Counting complex disordered states by efficient pattern matching: chromatic polynomials and Potts partition functions

Timme, Marc; Bussel, Frank Van; Fliegner, Denny; Stolzenberg, Sebastian

doi:10.1088/1367-2630/11/2/023001

Cited by 3 publications

(10 citation statements)

References 29 publications

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“…Ideally we would like to obtain results for larger N for additional verification. However, while the new method [17] does achieve a significant speedup, the problem of computing the chromatic polynomial remains difficult and so there is a limit to feasible N, P values.…”

Section: Discussionmentioning

confidence: 99%

“…of the chromatic polynomial in terms of sums over polynomials in Kronecker deltas δ σ i σ j seems particularly suited for computations [17] and also directly shows a link to Potts partition functions in statistical physics (see below). Here every σ = (σ 1 , .…”

Section: Chromatic Polynomials Partition Functions and Their Zerosmentioning

confidence: 99%

“…In this Communication, we start to fill this gap using a promising vertex based, symbolic pattern matching method developed recently [17] to compute chromatic polynomials of moderately sized graphs. Like other techniques developed for physics applications (cf.…”

Section: Introductionmentioning

confidence: 99%

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Chromatic polynomials of random graphs

Bussel¹,

Ehrlich²,

Fliegner³

et al. 2010

J. Phys. A: Math. Theor.

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Chromatic polynomials and related graph invariants are central objects in both graph theory and statistical physics. Computational difficulties, however, have so far restricted studies of such polynomials to graphs that were either very small, very sparse, or highly structured. Recent algorithmic advances (New J. Phys. 11:023001, 2009) now make it possible to compute chromatic polynomials for moderately sized graphs of arbitrary structure and number of edges. Here we present chromatic polynomials of ensembles of random graphs with up to 30 vertices, over the entire range of edge density. We specifically focus on the locations of the zeros of the polynomial in the complex plane. The results indicate that the chromatic zeros of random graphs have a very consistent layout. In particular, the crossing point, the point at which the chromatic zeros with non-zero imaginary part approach the real axis, scales linearly with the average degree over most of the density range. While the scaling laws obtained are purely empirical, if they continue to hold in general there are significant implications: the crossing points of chromatic zeros in the thermodynamic limit separate systems with zero ground state entropy from systems with positive ground state entropy, the latter an exception to the third law of thermodynamics.

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Section: Discussionmentioning

confidence: 99%

Section: Chromatic Polynomials Partition Functions and Their Zerosmentioning

confidence: 99%

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Chromatic polynomials of random graphs

Bussel¹,

Ehrlich²,

Fliegner³

et al. 2010

J. Phys. A: Math. Theor.

Self Cite

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show abstract

“…Timme et al's innovative techniques [29] have allowed scientists to consider the chromatic polynomials of certain large graphs. In this paper, the chromatic polynomial of the 4 × 4 × 4 grid graph with 64 vertices and 144 edges was reported to be computed exactly in 11 hours on a single Linux machine with an Intel Pentium 4, 2.8GHz-32 bit processor.…”

Section: Run-timementioning

confidence: 99%

“…Unfortunately, computing P (G, x) for a general graph G is known to be #P -hard [16,22] and deciding whether or not a graph is k-colorable is N P -hard [11]. Polynomial-time algorithms have been found for certain subclasses of graphs, including chordal graphs [21] and graphs of bounded clique-width [12,20], and recent advances have made it feasible to study P (G, x) for graphs of up to thirty vertices [29,30]. Still, the best known algorithm for computing P (G, x) for an arbitrary graph G of order n has complexity O(2 n n O (1) ) [4] and the best current implementation is limited to 2|E(G)| + |V (G)| < 950 and |V (G)| < 65 [14,15].…”

Section: Introductionmentioning

confidence: 99%

Approximating the Chromatic Polynomial

Kemper,

Beichl

2016

Preprint

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Chromatic polynomials are important objects in graph theory and statistical physics, but as a result of computational difficulties, their study is limited to graphs that are small, highly structured, or very sparse. We have devised and implemented two algorithms that approximate the coefficients of the chromatic polynomial P (G, x), where P (G, k) is the number of proper k-colorings of a graph G for k ∈ N. Our algorithm is based on a method of Knuth that estimates the order of a search tree. We compare our results to the true chromatic polynomial in several known cases, and compare our error with previous approximation algorithms.

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The multithreaded version of FORM

Tentyukov

Vermaseren

2010

Computer Physics Communications

227

211

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We present TFORM, the version of the symbolic manipulation system FORM that can make simultaneous use of several processors in a shared memory architecture. The implementation uses Posix threads, also called pthreads, and is therefore easily portable between various operating systems. Most existing FORM programs will be able to take advantage of the increased processing power, without the need for modifications. In some cases some minor additions may be needed. For a computer with two processors a typical improvement factor in the running time is 1.7 when compared to the traditional version of FORM. In the case of computers with 4 processors a typical improvement factor in the execution time is slightly above 3.

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Counting complex disordered states by efficient pattern matching: chromatic polynomials and Potts partition functions

Cited by 3 publications

References 29 publications

Chromatic polynomials of random graphs

Chromatic polynomials of random graphs

Approximating the Chromatic Polynomial

The multithreaded version of FORM

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