4,626 research outputs found
T=0 Partition Functions for Potts Antiferromagnets on Lattice Strips with Fully Periodic Boundary Conditions
We present exact calculations of the zero-temperature partition function for
the -state Potts antiferromagnet (equivalently, the chromatic polynomial)
for families of arbitrarily long strip graphs of the square and triangular
lattices with width 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, 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.Comment: 43 pages, latex, 4 postscript figure
An inertial lower bound for the chromatic number of a graph
Let ) and denote the chromatic and fractional chromatic
numbers of a graph , and let denote the inertia of .
We prove that:
1 + \max\left(\frac{n^+}{n^-} , \frac{n^-}{n^+}\right) \le \chi(G) \mbox{
and conjecture that } 1 + \max\left(\frac{n^+}{n^-} , \frac{n^-}{n^+}\right)
\le \chi_f(G)
We investigate extremal graphs for these bounds and demonstrate that this
inertial bound is not a lower bound for the vector chromatic number. We
conclude with a discussion of asymmetry between and , including some
Nordhaus-Gaddum bounds for inertia
Asymptotic Limits and Zeros of Chromatic Polynomials and Ground State Entropy of Potts Antiferromagnets
We study the asymptotic limiting function , where is the chromatic polynomial for a graph
with vertices. We first discuss a subtlety in the definition of
resulting from the fact that at certain special points , the
following limits do not commute: . We then
present exact calculations of and determine the corresponding
analytic structure in the complex plane for a number of families of graphs
, 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 with the zero-temperature Potts
antiferromagnet, we derive a theorem concerning the maximal finite real point
of non-analyticity in , denoted and apply this theorem to
deduce that and for the square and
honeycomb lattices. Finally, numerical calculations of and
are presented and compared with series expansions and bounds.Comment: 33 pages, Latex, 5 postscript figures, published version; includes
further comments on large-q serie
Exact Potts Model Partition Functions on Strips of the Honeycomb Lattice
We present exact calculations of the partition function of the -state
Potts model on (i) open, (ii) cyclic, and (iii) M\"obius strips of the
honeycomb (brick) lattice of width 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
plane for fixed temperature and in the complex temperature plane for fixed
values. We also give exact calculations of the zero-temperature partition
function (chromatic polynomial) and , the exponent of the ground-state
entropy, for the Potts antiferromagnet for honeycomb strips of type (iv)
, cyclic, (v) , M\"obius, (vi) , cylindrical, and (vii)
, open. In the infinite-length limit we calculate and determine
the continuous locus of points where it is nonanalytic. We show that our exact
calculation of the entropy for the strip with cylindrical boundary
conditions provides an extremely accurate approximation, to a few parts in
for moderate 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).Comment: 48 pages, latex, with encapsulated postscript figure
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