5,988 research outputs found
T=0 Partition Functions for Potts Antiferromagnets on Moebius Strips and Effects of Graph Topology
We present exact calculations of the zero-temperature partition function of
the -state Potts antiferromagnet (equivalently the chromatic polynomial) for
Moebius strips, with width 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.Comment: 9 pages, Latex, Phys. Lett. A, in pres
Chromatic Polynomials for Lattice Strips with Cyclic Boundary Conditions
The zero-temperature -state Potts model partition function for a lattice
strip of fixed width and arbitrary length 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
, i.e., cyclic, boundary conditions. In particular, the
chromatic polynomial of a family of generalized dodecahedra graphs is
calculated. The coefficient of degree in is
, where is the Chebyshev
polynomial of the second kind. We also present the chromatic polynomial for the
strip of the square lattice with , i.e., toroidal, boundary
conditions and width 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.Comment: 41 pages, latex, 18 figure
Bulk, surface and corner free energy series for the chromatic polynomial on the square and triangular lattices
We present an efficient algorithm for computing the partition function of the
q-colouring problem (chromatic polynomial) on regular two-dimensional lattice
strips. Our construction involves writing the transfer matrix as a product of
sparse matrices, each of dimension ~ 3^m, where m is the number of lattice
spacings across the strip. As a specific application, we obtain the large-q
series of the bulk, surface and corner free energies of the chromatic
polynomial. This extends the existing series for the square lattice by 32
terms, to order q^{-79}. On the triangular lattice, we verify Baxter's
analytical expression for the bulk free energy (to order q^{-40}), and we are
able to conjecture exact product formulae for the surface and corner free
energies.Comment: 17 pages. Version 2: added 4 further term to the serie
Boundary chromatic polynomial
We consider proper colorings of planar graphs embedded in the annulus, such
that vertices on one rim can take Q_s colors, while all remaining vertices can
take Q colors. The corresponding chromatic polynomial is related to the
partition function of a boundary loop model. Using results for the latter, the
phase diagram of the coloring problem (with real Q and Q_s) is inferred, in the
limits of two-dimensional or quasi one-dimensional infinite graphs. We find in
particular that the special role played by Beraha numbers Q=4 cos^2(pi/n) for
the usual chromatic polynomial does not extend to the case Q different from
Q_s. The agreement with (scarce) existing numerical results is perfect; further
numerical checks are presented here.Comment: 20 pages, 7 figure
A Penrose polynomial for embedded graphs
We extend the Penrose polynomial, originally defined only for plane graphs,
to graphs embedded in arbitrary surfaces. Considering this Penrose polynomial
of embedded graphs leads to new identities and relations for the Penrose
polynomial which can not be realized within the class of plane graphs. In
particular, by exploiting connections with the transition polynomial and the
ribbon group action, we find a deletion-contraction-type relation for the
Penrose polynomial. We relate the Penrose polynomial of an orientable
checkerboard colourable graph to the circuit partition polynomial of its medial
graph and use this to find new combinatorial interpretations of the Penrose
polynomial. We also show that the Penrose polynomial of a plane graph G can be
expressed as a sum of chromatic polynomials of twisted duals of G. This allows
us to obtain a new reformulation of the Four Colour Theorem
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