We study the asymptotic limiting function W ({G}, q) = lim n→∞ P (G, q) 1/n , where P (G, q)is the chromatic polynomial for a graph G with n vertices. We first discuss a subtlety in the definition of W ({G}, q) resulting from the fact that at certain special points q s , the following limits do not commute:We then present exact calculations of W ({G}, q) and determine the corresponding analytic structure in the complex q plane for a number of families of graphs {G}, including circuits, wheels, biwheels, bipyramids, and (cyclic and twisted) ladders. We study the zeros of the corresponding chromatic polynomials and prove a theorem that for certain families of graphs, all but a finite number of the zeros lie exactly on a unit circle, whose position depends on the family. Using the connection of P (G, q) with the zero-temperature Potts antiferromagnet, we derive a theorem concerning the maximal finite real point of non-analyticity in W ({G}, q), denoted q c and apply this theorem to deduce that q c (sq) = 3 and q c (hc) = (3 + √ 5)/2 for the square and honeycomb lattices. Finally, numerical calculations of W (hc, q) and W (sq, q) are presented and compared with series expansions and bounds. *

We calculate the chromatic polynomials P ((Gs)m, q) and, from these, the asymptotic limiting functions W ({Gs}, q) = limn→∞ P (Gs, q) 1/n for families of n-vertex graphs (Gs)m comprised of m repeated subgraphs H adjoined to an initial graph I. These calculations of W ({Gs}, q) for infinitely long strips of varying widths yield important insights into properties of W (Λ, q) for two-dimensional lattices Λ. In turn, these results connect with statistical mechanics, since W (Λ, q) is the ground state degeneracy of the q-state Potts model on the lattice Λ. For our calculations, we develop and use a generating function method, which enables us to determine both the chromatic polynomials of finite strip graphs and the resultant W ({Gs}, q) function in the limit n → ∞. From this, we obtain the exact continuous locus of points B where W ({Gs}, q) is nonanalytic in the complex q plane. This locus is shown to consist of arcs which do not separate the q plane into disconnected regions. Zeros of chromatic polynomials are computed for finite strips and compared with the exact locus of singularities B. We find that as the width of the infinitely long strips is increased, the arcs comprising B elongate and move toward each other, which enables one to understand the origin of closed regions that result for the (infinite) 2D lattice. 05.20.-y, 75

We describe a Monte Carlo algorithm for doing simulations in classical statistical physics in a different way. Instead of sampling the probability distribution at a fixed temperature, a random walk is performed in energy space to extract an estimate for the density of states. The probability can be computed at any temperature by weighting the density of states by the appropriate Boltzmann factor. Thermodynamic properties can be determined from suitable derivatives of the partition function and, unlike ''standard'' methods, the free energy and entropy can also be computed directly. To demonstrate the simplicity and power of the algorithm, we apply it to models exhibiting first-order or second-order phase transitions.

We prove a general rigorous lower bound for W (Λ, q) = exp(S 0 (Λ, q)/k B ), the exponent of the ground state entropy of the q-state Potts antiferromagnet, on an arbitrary Archimedean lattice Λ. We calculate large-q series expansions for the exact W r (Λ, q) = q −1 W (Λ, q) 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 W r (Λ, q) for large q on the various lattices Λ.Plots of W r (Λ, q) 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 W r (Λ, q) to be *

The q-state Potts antiferromagnet on a lattice Λ exhibits nonzero ground state entropy S 0 = k B ln W for sufficiently large q and hence is an exception to the third law of thermodynamics. An outstanding challenge has been the calculation of W (sq, q) on the square (sq) lattice. We present here an exact calculation of W on an infinite-length cyclic strip of the square lattice which embodies the expected analytic properties of W (sq, q). Similar results are given for the kagomé lattice.

We calculate the chromatic polynomials P for n-vertex strip graphs of the form J( From these results we compute the asymptotic limiting function W = lim n→∞ P 1/n ; for q ∈ Z + this has physical significance as the ground state degeneracy per site (exponent of the ground state entropy) of the q-state Potts antiferromagnet on the given strip. In the complex q plane, W is an analytic function except on a certain continuous locus B. In contrast to the (

Denoting P (G, q) as the chromatic polynomial for coloring an n-vertex graph G with q colors, and considering the limiting function W ({G}, q) = lim n→∞ P (G, q) 1/n , a fundamental question in graph theory is the following: is W r ({G}, q) = q −1 W ({G}, q) analytic or not at the origin of the 1/q plane?(where the complex generalization of q is assumed). This question is also relevant in statistical mechanics because W ({G}, q) = exp(S 0 /k B ), where S 0 is the ground state entropy of the q-state Potts antiferromagnet on the lattice graph {G}, and the analyticity of W r ({G}, q) at 1/q = 0 is necessary for the large-q series expansions of W r ({G}, q). Although W r is analytic at 1/q = 0 for many {G}, there are some {G} for which it is not; for these, W r has no large-q 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 W r ({G}, q) is analytic at 1/q = 0 and explains the nonanalyticity where it occurs. We also construct infinite families of graphs with W r * email: shrock@insti.physics.sunysb.edu † email: tsai@insti.physics.sunysb.edu 1 functions that are non-analytic at 1/q = 0 and investigate the properties of these functions. Our results are consistent with the conjecture that a sufficient condition for W r ({G}, q) to be analytic at 1/q = 0 is that {G} is a regular lattice graph Λ. (This is known not to be a necessary condition). 05.20.-y, 64.60.C, 75.10.H

We present exact calculations of the zero-temperature partition function (chromatic polynomial) and the (exponent of the) ground-state entropy S 0 for the q-state Potts antiferromagnet on families of cyclic and twisted cyclic (Möbius) strip graphs composed of p-sided polygons. Our results suggest a general rule concerning the maximal region in the complex q plane to which one can analytically continue from the physical interval where S 0 > 0. The chromatic zeros and their accumulation set B exhibit the rather unusual property of including support for Re(q) < 0 and provide further evidence for a relevant conjecture. *