605 research outputs found

    Ground State Entropy in Potts Antiferromagnets and Chromatic Polynomials

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    We discuss recent results on ground state entropy in Potts antiferromagnets and connections with chromatic polynomials. These include rigorous lower and upper bounds, Monte Carlo measurements, large--qq series, exact solutions, and studies of analytic properties. Some related results on Fisher zeros of Potts models are also mentioned.Comment: LATTICE98(spin) 3 pages, Late

    Exact T=0 Partition Functions for Potts Antiferromagnets on Sections of the Simple Cubic Lattice

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    We present exact solutions for the zero-temperature partition function of the qq-state Potts antiferromagnet (equivalently, the chromatic polynomial PP) on tube sections of the simple cubic lattice of fixed transverse size Lx×LyL_x \times L_y and arbitrarily great length LzL_z, for sizes Lx×Ly=2×3L_x \times L_y = 2 \times 3 and 2×42 \times 4 and boundary conditions (a) (FBCx,FBCy,FBCz)(FBC_x,FBC_y,FBC_z) and (b) (PBCx,FBCy,FBCz)(PBC_x,FBC_y,FBC_z), where FBCFBC (PBCPBC) denote free (periodic) boundary conditions. In the limit of infinite-length, LzL_z \to \infty, we calculate the resultant ground state degeneracy per site WW (= exponent of the ground-state entropy). Generalizing qq from Z+{\mathbb Z}_+ to C{\mathbb C}, we determine the analytic structure of WW and the related singular locus B{\cal B} which is the continuous accumulation set of zeros of the chromatic polynomial. For the LzL_z \to \infty limit of a given family of lattice sections, WW is analytic for real qq down to a value qcq_c. We determine the values of qcq_c for the lattice sections considered and address the question of the value of qcq_c for a dd-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 Km,mK_{m,m}.Comment: 28 pages, latex, six postscript figures, two Mathematica file

    Ground State Entropy of Potts Antiferromagnets on Cyclic Polygon Chain Graphs

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    We present exact calculations of chromatic polynomials for families of cyclic graphs consisting of linked polygons, where the polygons may be adjacent or separated by a given number of bonds. From these we calculate the (exponential of the) ground state entropy, WW, for the q-state Potts model on these graphs in the limit of infinitely many vertices. A number of properties are proved concerning the continuous locus, B{\cal B}, of nonanalyticities in WW. Our results provide further evidence for a general rule concerning the maximal region in the complex q plane to which one can analytically continue from the physical interval where S0>0S_0 > 0.Comment: 27 pages, Latex, 17 figs. J. Phys. A, in pres

    Families of Graphs with W_r({G},q) Functions That Are Nonanalytic at 1/q=0

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    Denoting P(G,q)P(G,q) as the chromatic polynomial for coloring an nn-vertex graph GG with qq colors, and considering the limiting function W({G},q)=limnP(G,q)1/nW(\{G\},q) = \lim_{n \to \infty}P(G,q)^{1/n}, a fundamental question in graph theory is the following: is Wr({G},q)=q1W({G},q)W_r(\{G\},q) = q^{-1}W(\{G\},q) analytic or not at the origin of the 1/q1/q plane? (where the complex generalization of qq is assumed). This question is also relevant in statistical mechanics because W({G},q)=exp(S0/kB)W(\{G\},q)=\exp(S_0/k_B), where S0S_0 is the ground state entropy of the qq-state Potts antiferromagnet on the lattice graph {G}\{G\}, and the analyticity of Wr({G},q)W_r(\{G\},q) at 1/q=01/q=0 is necessary for the large-qq series expansions of Wr({G},q)W_r(\{G\},q). Although WrW_r is analytic at 1/q=01/q=0 for many {G}\{G\}, there are some {G}\{G\} for which it is not; for these, WrW_r has no large-qq series expansion. It is important to understand the reason for this nonanalyticity. Here we give a general condition that determines whether or not a particular Wr({G},q)W_r(\{G\},q) is analytic at 1/q=01/q=0 and explains the nonanalyticity where it occurs. We also construct infinite families of graphs with WrW_r functions that are non-analytic at 1/q=01/q=0 and investigate the properties of these functions. Our results are consistent with the conjecture that a sufficient condition for Wr({G},q)W_r(\{G\},q) to be analytic at 1/q=01/q=0 is that {G}\{G\} is a regular lattice graph Λ\Lambda. (This is known not to be a necessary condition).Comment: 22 pages, Revtex, 4 encapsulated postscript figures, to appear in Phys. Rev.

    The Structure of Chromatic Polynomials of Planar Triangulation Graphs and Implications for Chromatic Zeros and Asymptotic Limiting Quantities

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    We present an analysis of the structure and properties of chromatic polynomials P(Gpt,m,q)P(G_{pt,\vec m},q) of one-parameter and multi-parameter families of planar triangulation graphs Gpt,mG_{pt,\vec m}, where m=(m1,...,mp){\vec m} = (m_1,...,m_p) is a vector of integer parameters. We use these to study the ratio of P(Gpt,m,τ+1)|P(G_{pt,\vec m},\tau+1)| to the Tutte upper bound (τ1)n5(\tau-1)^{n-5}, where τ=(1+5 )/2\tau=(1+\sqrt{5} \ )/2 and nn is the number of vertices in Gpt,mG_{pt,\vec m}. In particular, we calculate limiting values of this ratio as nn \to \infty for various families of planar triangulations. We also use our calculations to study zeros of these chromatic polynomials. We study a large class of families Gpt,mG_{pt,\vec m} with p=1p=1 and p=2p=2 and show that these have a structure of the form P(Gpt,m,q)=cGpt,1λ1m+cGpt,2λ2m+cGpt,3λ3mP(G_{pt,m},q) = c_{_{G_{pt}},1}\lambda_1^m + c_{_{G_{pt}},2}\lambda_2^m + c_{_{G_{pt}},3}\lambda_3^m for p=1p=1, where λ1=q2\lambda_1=q-2, λ2=q3\lambda_2=q-3, and λ3=1\lambda_3=-1, and P(Gpt,m,q)=i1=13i2=13cGpt,i1i2λi1m1λi2m2P(G_{pt,\vec m},q) = \sum_{i_1=1}^3 \sum_{i_2=1}^3 c_{_{G_{pt}},i_1 i_2} \lambda_{i_1}^{m_1}\lambda_{i_2}^{m_2} for p=2p=2. We derive properties of the coefficients cGpt,ic_{_{G_{pt}},\vec i} and show that P(Gpt,m,q)P(G_{pt,\vec m},q) has a real chromatic zero that approaches (1/2)(3+5 )(1/2)(3+\sqrt{5} \ ) as one or more of the mim_i \to \infty. The generalization to p3p \ge 3 is given. Further, we present a one-parameter family of planar triangulations with real zeros that approach 3 from below as mm \to \infty. Implications for the ground-state entropy of the Potts antiferromagnet are discussed.Comment: 57 pages, latex, 15 figure

    Families of Graphs With Chromatic Zeros Lying on Circles

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    We define an infinite set of families of graphs, which we call pp-wheels and denote (Wh)n(p)(Wh)^{(p)}_n, that generalize the wheel (p=1p=1) and biwheel (p=2p=2) graphs. The chromatic polynomial for (Wh)n(p)(Wh)^{(p)}_n is calculated, and remarkably simple properties of the chromatic zeros are found: (i) the real zeros occur at q=0,1,...p+1q=0,1,...p+1 for npn-p even and q=0,1,...p+2q=0,1,...p+2 for npn-p odd; and (ii) the complex zeros all lie, equally spaced, on the unit circle q(p+1)=1|q-(p+1)|=1 in the complex qq plane. In the nn \to \infty limit, the zeros on this circle merge to form a boundary curve separating two regions where the limiting function W({(Wh)(p)},q)W(\{(Wh)^{(p)}\},q) is analytic, viz., the exterior and interior of the above circle. Connections with statistical mechanics are noted.Comment: 8 pages, Late
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