605 research outputs found

### Ground State Entropy in Potts Antiferromagnets and Chromatic Polynomials

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--$q$ 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

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 $L_x \times
L_y$ and arbitrarily great length $L_z$, for sizes $L_x \times L_y = 2 \times
3$ and $2 \times 4$ 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, $L_z \to \infty$, we calculate the
resultant ground state degeneracy per site $W$ (= exponent of the ground-state
entropy). Generalizing $q$ from ${\mathbb Z}_+$ to ${\mathbb C}$, we determine
the analytic structure of $W$ and the related singular locus ${\cal B}$ which
is the continuous accumulation set of zeros of the chromatic polynomial. For
the $L_z \to \infty$ 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 $K_{m,m}$.Comment: 28 pages, latex, six postscript figures, two Mathematica file

### Ground State Entropy of Potts Antiferromagnets on Cyclic Polygon Chain Graphs

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, $W$, 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, ${\cal B}$, of nonanalyticities in $W$. 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 $S_0 > 0$.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

Denoting $P(G,q)$ as the chromatic polynomial for coloring an $n$-vertex
graph $G$ with $q$ colors, and considering the limiting function $W(\{G\},q) =
\lim_{n \to \infty}P(G,q)^{1/n}$, 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 $W_r$ 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 $\Lambda$. (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

We present an analysis of the structure and properties of chromatic
polynomials $P(G_{pt,\vec m},q)$ of one-parameter and multi-parameter families
of planar triangulation graphs $G_{pt,\vec m}$, where ${\vec m} =
(m_1,...,m_p)$ is a vector of integer parameters. We use these to study the
ratio of $|P(G_{pt,\vec m},\tau+1)|$ to the Tutte upper bound $(\tau-1)^{n-5}$,
where $\tau=(1+\sqrt{5} \ )/2$ and $n$ is the number of vertices in $G_{pt,\vec
m}$. In particular, we calculate limiting values of this ratio as $n \to
\infty$ 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 $G_{pt,\vec m}$ with $p=1$ and $p=2$ and show that these have
a structure of the form $P(G_{pt,m},q) = c_{_{G_{pt}},1}\lambda_1^m +
c_{_{G_{pt}},2}\lambda_2^m + c_{_{G_{pt}},3}\lambda_3^m$ for $p=1$, where
$\lambda_1=q-2$, $\lambda_2=q-3$, and $\lambda_3=-1$, and $P(G_{pt,\vec m},q) =
\sum_{i_1=1}^3 \sum_{i_2=1}^3 c_{_{G_{pt}},i_1 i_2}
\lambda_{i_1}^{m_1}\lambda_{i_2}^{m_2}$ for $p=2$. We derive properties of the
coefficients $c_{_{G_{pt}},\vec i}$ and show that $P(G_{pt,\vec m},q)$ has a
real chromatic zero that approaches $(1/2)(3+\sqrt{5} \ )$ as one or more of
the $m_i \to \infty$. The generalization to $p \ge 3$ is given. Further, we
present a one-parameter family of planar triangulations with real zeros that
approach 3 from below as $m \to \infty$. 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

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