Some variants of the exponential formula, with application to the multivariate Tutte polynomial (alias Potts model)
Alexander D. Scott, Alan D. Sokal
备注：
LaTeX2e, 39 pages. Dedicated to the memory of Pierre Leroux. Version 2 includes a new Appendix presenting a generalization of Mobius inversion on the partition lattice
发表信息：
Seminaire Lotharingien de Combinatoire 61A, article 61Ae (2009)
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.
The chromatic polynomial P_G(q) of a loopless graph G is known to be nonzero (with explicitly known sign) on the intervals (-\infty,0), (0,1) and (1,32/27]. Analogous theorems hold for the flow polynomial of bridgeless graphs and for the characteristic polynomial of loopless matroids. Here we exhibit all these results as special cases of more general theorems on real zero-free regions of the multivariate Tutte polynomial Z_G(q,v). The proofs are quite simple, and employ deletion-contraction together with parallel and series reduction. In particular, they shed light on the origin of the curious…
We find zero-free regions in the complex plane at large |q| for the multivariate Tutte polynomial (also known in statistical mechanics as the Potts-model partition function) Z_G(q,w) of a graph G with general complex edge weights w = {w_e}. This generalizes a result of Sokal (cond-mat/9904146) that applies only within the complex antiferromagnetic regime |1+w_e| \le 1. Our proof uses the polymer-gas representation of the multivariate Tutte polynomial together with the Penrose identity.
The multivariate Tutte polynomial (known to physicists as the Potts-model partition function) can be defined on an arbitrary finite graph G, or more generally on an arbitrary matroid M, and encodes much important combinatorial information about the graph (indeed, in the matroid case it encodes the full structure of the matroid). It contains as a special case the familiar two-variable Tutte polynomial -- and therefore also its one-variable specializations such as the chromatic polynomial, the flow polynomial and the reliability polynomial -- but is considerably more flexible. I begin by giving …
The classical relationship between the Tutte polynomial of graph theory and the Potts model of statistical mechanics has resulted in valuable interactions between the disciplines. Unfortunately, it does not include the external magnetic fields that appear in most Potts model applications. Here we define the V-polynomial, which lifts the classical relationship between the Tutte polynomial and the zero field Potts model to encompass external magnetic fields. The V-polynomial generalizes Nobel and Welsh's W-polynomial, which extends the Tutte polynomial by incorporating vertex weights and adaptin…
This is a short account of connections between the Tutte polynomial and the Ising, Potts, and random-cluster models. The four principal elements are the Ising model of 1925, the Tutte polynomial of 1947, the Potts model of 1952, and the random-cluster model of 1972.
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 invariant for a graph is NP-hard in general. In this paper, based on their self-similar structures, we recursively describe the Tutte polynomials of an infinite family of scale-free lattices. Furthermore, we give some exact analytical expressions of the Tutte polynomial for several special points at $(X,Y)$-plane.
We consider tiling rectangles of size 4m x 4n by T-shaped tetrominoes. Each tile is assigned a weight that depends on its orientation and position on the lattice. For a particular choice of the weights, the generating function of tilings is shown to be the evaluation of the multivariate Tutte polynomial Z\_G(Q,v) (known also to physicists as the partition function of the Q-state Potts model) on an (m-1) x (n-1) rectangle G, where the parameter Q and the edge weights v can take arbitrary values depending on the tile weights.
We prove that the Hessians of nonzero partial derivatives of the (homogenous) multivariate Tutte polynomial of any matroid have exactly one positive eigenvalue on the positive orthant when $0<q\leq 1$. Consequences are proofs of the strongest conjecture of Mason and negative dependence properties for $q$-state Potts model partition functions.
The n-sum graph Negami's splitting formula for the Tutte polynomial is not valid in the region $(x-1)(y-1)=q$ for $q=1,2,\ldots n-1$ with the additional region $y=1$ if $n>3$. This region corresponds to (up to prefactors and change of variables) the Ising model, the $q$-state Potts model, the number of spanning forest generator and particularizations of these. We show splitting formulas for these specializations.
Korteweg-de Vries Institute for Mathematics, University of Amsterdam（科特韦格-德弗里斯数学研究所，阿姆斯特丹大学）；Institute of Physics, University of Amsterdam（阿姆斯特丹大学物理研究所）；Dutch Institute for Emergent Phenomena（荷兰涌现现象研究所）
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…
We study the complexity of approximating the partition function of the $q$-state Potts model and the closely related Tutte polynomial for complex values of the underlying parameters. Apart from the classical connections with quantum computing and phase transitions in statistical physics, recent work in approximate counting has shown that the behaviour in the complex plane, and more precisely the location of zeros, is strongly connected with the complexity of the approximation problem, even for positive real-valued parameters. Previous work in the complex plane by Goldberg and Guo focused on $q…
We consider a specialization $Y_M(q,t)$ of the Tutte polynomial of a matroid $M$ which is inspired by analogy with the Potts model from statistical mechanics. The only information lost in this specialization is the number of loops of $M$. We show that the coefficients of $Y_M(1-p,t)$ are very simply related to the ranks of the Whitney homology groups of the opposite partial orders of the independent set complexes of the duals of the truncations of $M$. In particular, we obtain a new homological interpretation for the coefficients of the characteristic polynomial of a matroid.
Tutte proved that if $G_{pt}$ is a planar triangulation and $P(G_{pt},q)$ is its chromatic polynomial, then $|P(G_{pt},τ+1)| \le (τ-1)^{n-5}$, where $τ=(1+\sqrt{5} \,)/2$ and $n$ is the number of vertices in $G_{pt}$. Here we study the ratio $r(G_{pt})=|P(G_{pt},τ+1)|/(τ-1)^{n-5}$ for a variety of planar triangulations. We construct infinite recursive families of planar triangulations $G_{pt,m}$ depending on a parameter $m$ linearly related to $n$ and show that if $P(G_{pt,m},q)$ only involves a single power of a polynomial, then $r(G_{pt,m})$ approaches zero exponentially fast as $n \to \inft…
We introduce and study the notion of the $G$-Tutte polynomial for a list $\mathcal{A}$ of elements in a finitely generated abelian group $Γ$ and an abelian group $G$, which is defined by counting the number of homomorphisms from associated finite abelian groups to $G$. The $G$-Tutte polynomial is a common generalization of the (arithmetic) Tutte polynomial for realizable (arithmetic) matroids, the characteristic quasi-polynomial for integral arrangements, Brändén-Moci's arithmetic version of the partition function of an abelian group-valued Potts model, and the modified Tutte-Krushkal-Renhardy…
We study the partition function of Potts model in an external (magnetic) field, and its connections with the zero-field Potts model partition function. Using a deletion-contraction formulation for the partition function Z for this model, we show that it can be expanded in terms of the zero-field partition function. We also show that Z can be written as a sum over the spanning trees, and the spanning forests, of a graph G. Our results extend to Z the well-known spanning tree expansion for the zero-field partition function that arises though its connections with the Tutte polynomial.
We consider the Potts model in a magnetic field on an arbitrary graph $G$. Using a formula of F. Y. Wu for the partition function $Z$ of this model as a sum over spanning subgraphs of $G$, we prove some properties of $Z$ concerning factorization, monotonicity, and zeros. A generalization of the Tutte polynomial is presented that corresponds to this partition function. In this context we formulate and discuss two weighted graph-coloring problems. We also give a general structural result for $Z$ for cyclic strip graphs.
In this note we study some of the properties of the generating polynomial for homomorphisms from a graph to at complete weighted graph on $q$ vertices. We discuss how this polynomial relates to a long list of other well known graph polynomials and the partition functions for different spin models, many of which are specialisations of the homomorphism polynomial. We also identify the smallest graphs which are not determined by their homomorphism polynomials for $q=2$ and $q=3$ and compare this with the corresponding minimal examples for the $U$-polynomial, which generalizes the well known Tutte…
Originally in 1954 the Tutte polynomial was a bivariate polynomial associated to a graph in order to enumerate the colorings of this graph and of its dual graph at the same time. However the Tutte polynomial reveals more of the internal structure of a graph, and contains even other specializations from other sciences like the Jones polynomial in Knot theory, the partition function of the Pott model in statistical physics, and the reliability polynomial in network theory. In this article, we study the Tutte polynomial associated to more general objects which are the arrangements of hyperplanes.…
The deletion--contraction algorithm is perhaps the most popular method for computing a host of fundamental graph invariants such as the chromatic, flow, and reliability polynomials in graph theory, the Jones polynomial of an alternating link in knot theory, and the partition functions of the models of Ising, Potts, and Fortuin--Kasteleyn in statistical physics. Prior to this work, deletion--contraction was also the fastest known general-purpose algorithm for these invariants, running in time roughly proportional to the number of spanning trees in the input graph. Here, we give a substantially …
We introduce a polynomial invariant of graphs on surfaces, $P_G$, generalizing the classical Tutte polynomial. Topological duality on surfaces gives rise to a natural duality result for $P_G$, analogous to the duality for the Tutte polynomial of planar graphs. This property is important from the perspective of statistical mechanics, where the Tutte polynomial is known as the partition function of the Potts model. For ribbon graphs, $P_G$ specializes to the well-known Bollobas-Riordan polynomial, and in fact the two polynomials carry equivalent information in this context. Duality is also estab…
I show that there exist universal constants $C(r) < \infty$ such that, for all loopless graphs $G$ of maximum degree $\le r$, the zeros (real or complex) of the chromatic polynomial $P_G(q)$ lie in the disc $|q| < C(r)$. Furthermore, $C(r) \le 7.963906... r$. This result is a corollary of a more general result on the zeros of the Potts-model partition function $Z_G(q, {v_e})$ in the complex antiferromagnetic regime $|1 + v_e| \le 1$. The proof is based on a transformation of the Whitney-Tutte-Fortuin-Kasteleyn representation of $Z_G(q, {v_e})$ to a polymer gas, followed by verification of the …
I show that the zeros of the chromatic polynomials P_G(q) for the generalized theta graphs Θ^{(s,p)} are, taken together, dense in the whole complex plane with the possible exception of the disc |q-1| < 1. The same holds for their dichromatic polynomials (alias Tutte polynomials, alias Potts-model partition functions) Z_G(q,v) outside the disc |q+v| < |v|. An immediate corollary is that the chromatic zeros of not-necessarily-planar graphs are dense in the whole complex plane. The main technical tool in the proof of these results is the Beraha-Kahane-Weiss theorem on the limit sets of zeros for…
In this survey, we give a friendly introduction from a graph theory perspective to the q-state Potts model, an important statistical mechanics tool for analyzing complex systems in which nearest neighbor interactions determine the aggregate behavior of the system. We present the surprising equivalence of the Potts model partition function and one of the most renowned graph invariants, the Tutte polynomial, a relationship that has resulted in a remarkable synergy between the two fields of study. We highlight some of these interconnections, such as computational complexity results that have alte…
This paper deals with the partition function of the Ising model from statistical mechanics, which is used to study phase transitions in physical systems. A special case of interest is that of the Ising model with constant energies and external field. One may consider such an Ising system as a simple graph together with vertex and edge weights. When these weights are considered indeterminates, the partition function for the constant case is a trivariate polynomial Z(G;x,y,z). This polynomial was studied with respect to its approximability by L. A. Goldberg, M. Jerrum and M. Paterson in 2003. Z(…
partial abstract: The $q$-state Potts model partition function (equivalent to the Tutte polynomial) for a lattice strip of fixed width $L_y$ and arbitrary length $L_x$ has the form $Z(G,q,v)=\sum_{j=1}^{N_{Z,G,λ}}c_{Z,G,j}(λ_{Z,G,j})^{L_x}$, where $v$ is a temperature-dependent variable. The special case of the zero-temperature antiferromagnet ($v=-1$) is the chromatic polynomial $P(G,q)$. Using coloring and transfer matrix methods, we give general formulas for $C_{X,G}=\sum_{j=1}^{N_{X,G,λ}}c_{X,G,j}$ for $X=Z,P$ on cyclic and Möbius strip graphs of the square and triangular lattice. Combinin…
