5,920 research outputs found
A tree-decomposed transfer matrix for computing exact Potts model partition functions for arbitrary graphs, with applications to planar graph colourings
Combining tree decomposition and transfer matrix techniques provides a very
general algorithm for computing exact partition functions of statistical models
defined on arbitrary graphs. The algorithm is particularly efficient in the
case of planar graphs. We illustrate it by computing the Potts model partition
functions and chromatic polynomials (the number of proper vertex colourings
using Q colours) for large samples of random planar graphs with up to N=100
vertices. In the latter case, our algorithm yields a sub-exponential average
running time of ~ exp(1.516 sqrt(N)), a substantial improvement over the
exponential running time ~ exp(0.245 N) provided by the hitherto best known
algorithm. We study the statistics of chromatic roots of random planar graphs
in some detail, comparing the findings with results for finite pieces of a
regular lattice.Comment: 5 pages, 3 figures. Version 2 has been substantially expanded.
Version 3 shows that the worst-case running time is sub-exponential in the
number of vertice
Sabidussi Versus Hedetniemi for Three Variations of the Chromatic Number
We investigate vector chromatic number, Lovasz theta of the complement, and
quantum chromatic number from the perspective of graph homomorphisms. We prove
an analog of Sabidussi's theorem for each of these parameters, i.e. that for
each of the parameters, the value on the Cartesian product of graphs is equal
to the maximum of the values on the factors. We also prove an analog of
Hedetniemi's conjecture for Lovasz theta of the complement, i.e. that its value
on the categorical product of graphs is equal to the minimum of its values on
the factors. We conjecture that the analogous results hold for vector and
quantum chromatic number, and we prove that this is the case for some special
classes of graphs.Comment: 18 page
Zeros of the Potts Model Partition Function on Sierpinski Graphs
We calculate zeros of the -state Potts model partition function on
'th-iterate Sierpinski graphs, , in the variable and in a
temperature-like variable, . We infer some asymptotic properties of the loci
of zeros in the limit and relate these to thermodynamic
properties of the -state Potts ferromagnet and antiferromagnet on the
Sierpinski gasket fractal, .Comment: 6 pages, 8 figure
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