No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
Colouring Square Lattice Graphs
Bulletin of The London Mathematical Society
... In particular, in [22] we presented a number of results on acyclic orientations and their asymptotic behavior. A recursive family of graphs is a family of graphs such that the (m + 1)'th member, G m+1 , can be obtained from the m'th member, roughly speaking, by the addition of some subgraph [27][28][29]). For example, a square-lattice ladder strip of length m + 1 vertices with free boundary conditions can be obtained by adding a square to the end of the square-lattice strip of length m. ...
... (1. 29) In particular, in the n → ∞ limit, ...
... Related to this, we have noticed an interesting connection between our results for α(Λ) and α 0 (Λ) and the analytic expressions that were proved in [30] (see also [61,62]) to be lower bounds on W (Λ, q) for all Archimedean lattices (and dual Archimedean lattices) using a coloring-matrix method introduced in [29] to prove a lower bound on W (sq, q). We find that if one evaluates these expressions at q = −1 and q = 0, then the results are consistent with being upper bounds on α(Λ) and α 0 (Λ), respectively. ...
We calculate exponential growth constants describing the asymptotic behavior of several quantities enumerating classes of orientations of arrow variables on the bonds of several types of directed lattice strip graphs G of finite width and arbitrarily great length, in the infinite-length limit, denoted {G}. Specifically, we calculate the exponential growth constants for (i) acyclic orientations, , (ii) acyclic orientations with a single source vertex, , and (iii) totally cyclic orientations, . We consider several lattices, including square (sq), triangular (tri), and honeycomb (hc). From our calculations, we infer lower and upper bounds on these exponential growth constants for the respective infinite lattices. To our knowledge, these are the best current bounds on these quantities. Since our lower and upper bounds are quite close to each other, we can infer very accurate approximate values for the exponential growth constants, with fractional uncertainties ranging from to . Further, we present exact values of , , and and use them to show that our lower and upper bounds on these quantities are very close to these exact values, even for modest strip widths. Results are also given for a nonplanar lattice denoted . We show that , , and are monotonically increasing functions of vertex degree for these lattices. We also study the asymptotic behavior of the ratios of the quantities (i)-(iii) divided by the total number of edge orientations as the number of vertices goes to infinity. A comparison is given of these exponential growth constants with the corresponding exponential growth constant for spanning trees. Our results are in agreement with inequalities following from the Merino-Welsh and Conde-Merino conjectures.
... To determine these values and bounds, we use transfer matrix methods. These have been used extensively in statistical mechanics [1, 6] and combinatorics (see, e.g., [5, 7]) to evaluate and analyse many different functions defined on lattice graphs of various kinds. In statistical mechanics, they have been used to study partition functions [1, 6]. ...
... In statistical mechanics, they have been used to study partition functions [1, 6]. In graph theory, they have been used to count structures such as colourings [5] and self-avoiding walks [7]. There has been a number of attempts to estimate the number of legal Go positions on square lattice boards including the standard 19 × 19 one. ...
We use transfer matrix methods to determine bounds for the numbers of legal Go positions for various numbers of players on some planar lattice graphs, including square lattice graphs such as those on which the game is normally played. We also Þnd bounds on limiting constants that describe the behaviour of the number of legal positions on these lattice graphs as the dimensions of the lattices tend to inÞnity. These results amount to giving bounds for some speciÞc evaluations of Go polynomials on these graphs.
... Let us consider strips of various lattices with arbitrary length L x = m vertices and fixed width L y vertices (with the longitudinal and transverse directions taken to bex andŷ). The chromatic polynomials for the cyclic and Möbius strip graphs of the square lattice were calculated for L y = 2 in [8] by means of iterative deletion-contraction operations, and subsequently via a transfer matrix method in [18] and a coloring matrix method in [19] (see also [40]). We have extended this to the corresponding cyclic [36] and Möbius [39] L y = 3 strips. ...
... Using a coloring matrix method, Biggs [19] proved upper and lower bounds for W for the square lattice. We applied this method to prove such bounds for the triangular and honeycomb lattices [26]. ...
Let P(G,q) be the chromatic polynomial for coloring the n-vertex graph G with q colors, and define . Besides their mathematical interest, these functions are important in statistical physics. We give a comparative discussion of exact calculations of P and W for a variety of recursive families of graphs, including strips of regular lattices with various boundary conditions and homeomorphic expansions thereof. Generalizing to , we determine the accumulation sets of the chromatic zeros constituting the continuous loci of points on which W is nonanalytic. Various families of graphs with the property that the chromatic zeros and/or their accumulation sets (i) include support for ; (ii) bound regions and pass through q=0; and (iii) are noncompact are discussed, and the role of boundary conditions is analyzed. Some corresponding results are presented for Potts model partition functions for nonzero temperature, equivalent to the full Tutte polynomials for various families of graphs.
... In [1] we noted our observation concerning analytic expressions that were proved to be lower bounds on W ( , q) for Archimedean lattices in [26] (see also [29][30][31][32]) with q ≥ χ( ), using a coloring-matrix method that had been applied to derive a lower bound on W (sq, q) in [33]. The observation was that, for a given Archimedean lattice , if one sets q = −1 or q = 0 in the analytic expressions that had been proved in [26] to be lower bounds on W ( , q) for q ≥ χ( ), then the resultant values are consistent with being upper bounds on α( ) and α 0 ( ), respectively. ...
We infer upper and lower bounds on the exponential growth constants , , and describing the large-n behavior of, respectively, the number of acyclic orientations, acyclic orientations with a unique source vertex, and totally cyclic orientations of arrows on bonds of several n-vertex heteropolygonal Archimedean lattices . These are, to our knowledge, the best bounds on these growth constants. The inferred upper and lower bounds on the growth constants are quite close to each other, which enables us to infer rather accurate estimates for the actual exponential growth constants. Our new results for heteropolygonal Archimedean lattices, combined with our recent results for homopolygonal Archimedean lattices, are consistent with the inference that the exponential growth constants , , and on these lattices are monotonically increasing functions of the lattice coordination number. Comparisons are made with the corresponding growth constants for spanning trees on these lattices. Our findings provide further support for the Merino–Welsh and Conde–Merino conjectures.
... In [1] we noted our observation concerning analytic expressions that were proved to be lower bounds on W (Λ, q) for Archimedean lattices in [18] (see also [24]- [27]) with q ≥ χ(Λ), using a coloring-matrix method that had been applied to derive a lower bound on W (sq, q) in [28]. The observation was that, for a given Archimedean lattice Λ, if one sets q = −1 or q = 0 in the analytic expressions that had been proved in [18] to be lower bounds on W (Λ, q) for q ≥ χ(Λ), then the resultant values are consistent with being upper bounds on α(Λ) and α 0 (Λ), respectively. ...
We infer upper and lower bounds on the exponential growth constants , , and describing the large-n behavior of, respectively, the number of acyclic orientations, acyclic orientations with a unique source vertex, and totally cyclic orientations of arrows on bonds of several n-vertex heteropolygonal Archimedean lattices . These are, to our knowledge, the best bounds on these growth constants. The inferred upper and lower bounds on the growth constants are quite close to each other, which enables us to derive rather accurate values for the actual exponential growth constants. Combining our new results for heteropolygonal Archimedean lattices with our recent results for homopolygonal Archimedean lattices, we show that the exponential growth constants , , and on these lattices are monotonically increasing functions of the lattice coordination number. Comparisons are made with the corresponding growth constants for spanning trees on these lattices. Our findings provide further support for the Merino-Welsh and Conde-Merino conjectures.
... In almost all cases the commonly used technique follows a transfer matrix formulation or something equivalent. This technique was also used in many enumeration problems dealing with square lattices, e.g., [5][6][7]. In [8], the authors developed a method for computing the Tutte polynomial of the square lattice based on transfer matrices. ...
Benzenoid systems are natural graph representation of benzenoid hydrocarbons. Many chemically and combinatorially interesting indices and polynomials for bezenoid systems have been widely researched by both chemists and graph theorists. The Tutte polynomial of benzenoid chains without branched hexagons has already been computed by the recursive method. In this paper, by multiple recursion schema, an explicit expression for the Tutte polynomial of benzenoid systems with exactly one branched hexagon is obtained in terms of the number of hexagons on three linear or kinked chains. As a by-product, the number of spanning trees for these kind of benzenoid systems is determined.
... For another basic case of p = q (i.e., it is equivalent to L(1, 1)-labeling problem), it is also shown to be NP-hard in [54], where the problem is called Distance-2 Graph Coloring Problem. The L(1, 1)-labeling problem is well studied also from the viewpoint of restricted classes of graphs, such as planar graphs [2,10,55,60,66], outerplanar graphs [3], and so on [8,53]. ...
... For another basic case of p = q (i.e., it is equivalent to L(1, 1)-labeling problem), it is also shown to be NP-hard in [54], where the problem is called Distance-2 Graph Coloring Problem. The L(1, 1)-labeling problem is well studied also from the viewpoint of restricted classes of graphs, such as planar graphs [2,10,55,60,66], outerplanar graphs [3], and so on [8,53]. ...
Distance constrained labeling problems, e.g., L(p,q)-labeling and (p,q)-total labeling, are originally motivated by the frequency assignment. From the viewpoint of theory, the upper bounds on the labeling numbers and the time complexity of finding a minimum labeling are intensively and extensively studied. In this paper, we survey the distance constrained labeling problems from algorithmic aspects, that is, computational complexity, approximability, exact computation, and so on.
... If we now assume L n T is self-dual, which is not strictly true because it is nonplanar, it is generally accepted ͑Ref. 56 Our objective is to find exact results for other evaluations T(L n ;x,y) but this includes some very difficult problems. ...
This is an invited survey on the relation between the partition function of the Potts model and the Tutte polynomial. On the assumption that the Potts model is more familiar we have concentrated on the latter and its interpretations. In particular we highlight the connections with Abelian sandpiles, counting problems on random graphs, error correcting codes, and the Ehrhart polynomial of a zonotope. Where possible we use the mean field and square lattice as illustrations. We also discuss in some detail the complexity issues involved. © 2000 American Institute of Physics.
... We also consider vertex colourings of the lattice; each vertex is coloured using labels from 0 to q − 1 for some fixed q, and again we prohibit certain local configurations. This is related to the chromatic polynomial and the anti-ferromagnetic Potts model (see for example [7,29,23,19]). When q = 2, there are only two possible configurations, while for q = 3 it is known that the growth rate is exactly (4/3) 3/2 = 1.5396007178 . . . ...
We analyse the capacity of several two-dimensional constraint families - the
exclusion, colouring, parity and charge model families. Using Baxter's corner
transfer matrix formalism combined with the corner transfer matrix
renormalisation group method of Nishino and Okunishi, we calculate very tight
lower bounds and estimates on the growth rate of these models. Our results
strongly improve previous known lower bounds, and lead to the surprising
conjecture that the capacity of the even and charge(3) constraints are
identical.
... The use of transfer-matrices is common in enumeration problems dealing with square lattices (see [2, 5]) but our approach is novel for computing Tutte polynomials. Let us mention that a different transfer-matrix approach is used in [1] for computing chromatic polynomials of square lattices. ...
In a recent paper Merino and Welsh (1999) studied several counting problems on the square lattice . The authors gave the following bounds for the asymptotics of f(n), the number of forests of , and , the number of acyclic orientations of : and . In this paper we improve these bounds as follows: and . We obtain this by developing a method for computing the Tutte polynomial of the square lattice and other related graphs based on transfer matrices.
We study properties of the Potts model partition function on m’th iterates of Hanoi graphs, , and use the results to draw inferences about the limit that yields a self-similar Hanoi fractal, . We also calculate the chromatic polynomials . From calculations of the configurational degeneracy, per vertex, of the zero-temperature Potts antiferromagnet on , denoted , estimates of , are given for q=3 and q=4 and compared with known values on other lattices. We compute the zeros of in the complex q plane for various values of the temperature-dependent variable and in the complex y plane for various values of q. These are consistent with accumulating to form loci denoted and , or equivalently, , in the limit. Our results motivate the inference that the maximal point at which crosses the real q axis, denoted , has the value and correspondingly, if , then crosses the real y axis at y=0, i.e., the Potts antiferromagnet on with has a T=0 critical point. Finally, we analyze the partition function zeros in the y plane for and show that these accumulate approximately along parts of the sides of an equilateral triangular with apex points that scale like and . Some comparisons are presented of these findings for Hanoi graphs with corresponding results on m’th iterates of Sierpinski gasket graphs and the limit yielding the Sierpinski gasket fractal.
We prove several theorems concerning Tutte polynomials T(G,x,y) for recursive families of graphs. In addition to its interest in mathematics, the Tutte polynomial is equivalent to an important function in statistical physics, the Potts model partition function of the q-state Potts model, Z(G,q,v), where v is a temperature-dependent variable. We determine the structure of the Tutte polynomial for a cyclic clan graph comprised of a chain of m copies of the complete graph such that the linkage L between each successive pair of 's is a join jn, and r and m are arbitrary. The explicit calculation of the case r=3 (for arbitrary m) is presented. The continuous accumulation set of the zeros of Z in the limit is considered. Further, we present calculations of two special cases of Tutte polynomials, namely, flow and reliability polynomials, for cyclic clan graphs and discuss the respective continuous accumulation sets of their zeros in the limit . Special valuations of Tutte polynomials give enumerations of spanning trees and acyclic orientations. Two theorems are presented that determine the number of spanning trees on and , where L=id means that the identity linkage. We report calculations of the number of acyclic orientations for strips of the square lattice and use these to obtain an improved lower bound on the exponential growth rate of the number of these acyclic orientations.
We study properties of the Potts model partition function on m'th iterates of Hanoi graphs, , and use the results to draw inferences about the limit that yields a self-similar Hanoi fractal, . We also calculate the chromatic polynomials . From calculations of the configurational degeneracy, per vertex, of the zero-temperature Potts antiferromagnet on , denoted , estimates of , are given for q=3 and q=4 and compared with known values on other lattices. We compute the zeros of in the complex q plane for various values of the temperature-dependent variable and in the complex y plane for various values of q. These are consistent with accumulating to form loci denoted and , or equivalently, , in the limit. Our results motivate the inference that the maximal point at which crosses the real q axis, denoted , has the value and correspondingly, if , then crosses the real y axis at y=0, i.e., the Potts antiferromagnet on with has a T=0 critical point. Finally, we analyze the partition function zeros in the y plane for and show that these accumulate approximately along parts of the sides of an equilateral triangular with apex points that scale like and . Some comparisons are presented of these findings for Hanoi graphs with corresponding results on m'th iterates of Sierpinski gasket graphs and the limit yielding the Sierpinski gasket fractal.
The transfer matrix is a powerful technique that can be applied to statistical mechanics systems as, for example, in the calculus of the entropy of the ice model. One interesting way to study such systems is to map it onto a three-color problem. In this paper, we explicitly build the transfer matrix for the three-color problem in order to calculate the number of possible configurations for finite systems with free, periodic in one direction and toroidal boundary conditions (periodic in both directions)
The ice-type model proposed by Linus Pauling to explain its entropy at low temperatures is here approached in a didactic way. We first present a theoretically estimated low-temperature entropy and compare it with numerical results. Then, we consider the mapping between this model and the three-colour problem, i.e. colouring a regular graph with coordination equal to 4 (a two-dimensional lattice) with three colours, for which we apply the transfer-matrix method to calculate all allowed configurations for two-dimensional square lattices of N oxygen atoms ranging from 4 to 225. Finally, from a linear regression of the transfer matrix results, we obtain an estimate for the case N → ∞ which is compared with the exact solution by Lieb.
We calculate exponential growth constants describing the asymptotic behavior of several quantities enumerating classes of orientations of arrow variables on the bonds of several types of directed lattice strip graphs G of finite width and arbitrarily great length, in the infinite-length limit, denoted {G}. Specifically, we calculate the exponential growth constants for (i) acyclic orientations, α({G}), (ii) acyclic orientations with a single source vertex, α0({G}), and (iii) totally cyclic orientations, β({G}). We consider several lattices, including square (sq), triangular (tri), and honeycomb (hc). From our calculations, we infer lower and upper bounds on these exponential growth constants for the respective infinite lattices. To our knowledge, these are the best current bounds on these quantities. Since our lower and upper bounds are quite close to each other, we can infer very accurate approximate values for the exponential growth constants, with fractional uncertainties ranging from O(10−4) to O(10−2). Further, we present exact values of α(tri), α0(tri), and β(hc) and use them to show that our lower and upper bounds on these quantities are very close to these exact values, even for modest strip widths. Results are also given for a nonplanar lattice denoted sqd. We show that α({G}), α0({G}), and β({G}) are monotonically increasing functions of vertex degree for these lattices. A comparison is given of these exponential growth constants with the corresponding exponential growth constant τ({G}) for spanning trees. Our results are in agreement with inequalities following from the Merino–Welsh and Conde–Merino conjectures.
We present generalized methods for calculating lower bounds on the
ground-state entropy per site, , or equivalently, the ground-state
degeneracy per site, , of the antiferromagnetic Potts model. We
use these methods to derive improved lower bounds on W for several lattices.
The q-state Potts antiferromagnet exhibits nonzero ground state entropy S0({G},q)≠0 for sufficiently large q on a given n-vertex lattice Λ or graph G and its n→∞ limit {G}. We present exact calculations of the zero-temperature partition function Z(G,q,T=0) and W({G},q), where , for this model on a number of families G. These calculations have interesting connections with graph theory, since Z(G,q,T=0)=P(G,q), where the chromatic polynomial P(G,q) gives the number of ways of coloring the vertices of the graph G such that no adjacent vertices have the same color. Generalizing q from to , we determine the accumulation set of the zeros of P(G,q), which constitute the continuous loci of points on which W is nonanalytic. The Potts antiferromagnet has a zero-temperature critical point at the maximal value qc where crosses the real q-axis. In particular, exact solutions for W and are given for infinitely long, finite-width strips of various lattices; in addition to their intrinsic interest, these yield insight into the approach to the 2D thermodynamic limit. Some corresponding results are presented for the exact finite-temperature Potts free energy on families of graphs. Finally, we present rigorous upper and lower bounds on W for 2D lattices.
Let T(Lm,n;x,y) be the Tutte polynomial of the square lattice Lm,n, for integers m,n∈Z>0. Using a family of Tutte polynomial inequalities established by the author in a previous work, we study the analytical properties of the sequences (T(Lm,n;x,y)1/mn:n∈Z>0) for a fixed m∈Z>0, and (T(Ln,n;x,y)1/n2:n∈Z>0), in the region x,y⩾1. We show that these sequences are monotonically increasing when (x−1)(y−1)>1. We also compute lower bounds for these limits when (x−1)(y−1)>1, and upper bounds when (x−1)(y−1)1. At the point (x=2, y=1), where the Tutte polynomial is known to count the number of forests, we compute limn→∞T(Ln,n;2,1)1/n2⩽3.705603, which improves upon the previous best upper bound of 3.74101 obtained by Calkin, Merino, Noble and Noy (2003).
The problem of nonzero ground-state entropy has been one of the most important subjects in statistical physics, condensed-matter physics, and mathematics. For the square-lattice Q-state Potts antiferromagnets with Q > 2, the ground-states are highly degenerate, and the nonzero ground-state entropies of these models occur without frustration. However, the exact values of the ground-state entropies for the square-lattice Q-state Potts antiferromagnets are not known except for Q = 3. We evaluate the exact integer values for the numbers of antiferromagnetic ground states on L × L square lattices by using the microcanonical transfer matrix (Q ≤ 10) and the random-cluster transfer matrix (Q > 10) and we estimate the values of the ground-state entropies in the limit L → ∞.
The ergodic theory of subadditive stochastic processes is used to prove a limit theorem for the rank polynomials of expanding finite subsections of an arbitrary lattice from which vertices have been deleted at random. A similar result holds for the chromatic polynomials of these subsections.
This paper is concerned with the chromatic polynomials of ‘bracelets’: specifically, graphs constructed by
taking n copies of a complete graph and linking them together in a ring. Using a sieve method, explicit
formulae for the dominant and subdominant terms of the chromatic polynomial are obtained. Finally, a
simple description of the effect of twisting the links is obtained.
There is no known polynomial time algorithm which generates a random forest or counts forests or acyclic orientations in general graphs. On the other hand, there is no technical reason why such algorithms should not exist. These are key questions in the theory of approximately evaluating the Tutte polynomial which in turn contains several other specializations of interest to statistical physics, such as the Ising, Potts, and random cluster models.Here, we consider these problems on the square lattice, which apart from its interest to statistical physics is, as we explain, also a crucial structure in complexity theory. We obtain some asymptotic counting results about these quantities on then n section of the square lattice together with some properties of the structure of the random forest. There are, however, many unanswered questions.
The chromatic polynomials of certain families of graphs can be expressed in terms of the eigenspaces of a linear operator. The operator is represented by a matrix, which is referred to here as the compatibility matrix. In this paper complete sets of eigenfunctions are obtained for several related families, and the results are used to provide information about the location of the zeros of the associated chromatic polynomials. A number of uniform features are observed, and these are explained in terms of general properties of the underlying construction.
We present exact calculations of the zero-temperature partition function for the q-state Potts antiferromagnet (equivalently, the chromatic polynomial) for families of arbitrarily long strip graphs of the square and triangular lattices with width Ly=4 and boundary conditions that are doubly periodic or doubly periodic with reversed orientation (i.e., of torus or Klein bottle type). These boundary conditions have the advantage of removing edge effects. In the limit of infinite length, we calculate the exponent of the entropy, W(q) and determine the continuous locus where it is singular. We also give results for toroidal strips involving “crossing subgraphs”; these make possible a unified treatment of torus and Klein bottle boundary conditions and enable us to prove that for a given strip, the locus is the same for these boundary conditions.
The zero-temperature q-state Potts model partition function for a lattice strip of fixed width Ly and arbitrary length Lx has the form , and is equivalent to the chromatic polynomial for this graph. We present exact zero-temperature partition functions for strips of several lattices with (FBCy,PBCx), i.e., cyclic, boundary conditions. In particular, the chromatic polynomial of a family of generalized dodecahedra graphs is calculated. The coefficient of degree d in q is , where Un(x) is the Chebyshev polynomial of the second kind. We also present the chromatic polynomial for the strip of the square lattice with (PBCy,PBCx), i.e., toroidal, boundary conditions and width Ly=4 with the property that each set of four vertical vertices forms a tetrahedron. A number of interesting and novel features of the continuous accumulation set of the chromatic zeros, are found.
We consider the q-state Potts model on families of self-dual strip graphs GD of the square lattice of width Ly and arbitrarily great length Lx, with periodic longitudinal boundary conditions. The general partition function Z and the T=0 antiferromagnetic special case P (chromatic polynomial) have the respective forms ∑j=1NF,Ly,λcF,Ly,j(λF,Ly,j)Lx, with F=Z,P. For arbitrary Ly, we determine (i) the general coefficient cF,Ly,j in terms of Chebyshev polynomials, (ii) the number nF(Ly,d) of terms with each type of coefficient, and (iii) the total number of terms NF,Ly,λ. We point out interesting connections between the nZ(Ly,d) and Temperley–Lieb algebras, and between the NF,Ly,λ and enumerations of directed lattice animals. Exact calculations of P are presented for 2⩽Ly⩽4. In the limit of infinite length, we calculate the ground state degeneracy per site (exponent of the ground state entropy), W(q). Generalizing q from to , we determine the continuous locus in the complex q plane where W(q) is singular. We find the interesting result that for all Ly values considered, the maximal point at which crosses the real q-axis, denoted qc, is the same, and is equal to the value for the infinite square lattice, qc=3. This is the first family of strip graphs of which we are aware that exhibits this type of universality of qc.
The q-state Potts model partition function (equivalent to the Tutte polynomial) for a lattice strip of fixed width Ly and arbitrary length Lx has the form , 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 for X=Z,P on cyclic and Möbius strip graphs of the square and triangular lattice. Combining these with a general expression for the (unique) coefficient cZ,G,j of degree d in q: , where Un(x) is the Chebyshev polynomial of the second kind, we determine the number of λZ,G,j's with coefficient c(d) in Z(G,q,v) for these cyclic strips of width Ly to be for 0⩽d⩽Ly and zero otherwise. For both cyclic and Möbius strips of these lattices, the total number of distinct eigenvalues λZ,G,j is calculated to be . Results are also presented for the analogous numbers nP(Ly,d) and NP,Ly,λ for P(G,q). We find that nP(Ly,0)=nP(Ly−1,1)=MLy−1 (Motzkin number), nZ(Ly,0)=CLy (the Catalan number), and give an exact expression for NP,Ly,λ. Our results for NZ,Ly,λ and NP,Ly,λ apply for both the cyclic and Möbius strips of both the square and triangular lattices; we also point out the interesting relations NZ,Ly,λ=2NDA,tri,Ly and NP,Ly,λ=2NDA,sq,Ly, where NDA,Λ,n denotes the number of directed lattice animals on the lattice Λ. We find the asymptotic growths NZ,Ly,λ∼Ly−1/24Ly and NP,Ly,λ∼Ly−1/23Ly as Ly→∞. Some general geometric identities for Potts model partition functions are also presented.
We present exact calculations of the partition function Z of the q-state Potts model and its generalization to real q, for arbitrary temperature on n-vertex ladder graphs, i.e., strips of the square lattice with width Ly=2 and arbitrary length Lx, with free, cyclic, and Möbius longitudinal boundary conditions. These partition functions are equivalent to Tutte/Whitney polynomials for these graphs. The free energy is calculated exactly for the infinite-length limit of these ladder graphs and the thermodynamics is discussed. By comparison with strip graphs of other widths, we analyze how the singularities at the zero-temperature critical point of the ferromagnet on infinite-length, finite-width strips depend on the width. We point out and study the following noncommutativity at certain special values . It is shown that the Potts antiferromagnet on both the infinite-length line and ladder graphs with cyclic or Möbius boundary conditions exhibits a phase transition at finite temperature if 0<q<2, but with unphysical properties, including negative specific heat and non-existence, in the low-temperature phase, of an n→∞ limit for thermodynamic functions that is independent of boundary conditions. Considering the full generalization to arbitrary complex q and temperature, we determine the singular locus in the corresponding space, arising as the accumulation set of partition function zeros as n→∞. In particular, we study the connection with the T=0 limit of the Potts antiferromagnet where reduces to the accumulation set of chromatic zeros. Certain properties of the complex-temperature phase diagrams are shown to exhibit close connections with those of the model on the square lattice, showing that exact solutions on infinite-length strips provide a way of gaining insight into these complex-temperature phase diagrams.
In this paper we present exact calculations of the partition function Z of the q-state Potts model and its generalization to real q, for arbitrary temperature on n-vertex strip graphs, of width Ly=2 and arbitrary length, of the triangular lattice with free, cyclic, and Möbius longitudinal boundary conditions. These partition functions are equivalent to Tutte/Whitney polynomials for these graphs. The free energy is calculated exactly for the infinite-length limit of the graphs, and the thermodynamics is discussed. Considering the full generalization to arbitrary complex q and temperature, we determine the singular locus in the corresponding space, arising as the accumulation set of partition function zeros as n→∞. In particular, we study the connection with the T=0 limit of the Potts antiferromagnet where reduces to the accumulation set of chromatic zeros. Comparisons are made with our previous exact calculation of Potts model partition functions for the corresponding strips of the square lattice. Our present calculations yield, as special cases, several quantities of graph-theoretic interest.
We present exact calculations of the partition function of the zero-temperature Potts antiferromagnet (equivalently, the chromatic polynomial) for graphs of arbitrarily great length composed of repeated complete subgraphs Kb with b=5,6 which have periodic or twisted periodic boundary condition in the longitudinal direction. In the Lx→∞ limit, the continuous accumulation set of the chromatic zeros is determined. We give some results for arbitrary b including the extrema of the eigenvalues with coefficients of degree b−1 and the explicit forms of some classes of eigenvalues. We prove that the maximal point where crosses the real axis, qc, satisfies the inequality qc⩽b for 2⩽b, the minimum value of q at which crosses the real q axis is q=0, and we make a conjecture concerning the structure of the chromatic polynomial for Klein bottle strips.
We present exact calculations of the partition function of the q-state Potts model for general q and temperature on strips of the square lattice of width Ly=3 vertices and arbitrary length Lx with periodic longitudinal boundary conditions, of the following types: (i) cyclic, (ii) Möbius, (iii) toroidal, and (iv) Klein bottle, where FBC and (T)PBC refer to free and (twisted) periodic boundary conditions. Results for the Ly=2 torus and Klein bottle strips are also included. In the infinite-length limit the thermodynamic properties are discussed and some general results are given for low-temperature behavior on strips of arbitrarily great width. We determine the submanifold in the space of q and temperature where the free energy is singular for these strips. Our calculations are also used to compute certain quantities of graph-theoretic interest.
We present exact solutions for the zero-temperature partition function (chromatic polynomial P) and the ground state degeneracy per site W(= exponent of the ground-state entropy) for the q-state Potts antiferromagnet on strips of the square lattice of width Ly vertices and arbitrarily great length Lx vertices. The specific solutions are for (a) Ly=4, (FBCy,PBCx) (cyclic); (b) Ly=4,(FBCy,TPBCx) (Möbius); (c) Ly=5,6,(PBCy,FBCx) (cylindrical); and (d) (open), where FBC,PBC, and TPBC denote free, periodic, and twisted periodic boundary conditions, respectively. In the Lx→∞ limit of each strip we discuss the analytic structure of W in the complex q plane. The respective W functions are evaluated numerically for various values of q. Several inferences are presented for the chromatic polynomials and analytic structure of W for lattice strips with arbitrarily great Ly. The absence of a nonpathological Lx→∞ limit for real nonintegral q in the interval 0<q<3(0<q<4) for strips of the square (triangular) lattice is discussed.
We present transfer matrices for the zero-temperature partition function of the q-state Potts antiferromagnet (equivalently, the chromatic polynomial) on cyclic and Möbius strips of the square, triangular, and honeycomb lattices of width Ly and arbitrarily great length Lx. We relate these results to our earlier exact solutions for square-lattice strips with Ly=3,4,5, triangular-lattice strips with Ly=2,3,4, and honeycomb-lattice strips with Ly=2,3 and periodic or twisted periodic boundary conditions. We give a general expression for the chromatic polynomial of a Möbius strip of a lattice Λ and exact results for a subset of honeycomb-lattice transfer matrices, both of which are valid for arbitrary strip width Ly. New results are presented for the Ly=5 strip of the triangular lattice and the Ly=4 and Ly=5 strips of the honeycomb lattice. Using these results and taking the infinite-length limit Lx→∞, we determine the continuous accumulation locus of the zeros of the above partition function in the complex q plane, including the maximal real point of nonanalyticity of the degeneracy per site, W as a function of q.
We present exact calculations of the partition function of the q-state Potts model on (i) open, (ii) cyclic, and (iii) Möbius strips of the honeycomb (brick) lattice of width Ly=2 and arbitrarily great length. In the infinite-length limit the thermodynamic properties are discussed. The continuous locus of singularities of the free energy is determined in the q plane for fixed temperature and in the complex temperature plane for fixed q values. We also give exact calculations of the zero-temperature partition function (chromatic polynomial) and W(q), the exponent of the ground-state entropy, for the Potts antiferromagnet for honeycomb strips of type (iv) Ly=3, cyclic, (v) Ly=3, Möbius, (vi) Ly=4, cylindrical, and (vii) Ly=4, open. In the infinite-length limit we calculate W(q) and determine the continuous locus of points where it is nonanalytic. We show that our exact calculation of the entropy for the Ly=4 strip with cylindrical boundary conditions provides an extremely accurate approximation, to a few parts in 105 for moderate q values, to the entropy for the full 2D honeycomb lattice (where the latter is determined by Monte Carlo measurements since no exact analytic form is known).
We present exact calculations of the zero-temperature partition function (chromatic polynomial) P for the q-state Potts antiferromagnet on triangular lattice strips of arbitrarily great length Lx vertices and of width Ly vertices and, in the Lx→∞ limit, the exponent of the ground state entropy, W=eS0/kB. The strips considered, with their boundary conditions (BC), are (a) (FBCy, PBCx) = cyclic for Ly=3, 4, (b) (FBCy, TPBCx) = Möbius, Ly=3, (c) (PBCy, PBCx) = toroidal, Ly=3, (d) (PBCy, TPBCx) = Klein bottle, Ly=3, (e) (PBCy, FBCx) = cylindrical, Ly=5, 6, and (f) (FBCy, FBCx) = free, Ly=5, where F, P, and TP denote free, periodic, and twisted periodic. Several interesting features are found, including the presence of terms in P proportional to cos(2πLx/3) for case (c). The continuous locus of points where W is nonanalytic in the q plane is discussed for each case and a comparative discussion is given of the respective loci for families with different boundary conditions. Numerical values of W are given for infinite-length strips of various widths and are shown to approach values for the 2D lattice rapidly. A remark is also made concerning a zero-free region for chromatic zeros. Some results are given for strips of other lattices.
We present exact calculations of the zero-temperature partition function of the q-state Potts antiferromagnet (equivalently the chromatic polynomial) for Möbius strips, with width Ly=2 or 3, of regular lattices and homeomorphic expansions thereof. These are compared with the corresponding partition functions for strip graphs with (untwisted) periodic longitudinal boundary conditions.
We prove several theorems concerning Tutte polynomials T(G,x,y) for recursive families of graphs. In addition to its interest in mathematics, the Tutte polynomial is equivalent to an important function in statistical physics, the partition function of the q-state Potts model, Z(G,q,v), where v is a temperature-dependent variable. These theorems determine the general structure of the Tutte polynomial for a homogeneous cyclic clan graph Jm(Kr) comprised of a chain of m copies of the complete graph Kr such that the linkage L between each successive pair of Kr's is a join, and r and m are arbitrary. The explicit calculation of the case r=3 (for arbitrary m) is presented. The continuous accumulation set of the zeros of Z in the limit m→∞ is considered. Further, we present calculations of two special cases of Tutte polynomials, namely flow and reliability polynomials, for homogeneous cyclic clan graphs and discuss the respective continuous accumulation sets of their zeros in the limit m→∞. Special valuations of Tutte polynomials give enumerations of spanning trees and acyclic orientations. Two theorems are presented that determine the number of spanning trees on Jm(Kr) and the graph Im(Kr) comprised of a chain of m copies of the complete graph Kr such that the linkage between each successive pair of Kr's is the identity linkage, and r and m are arbitrary. We report calculations of the number of acyclic orientations for strips of the square lattice and use these to suggest an improved lower bound on the exponential growth rate of the number of these acyclic orientations.
We present a set of general results on structural features of the q-state Potts model partition function Z(G,q,v) for arbitrary q and temperature Boltzmann variable v for various lattice strips of arbitrarily great width Ly vertices and length Lx vertices, including (i) cyclic and Möbius strips of the square and triangular lattices, and (ii) self-dual cyclic strips of the square lattice. We also present an exact solution for the chromatic polynomial for the cyclic and Möbius strips of the square lattice with width Ly=5 (the greatest width for which an exact solution has been obtained so far for these families). In the Lx→∞ limit, we calculate the ground-state degeneracy per site, W(q) and determine the boundary across which W(q) is singular in the complex q plane.
It is well known that counting -colourings () is #P-complete for general graphs, and also for several restricted classes such as bipartite planar graphs. On the other hand, it is known to be polynomial time computable for graphs of bounded tree-width. There is often special interest in counting colourings of square grids, and such graphs can be regarded as borderline graphs of unbounded tree-width in a specific sense. We are thus motivated to consider the complexity of counting colourings of subgraphs of the square grid. We show that the problem is #P-complete when . It remains #P-complete when restricted to induced subgraphs with maximum degree 3.
An explicit formula for the chromatic polynomials of certain families of graphs, called 'bracelets', is obtained. The terms correspond to irreducible representations of symmetric groups. The theory is developed using the standard bases for the Specht modules of representation theory, and leads to an eective means of calculation.
We present exact calculations of the chromatic polynomial and resultant ground state entropy of the q-state Potts antiferromagnet on lattice strips that are homeomorphic expansions of a strip of the kagomé lattice. The dependence of the ground state entropy on the form of homeomorphic expansion is elucidated.
We calculate rigorous lower bounds for the ground-state degeneracy per site, W, of the q-state Potts antiferromagnet on slabs of the simple cubic lattice that are infinite in two directions and finite in the third and that thus interpolate between the square (sq) and simple cubic (sc) lattices. We give a comparison with large-q series expansions for the sq and sc lattices and also present numerical comparisons.
We present exact calculations of the zero-temperature partition function (chromatic polynomial) and W(q), the exponent of the ground-state entropy, for the q-state Potts antiferromagnet with next-nearest-neighbor spin-spin couplings on square lattice strips, of width L(y)=3 and L(y)=4 vertices and arbitrarily great length Lx vertices, with both free and periodic boundary conditions. The resultant values of W for a range of physical q values are compared with each other and with the values for the full two-dimensional lattice. These results give insight into the effect of such nonnearest-neighbor couplings on the ground-state entropy. We show that the q=2 (Ising) and q=4 Potts antiferromagnets have zero-temperature critical points on the Lx-->infinity limits of the strips that we study. With the generalization of q from Z+ to C, we determine the analytic structure of W(q) in the q plane for the various cases.
We present exact solutions for the zero-temperature partition function of the q-state Potts antiferromagnet (equivalently, the chromatic polynomial P) on tube sections of the simple cubic lattice of fixed transverse size Lx x L(y) and arbitrarily great length L(z), for sizes Lx x L(y)=2 x 3 and 2 x 4 and boundary conditions (a) (FBC(x),FBC(y),FBC(z)) and (b) (PBC(x),FBC(y),FBC(z)), where FBC (PBC) denote free (periodic) boundary conditions. In the limit of infinite length, L(z)-->infinity, we calculate the resultant ground-state degeneracy per site W (=exponent of the ground-state entropy). Generalizing q from Z+ to C, we determine the analytic structure of W and the related singular locus Beta which is the continuous accumulation set of zeros of the chromatic polynomial. For the L(z)-->infinity limit of a given family of lattice sections, W is analytic for real q down to a value q(c). We determine the values of q(c) for the lattice sections considered and address the question of the value of q(c) for a d-dimensional Cartesian lattice. Analogous results are presented for a tube of arbitrarily great length whose transverse cross section is formed from the complete bipartite graph K(m,m).
We study the ground state degeneracy per site (exponent of the ground state entropy) for the q-state Potts antiferromagnet on infinitely long strips with width of 2D lattices with free and periodic boundary conditions in the y direction, denoted FBC and PBC. We show that the approach of W to its 2D thermodynamic limit as increases is quite rapid; for moderate values of q and , is within about 5 % and of the 2D value for FBC and PBC, respectively. The approach of W to the 2D thermodynamic limit is proved to be monotonic (non-monotonic) for FBC (PBC). It is noted that ground state entropy determinations on infinite strips can be used to obtain the central charge for cases with critical ground states. Comment: 21 pages, Latex, 1 figures; some misprints in Table 1 corrected
We prove a general rigorous lower bound for , the exponent of the ground state entropy of the q-state Potts antiferromagnet, on an arbitrary Archimedean lattice . We calculate large-q series expansions for the exact and compare these with our lower bounds on this function on the various Archimedean lattices. It is shown that the lower bounds coincide with a number of terms in the large-q expansions and hence serve not just as bounds but also as very good approximations to the respective exact functions for large q on the various lattices . Plots of are given, and the general dependence on lattice coordination number is noted. Lower bounds and series are also presented for the duals of Archimedean lattices. As part of the study, the chromatic number is determined for all Archimedean lattices and their duals. Finally, we report calculations of chromatic zeros for several lattices; these provide further support for our earlier conjecture that a sufficient condition for to be analytic at 1/q=0 is that is a regular lattice. Comment: 39 pages, Revtex, 9 encapsulated postscript figures, to appear in Phys. Rev. E
We report several results concerning , the exponent of the ground state entropy of the Potts antiferromagnet on a lattice . First, we improve our previous rigorous lower bound on W(hc,q) for the honeycomb (hc) lattice and find that it is extremely accurate; it agrees to the first eleven terms with the large-q series for W(hc,q). Second, we investigate the heteropolygonal Archimedean lattice, derive a rigorous lower bound, on , and calculate the large-q series for this function to where . Remarkably, these agree exactly to all thirteen terms calculated. We also report Monte Carlo measurements, and find that these are very close to our lower bound and series. Third, we study the effect of non-nearest-neighbor couplings, focusing on the square lattice with next-nearest-neighbor bonds. Comment: 13 pages, Latex, to appear in Phys. Rev. E
We derive rigorous upper and lower bounds for the ground state entropy of the q-state Potts antiferromagnet on the honeycomb and triangular lattices. These bounds are quite restrictive, especially for large q. Comment: 8 pages, latex, with two eps figures
At low temperatures, ice has a residual entropy, presumably caused by an indeterminacy in the positions of the hydrogen atoms. While the oxygen atoms are in a regular lattice, each O-H-O bond permits two possible positions for the hydrogen atom, subject to certain constraints called the “ice condition.” The statement of the problem in two dimensions is to find the number of ways of drawing arrows on the bonds of a square planar net so that precisely two arrows point into each vertex. If N is the number of molecules and (for large N) W
N
is the number of arrangements, then S = Nk lnW. Our exact result is W = (4/3)3/2.
The evaluation of the partition function of the 2‐dimensional ice model is equivalent to counting the number of ways of coloring the faces of the square lattice with three colors so that no two adjacent faces are colored alike. In this paper we solve a generalized problem in which activities are associated with the colors. If one of the colors is regarded as a particle and the others as forming a background, then the model is reminiscent of the hard‐square lattice gas. It is found to undergo a phase transition with infinite compressibility at the density ρ = 1∕3.
It is shown that the free and periodic boundary conditions are completely equivalent for the ice‐rule (six‐vertex) models in zero field. With an external direct or staggered field, we establish that in an ice‐rule model the free and periodic boundary conditions are equivalent, and also equal to some special boundary conditions, either at sufficiently low temperatures or with sufficiently high fields in the appropriate direction. Regions of constant direct polarization are found. We also establish the existence of the spontaneous staggered polarization in an antiferroelectric using the Peierls argument.
A recursive family of graphs is defined as a sequence of graphs whose Tutte polynomials satisfy a homogeneous linear recurrence relation. Some necessary conditions for a family to be recursive are proved, and the theory is applied to the families of graphs known as the prisms and the Möbius ladders to give the chromatic polynomials and complexity of these graphs. It is conjectured that there is some function B(k) such that the chromatic roots of all regular graphs of valency k lie in the disc | u | ≤ B(k). For the prisms and Möbius ladders it is shown that the chromatic roots lie in the disc | u | ≤ 3.
This is a substantial revision of a much-quoted monograph, first published in 1974. The structure is unchanged, but the text has been clarified and the notation brought into line with current practice. A large number of 'Additional Results' are included at the end of each chapter, thereby covering most of the major advances in the last twenty years. Professor Biggs' basic aim remains to express properties of graphs in algebraic terms, then to deduce theorems about them. In the first part, he tackles the applications of linear algebra and matrix theory to the study of graphs; algebraic constructions such as adjacency matrix and the incidence matrix and their applications are discussed in depth. There follows an extensive account of the theory of chromatic polynomials, a subject which has strong links with the 'interaction models' studied in theoretical physics, and the theory of knots. The last part deals with symmetry and regularity properties. Here there are important connections with other branches of algebraic combinatorics and group theory. This new and enlarged edition this will be essential reading for a wide range of mathematicians, computer scientists and theoretical physicists.