View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Applied Mathematics 157 (2009) 2515–2524 Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam The bivariate Ising polynomial of a graph Daniel Andrén, Klas Markström ∗ Department of Mathematics and Mathematical Statistics, Umeå universitet, SE-901 87 Umeå, Sweden article info a b s t r a c t Article history: In this paper we discuss the two variable Ising polynomials in a graph theoretical setting. Received 29 October 2007 This polynomial has its origin in physics as the partition function of the Ising model with Received in revised form 2 December 2008 an external field. We prove some basic properties of the Ising polynomial and demonstrate Accepted 27 February 2009 that it encodes a large amount of combinatorial information about a graph. We also give Available online 28 March 2009 examples which prove that certain properties, such as the chromatic number, are not determined by the Ising polynomial. Finally we prove that there exist large families of non- Keywords: isomorphic planar triangulations with identical Ising polynomial. Graph polynomials Ising polynomial ' 2009 Published by Elsevier B.V. Graph invariants 1. Introduction In graph theory a number of different polynomials associated to a graph have been introduced over the years, and it has repeatedly turned out that they are in fact specialisations of the flagship of this armada of polynomials, the Tutte polynomial. The Tutte polynomial was introduced by Tutte in 1947 [19] after he had observed that a number of interesting graph parameters satisfied similar recursive identities. The Tutte polynomial contains important polynomials such as the chromatic polynomial [12], the Jones polynomial of a knot and the reliability polynomial of a network, see [22] for a survey. In the 1970s it was realised that the Tutte polynomial had an important role to play in statistical physics as well. In 1925 Ising and his thesis advisor Lenz [7] introduced the Ising model for magnetism. In this model a ``spin'' of value ±1 is assigned to every vertex in a graph G. An edge with equal spins at the endpoints is given an energy of 1 and one with unequal endpoints an energy of −1. The total sum of the spin is called the magnetisation. Summing over all such spin assignments we get a generating function Z.G; x; y/, here called the Ising polynomial, where the coefficient of xiyj counts the assignments with energy i and magnetisation j. The Ising model studies how magnetisation and energy are correlated under a suitable probability measure on the set of spin assignments. This model was later extended to the Potts model, which allows more than just two values of the spin. In 1972 Fortuin and Kasteleyn [2] introduced a new representation for the Potts model, where the magnetisation was not included, called the random-cluster model, see [6] for a textbook treatment. It was then realised that the generating function controlling this model is in fact equivalent to the Tutte polynomial. See [18] for a recent survey of the many results connecting the Tutte polynomial and the random-cluster model to various topics in graph theory and physics. For the Ising model we can find the restricted polynomial Z.G; x; 1/ from the Tutte polynomial of G. However, since this polynomial no longer contains the total magnetisation of the spins the random-cluster model does not capture all properties of the original Ising model. There has been many interesting results specifically for the Ising model in the physics literature over the years, such as [8] where it is shown that for a fixed x the zeros of Z.G; x; y/ lie on a circle in the complex plane. However the full Ising polynomial Z.G; x; y/ has not received the same attention in the graph theoretical literature. Our current aim is to demonstrate that the Ising polynomial Z.G; x; y/ is an interesting polynomial from a graph theoretical perspective and study some of its properties. Interestingly we will show that although Z.G; x; 1/ is determined ∗ Corresponding author. Fax: +46 907865222. E-mail address: [email protected] (K. Markström). 0166-218X/$ – see front matter ' 2009 Published by Elsevier B.V. doi:10.1016/j.dam.2009.02.021 2516 D. Andrén, K. Markström / Discrete Applied Mathematics 157 (2009) 2515–2524 by the Tutte polynomial of G the full polynomial Z.G; x; y/ is not, e.g. unlike the Tutte polynomial it is not a trivial function when G is a tree. In the other direction we will also show that the Tutte polynomial is not determined by the Ising polynomial. We thus have a graph polynomial which, although related to, is essentially different from the Tutte polynomial. Just like for the Tutte polynomial there are known graph polynomials which are contained in the Ising polynomial, e.g. the matching polynomial and for regular graphs the independence polynomial. Perhaps the first two members of a new armada. 2. Definitions and relations We will now give the formal definition of the Ising polynomial. In fact we will give two equivalent definitions and demonstrate that the Ising Polynomial is also equivalent to a second polynomial related to eulerian subgraphs of G. The first definition is the original physics definition in terms of ``spin states'' on the vertex set of G and the second is a reformulation of this definition in terms of edge cuts in G. 2.1. The state sum definition We first define a few terms needed for the definition of the Ising polynomial, due to the origin of the Ising polynomial as a physical model for phase transitions in magnetic materials the terminology has a physical flavour. We let G be a simple graph, V .G/ its vertex set, and E.G/ its edge set. We will also use n D jV .G/j and m D jE.G/j. A state σ on G is a function σ V V .G/ ! {−1; 1g, the value of σ at a vertex v is called the magnetisation of v. Definition 2.1. Given a state σ the energy E(σ ; e/ of an edge e D .u; v/ in G is E(σ ; e/ D σ .u)σ .v/, and the energy E(σ /of the state σ is the sum of the energies of the edges, that is X E(σ / D E(σ ; e/: e2E.G/ Definition 2.2. The magnetisation M(σ / of a state σ is the sum of the magnetisations of all the vertices in G, that is, X M(σ / D σ .u/: u2V .G/ Let Ω denote the set of all states on G. We can now define the Ising polynomial: Definition 2.3 (The Ising Polynomial). The Ising polynomial is X E(σ / M(σ / X i j Z.G; x; y/ D x y D ai;jx y : σ2Ω i;j Here we can note two things about Z.G; x; y/. First, Z.G; x; y/ is a Laurent polynomial rather than a polynomial, it can have monomials with negative, but integer, powers. Second, it is also the generating function for the number of states on G with given magnetisation and energy. The following simple lemma will later be useful. Lemma 2.4. The exponents of x in Z.G; x; y/ are at most m, and at least −m. The exponents of y in Z.G; x; y/ are at most n, and at least −n. Another way to look at the state sum definition is to consider a state on G as a graph homomorphism from G to a weighted −1 −1 K2 with loops on both vertices, the vertices have weights y and y , the loops have weight x and the ordinary edge weight x . Recently [3] has shown that a large set of models from statistical mechanics, having a property called reflection positivity, are equivalent to counting weighted graph homomorphisms in this way. In [9] the transfer matrix methods often used for computing partition functions of spin models on lattice graphs were also studied in this general setting. 2.2. Cuts and T -joins A more graph theoretical interpretation of the Ising polynomial can be given in terms of cuts. The values of a state σ defines a bipartition of the graph G such that all the edges with negative energy have one endpoint in each partition. Thus the coefficient ai;j enumerates edge cuts in G such that there are .m − i/=2 edges in the cut and .n − j/=2 vertices in one part of the bipartition. Definition 2.5 (Cut). A cut TS; SNU, or equivalently TSN; SU, where SN D V n S, in a graph G D .V ; E/ is a subset of edges, induced by a partition S [SN D V of the vertices of G, that have one endpoint in S and the other in SN. We let TS; SNU denote the number of edges with exactly one endpoint in S. D. Andrén, K. Markström / Discrete Applied Mathematics 157 (2009) 2515–2524 2517 In this notation we can now give the following equivalent definition of the Ising polynomial Definition 2.6 (The Ising Polynomial). The Ising polynomial is now defined as X n−2i m−2j Z.G; x; y/ D ai;jx y ; i;j where ai;j counts the number of cuts TA; BU of V .G/ such that jAj D j and jTA; BUj D i.