We define a graph Laplacian with vertex weights in addition to the more classical edge weights, which unifies the combinatorial Laplacian and the normalised Laplacian. Moreover, we give a combinatorial interpretation for the coefficients of the weighted Laplacian characteristic polynomial in terms of weighted spanning forests and use this to prove a deletion-contraction relation. We prove various interlacing theorems relating to deletion and contraction, as well as to rectangular tilings, drawing on the work of Brooks, Smith, Stone and Tutte on square tilings. Additionally, we show that the we…
For any matroid $M$, we compute the Tutte polynomial $T_M(x,y)$ using the mixed intersection numbers of certain classes in the combinatorial Chow ring $A^\bullet(M)$ arising from hypersimplices. Using the mixed Hodge-Riemann relations, we deduce a strengthening of the log-concavity of the $h$-vector of a matroid complex, improving on an old conjecture of Dawson.
A Complete Grammar for Decomposing a Family of Graphs into 3-connected Components
Tutte在《Connectivity in graphs》一书中描述了将任何图分解为3-连通分量的规范分解。在这篇文章中，我们翻译（使用语言的符号组合学）Tutte的分解成一个一般的语法表示任何家庭的图（与一些稳定性条件）的3-连通子家庭。我们使用的一个关键成分是所谓的不对称定理的扩展，它在语法中产生负号。作为一个主要的应用程序，我们恢复在一个纯粹的组合的方式Giménez和Noy发现的一系列计数标记可平面图的解析表达式（这样的表达式是至关重要的做渐近计数和获得各种参数的限制法律的随机可平面图）。除了语法，我们的方法的一个重要组成部分是最近的双射建设的平面地图的布蒂尔，迪弗朗西斯科和Guitter。
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 obtain a generalization of Mason's conjecture on the $f$-vectors of independent subsets of matroids to arbitrary morphisms of matroids. To establish this, we define multivariate Tutte polynomials of morphisms of matroids, and show that they are Lorentzian in the sense of [BH19] for sufficiently small positive parameters.
