Counting Complex Disordered States by Efficient Pattern Matching: Chromatic Polynomials and Potts Partition Functions

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dc.creatorTimme, Marc
dc.creatorvan Bussel, Frank
dc.creatorFliegner, Denny
dc.creatorStolzenberg, Sebastian
dc.date2009-04-27
dc.date.accessioned2026-07-07T13:14:52Z
dc.date.available2026-07-07T13:14:52Z
dc.descriptionCounting problems, determining the number of possible states of a large system under certain constraints, play an important role in many areas of science. They naturally arise for complex disordered systems in physics and chemistry, in mathematical graph theory, and in computer science. Counting problems, however, are among the hardest problems to access computationally. Here, we suggest a novel method to access a benchmark counting problem, finding chromatic polynomials of graphs. We develop a vertex-oriented symbolic pattern matching algorithm that exploits the equivalence between the chromatic polynomial and the zero-temperature partition function of the Potts antiferromagnet on the same graph. Implementing this bottom-up algorithm using appropriate computer algebra, the new method outperforms standard top-down methods by several orders of magnitude, already for moderately sized graphs. As a first application, we compute chromatic polynomials of samples of the simple cubic lattice, for the first time computationally accessing three-dimensional lattices of physical relevance. The method offers straightforward generalizations to several other counting problems.
dc.description7 pages, 4 figures
dc.identifierhttps://arxiv.org/abs/0904.4010
dc.identifierhttp://arxiv.org/abs/0904.4010
dc.identifierNew J. Phys. 11:023001 (2009); freely available online
dc.identifierdoi:10.1088/1367-2630/11/2/023001
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/230313
dc.subjectStatistical Mechanics
dc.subjectDisordered Systems and Neural Networks
dc.subjectSymbolic Computation
dc.subjectCombinatorics
dc.titleCounting Complex Disordered States by Efficient Pattern Matching: Chromatic Polynomials and Potts Partition Functions
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