Tutte Polynomial Research Articles

Polynomial invariants for cactuses

Graph invariants are a useful tool in graph theory. Not only do they encode useful information about the graphs to which they are associated, but complete invariants can be used to distinguish between non-isomorphic graphs. Polynomial invariants for graphs such as the well-known Tutte polynomial have been studied for several years, and recently there has been interest to also define such invariants for phylogenetic networks, a special type of graph that arises in the area of evolutionary biology. Recently Liu gave a complete invariant for (phylogenetic) trees. However, the polynomial invariants defined thus far for phylogenetic networks that are not trees require vertex labels and either contain a large number of variables, or they have exponentially many terms in the number of reticulations. This can make it difficult to compute these polynomials and to use them to analyse unlabelled networks. In this paper, we shall show how to circumvent some of these difficulties for rooted cactuses and cactuses. As well as being important in other areas such as operations research, rooted cactuses contain some common classes of phylogenetic networks such phylogenetic trees and level-1 networks. More specifically, we define a polynomial F that is a complete invariant for the class of rooted cactuses without vertices of indegree 1 and outdegree 1 that has 5 variables, and a polynomial Q that is a complete invariant for the class of rooted cactuses that has 6 variables whose degree can be bounded linearly in terms of the size of the rooted cactus. We also explain how to extend the Q polynomial to define a complete invariant for leaf-labelled rooted cactuses as well as (unrooted) cactuses.

Matroid Chern-Schwartz-MacPherson cycles and Tutte activities

Lopéz de Medrano-Rincón-Shaw defined Chern-Schwartz-MacPher-son cycles for an arbitrary matroid M {M} and proved by an inductive geometric argument that the unsigned degrees of these cycles agree with the coefficients of T ( M ; x , 0 ) T({M};x,0) , where T ( M ; x , y ) T({M};x,y) is the Tutte polynomial associated to M {M} . Ardila-Denham-Huh recently utilized this interpretation of these coefficients in order to demonstrate their log-concavity. In this note we provide a direct calculation of the degree of a matroid Chern-Schwartz-MacPherson cycle by taking its stable intersection with a generic tropical linear space of the appropriate codimension and showing that the weighted point count agrees with the Gioan-Las Vergnas refined activities expansion of the Tutte polynomial.

Harmonic Tutte polynomials of matroids

In the present paper, we introduce the concept of harmonic Tutte polynomials of matroids and discuss some of their properties. In particular, we generalize Greene’s theorem, thereby expressing harmonic weight enumerators of codes as evaluations of harmonic Tutte polynomials.

Parking Functions: From Combinatorics to Probability

Suppose that m drivers each choose a preferred parking space in a linear car park with n spots. In order, each driver goes to their chosen spot and parks there if possible, and otherwise takes the next available spot if it exists. If all drivers park successfully, the sequence of choices is called a parking function. Classical parking functions correspond to the case $$m=n$$ ; we study here combinatorial and probabilistic aspects of this generalized case. We construct a family of bijections between parking functions $$\text {PF}(m, n)$$ with m cars and n spots and spanning forests $$\mathscr {F}(n+1, n+1-m)$$ with $$n+1$$ vertices and $$n+1-m$$ distinct trees having specified roots. This leads to a bijective correspondence between $$\text {PF}(m, n)$$ and monomial terms in the associated Tutte polynomial of a disjoint union of $$n-m+1$$ complete graphs. We present an identity between the “inversion enumerator” of spanning forests with fixed roots and the “displacement enumerator” of parking functions. The displacement is then related to the number of graphs on $$n+1$$ labeled vertices with a fixed number of edges, where the graph has $$n+1-m$$ disjoint rooted components with specified roots. We investigate various probabilistic properties of a uniform parking function, giving a formula for the law of a single coordinate. As a side result we obtain a recurrence relation for the displacement enumerator. Adapting known results on random linear probes, we further deduce the covariance between two coordinates when $$m=n$$ .

Short proof of a theorem of Brylawski on the coefficients of the Tutte polynomial

In this short note we show that a system M=(E,r) with a ground set E of size m and (rank) function r:2E→Z≥0 satisfying r(S)≤min(r(E),|S|) for every set S⊆E, the Tutte polynomial TM(x,y)≔∑S⊆E(x−1)r(E)−r(S)(y−1)|S|−r(S),written as TM(x,y)=∑i,jtijxiyj, satisfies that for any integer h≥0, we have ∑i=0h∑j=0h−ih−ij(−1)jtij=(−1)m−rh−rh−m,where r=r(E), and we use the convention that when h<m, the binomial coefficient h−rh−m is interpreted as 0.This generalizes a theorem of Brylawski on matroid rank functions and h<m, and a theorem of Gordon for h≤m with the same assumptions on the rank function.The proof presented here is significantly shorter than the previous ones. We only use the fact that the Tutte polynomial TM(x,y) simplifies to (x−1)r(E)y|E| along the hyperbola (x−1)(y−1)=1.

Graph polynomials associated with Dyson-Schwinger equations

Quantum motions are encoded by a particular family of recursive Hochschild equations in the renormalization Hopf algebra which represent Dyson-Schwinger equations, combinatorially. Feynman graphons, which topologically complete the space of Feynman diagrams of a gauge field theory, are considered to formulate some random graph representations for solutions of quantum motions. This framework leads us to explain the structures of Tutte and Kirchhoff-Symanzik polynomials associated with solutions of Dyson-Schwinger equations. These new graph polynomials are applied to formulate a new parametric representation for large Feynman diagrams and their corresponding Feynman rules.

Stellahedral geometry of matroids

Abstract We use the geometry of the stellahedral toric variety to study matroids. We identify the valuative group of matroids with the cohomology ring of the stellahedral toric variety and show that valuative, homological and numerical equivalence relations for matroids coincide. We establish a new log-concavity result for the Tutte polynomial of a matroid, answering a question of Wagner and Shapiro–Smirnov–Vaintrob on Postnikov–Shapiro algebras, and calculate the Chern–Schwartz–MacPherson classes of matroid Schubert cells. The central construction is the ‘augmented tautological classes of matroids’, modeled after certain toric vector bundles on the stellahedral toric variety.

Topological quantum field theory and polynomial identities for graphs on the torus

We establish a relation between the trace evaluation in SO(3) topological quantum field theory and evaluations of a topological Tutte polynomial. As an application, a generalization of the Tutte golden identity is proved for graphs on the torus.

Irreducibility of the Tutte polynomial of an embedded graph

We prove that the ribbon graph polynomial of a graph embedded in an orientable surface is irreducible if and only if the embedded graph is neither the disjoint union nor the join of embedded graphs. This result is analogous to the fact that the Tutte polynomial of a graph is irreducible if and only if the graph is connected and non-separable.

The Merino–Welsh Conjecture for Split Matroids

In 1999, Merino and Welsh conjectured that evaluations of the Tutte polynomial of a graph satisfy an inequality. In this short article, we show that the conjecture generalized to matroids holds for the large class of all split matroids by exploiting the structure of their lattice of cyclic flats. This class of matroids strictly contains all paving and copaving matroids.

Interpretations for the Tutte polynomials of morphisms of matroids

We study interpretations of the Tutte polynomials of set-pointed matroids and matroid perspectives. Suppose that M is a matroid whose ground set E is given with a linear ordering < . We show that the Tutte polynomial of M pointed by A ⊆ E equals ∑ x r ( M / ( A ∪ X ) ) y r ∗ ( M − ( A ∪ Y ) ) z r ( ( M / X ) − Y ) where the sum runs through the set D ( M ; A ; < ) of couples ( X , Y ) such that X ⊆ E ∖ A , Y = ( E ∖ A ) ∖ X , and X ∩ C (resp. Y ∩ C ) differs from { min ( C ) } for each circuit C of M ∗ − A (resp. M − A ). Furthermore, we characterize D ( M ; A ; < ) by a contraction–deletion rule. Analogous results are introduced for the Tutte polynomials of matroid perspectives.

Hamming polynomial of a demimatroid

Johnsen, Roksvold and Verdure have shown how fundamental invariants and concepts of linear coding theory, such as weight enumerators and higher weights, can be studied in the context of matroid theory. They have shown that Betti numbers, of a matroid and its elongations, associated to a linear code, determine both the weight enumerator and the higher weights of the matroid. In the same vein, Britz, Johnsen, Mayhew and Shiromoto, introduced demimatroids as a natural generalization of matroids. They have shown that demimatroids are the appropriate combinatorial objects for studying Wei’s duality. Continuing in this line of research, we study the Hamming polynomial of a demimatroid, which is a generalization of the extended weight enumerator of a matroid. It results that the Hamming polynomial is a specialization of the Tutte polynomial of the demimatroid, and actually is equivalent to it. In addition, guided by the work of Johnsen, Roksvold and Verdure for matroids, we prove that Betti numbers of, a demimatroid and its elongations, determine the Hamming polynomial. Our results may be applied to simplicial complexes since in a canonical way they can be viewed as demimatroids. Furthermore, following work of Brylawski and Gordon, we show how demimatroids may be generalized one step further, to Brylawski structures. All this largely extends notions such as deletion, contraction, duality, nullity, minors, weight enumerators, higher weights, Tutte polynomial and MacWilliams identity, to non-matroidal structures.

Tutte Polynomials and Graph Symmetries

The Tutte polynomial is an isomorphism invariant of graphs that generalizes the chromatic and the flow polynomials. This two-variable polynomial with integral coefficients is known to carry important information about the properties of the graph. It has been used to prove long-standing conjectures in knot theory. Furthermore, it is related to the Potts and Ising models in statistical physics. The purpose of this paper is to study the interaction between the Tutte polynomial and graph symmetries. More precisely, we prove that if the automorphism group of the graph G contains an element of prime order p, then the coefficients of the Tutte polynomial of G satisfy certain necessary conditions.

Generalized bijective maps between G-parking functions, spanning trees, and the Tutte polynomial

We introduce an object called a tree growing sequence (TGS) in an effort to generalize bijective correspondences between G -parking functions, spanning trees, and the set of monomials in the Tutte polynomial of a graph G . A tree growing sequence determines an algorithm which can be applied to a single function, or to the set P G , q of G -parking functions. When the latter is chosen, the algorithm uses splitting operations – inspired by the recursive definition of the Tutte polynomial – to iteratively break P G , q into disjoint subsets. This results in bijective maps τ and ρ from P G , q to the spanning trees of G and Tutte monomials, respectively. We compare the TGS algorithm to Dhar’s algorithm and the family described by Chebikin and Pylyavskyy in 2005. Finally, we compute a Tutte polynomial of a zonotopal tiling using analogous splitting operations.

Proving identities on weight polynomials of tiered trees via Tutte polynomials

A tiered graphG=(V,E) with m tiers is a simple graph with V⊆〚n〛, where 〚n〛={1,2,⋯,n}, and with a surjective map t from V to 〚m〛 such that if v is a vertex adjacent to v′ in G with v>v′, then t(v)>t(v′). For any ordered partition p=(p1,p2,⋯,pm) of n, let Tp denote the set of tiered trees with vertex set 〚n〛 and with a map t:〚n〛→〚m〛 such that |t−1(i)|=pi for all i=1,2,…,m. For any T∈Tp, let KT denote the complete tiered graph whose vertex set and tiering map are the same as those of T. If the edges of KT are ordered lexicographically by their endpoints, then the weight w(T) of T is the external activity of T in KT, i.e., the number of edges e∈E(KT)∖E(T) such that e is the least element in the unique cycle determined by T∪e. Let Pp(q)=∑T∈Tpqw(T). Dugan, Glennon, Gunnells and Steingrímsson (2019) [4] asked for an elementary proof of the identity Pp(q)=Pπ(p)(q) for any permutation π of 1,2,⋯,m, where π(p)=(pπ(1),pπ(2),⋯,pπ(m)). In this article, we will prove an extension of this identity by applying Tutte polynomials. Furthermore, we also provide a proof of the identity P(1,p1,p2)(q)=P(p1+1,p2+1)(q) via Tutte polynomials.

The Free m-Cone of a Matroid and Its $${\mathcal {G}}$$-Invariant

For a matroid, its configuration determines its \({\mathcal {G}}\)-invariant. Few examples are known of pairs of matroids with the same \({\mathcal {G}}\)-invariant but different configurations. In order to produce new examples, we introduce the free m-cone \(Q_m(M)\) of a loopless matroid M, where m is a positive integer. We show that the \({\mathcal {G}}\)-invariant of M determines the \({\mathcal {G}}\)-invariant of \(Q_m(M)\), and that the configuration of \(Q_m(M)\) determines M; so if M and N are nonisomorphic and have the same \({\mathcal {G}}\)-invariant, then \(Q_m(M)\) and \(Q_m(N)\) have the same \({\mathcal {G}}\)-invariant but different configurations. We prove analogous results for several variants of the free m-cone. We also define a new matroid invariant of M, and show that it determines the Tutte polynomial of \(Q_m(M)\).

On the Tutte and Matching Polynomials for Complete Graphs

Let T(G; X, Y) be the Tutte polynomial for graphs. We study the sequence ta,b(n) = T(Kn; a, b) where a, b are integers, and show that for every μ ∈ ℕ the sequence ta,b(n) is ultimately periodic modulo μ provided a ≠ 1 mod μ and b ≠ 1 mod μ. This result is related to a conjecture by A. Mani and R. Stones from 2016. The theorem is a consequence of a more general theorem which holds for a wide class of graph polynomials definable in Monadic Second Order Logic. This gives also similar results for the various substitution instances of the bivariate matching polynomial and the trivariate edge elimination polynomial ξ(G; X, Y, Z) introduced by I. Averbouch, B. Godlin and the second author in 2008. All our results depend on the Specker-Blatter Theorem from 1981, which studies modular recurrence relations of combinatorial sequences which count the number of labeled graphs.

Evaluations of Tutte polynomials of regular graphs

Let TG(x,y) be the Tutte polynomial of a graph G. In this paper we show that if (Gn)n is a sequence of d-regular graphs with girth g(Gn)→∞, then for x≥1 and 0≤y≤1 we havelimn→∞⁡TGn(x,y)1/v(Gn)=td(x,y), wheretd(x,y)={(d−1)((d−1)2(d−1)2−x)d/2−1ifx≤d−1and0≤y≤1,x(1+1x−1)d/2−1ifx>d−1and0≤y≤1. If (Gn)n is a sequence of random d-regular graphs, then the same statement holds true asymptotically almost surely.This theorem generalizes results of McKay (x=1,y=1, spanning trees of random d-regular graphs) and Lyons (x=1,y=1, spanning trees of large-girth d-regular graphs). Interesting special cases are TG(2,1) counting the number of spanning forests, and TG(2,0) counting the number of acyclic orientations.

Tutte short exact sequences of graphs

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 exact sequences. For instance, we reprove a theorem of Merino that the critical polynomial of a graph is an evaluation of its Tutte polynomial, and relate the vanishing of certain combinatorial invariants (the number of acyclic orientations on connected partition graphs satisfying a unique sink property) of \(G/e\) to the equality of the corresponding invariants of \(G\) and \(G \setminus e\).Mathematics Subject Classifications: 13D02, 05E40Keywords: Tutte polynomials, chip firing games, toppling ideals, \(G\)-parking function ideals, canonical modules

The Tutte polynomial of phenylene systems with given number of branching hexagons

AbstractPolynomial graph invariants have been confirmed to have important applications in quantum chemistry and biological information. One of the famous polynomial graph invariants is the Tutte polynomial which gives abundant graph‐theoretical information of the underlying graph. In this paper, we first give a simpler and more efficient method to get the Tutte polynomials of alternating polycyclic chains. Then we obtain the explicit expressions for the Tutte polynomials and the number of spanning trees of phenylene systems with given number of branching hexagons. Moreover, we determine the extremal values of the number of spanning trees among the phenylene systems with given one branching hexagon and two branching hexagons. The corresponding extremal phenylene systems are characterized, respectively.