Paper Info
Title
Counting complex disordered states by efficient pattern matching: chromatic polynomials and Potts partition functions
Abstract
Counting 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.
Year
DOI
Venue
2009
10.1088/1367-2630/11/2/023001
NEW JOURNAL OF PHYSICS
Keywords
Field
DocType
simple cubic,bottom up,top down,three dimensional,graph theory,chromatic polynomial,pattern matching,computer algebra,partition function
Discrete mathematics,Partition function (mathematics),Polynomial,Chromatic scale,Quantum mechanics,Computer science,Symbolic computation,Counting problem,Chromatic polynomial,Pattern matching,Critical graph
Journal
Volume
Issue
ISSN
11
2
1367-2630
Citations 
PageRank 
References 
1
0.37
5
Authors
4
Name
Order
Citations
PageRank
Marc Timme113615.21
Frank Van Bussel2264.49
Denny Fliegner310.71
Sebastian Stolzenberg410.37