Clique polynomials and independent set polynomials of graphs

Hoede, C.; Li, Xueliang

doi:10.1016/0012-365x(94)90163-5

Cited by 128 publications

(80 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Traditionally the independent-set polynomial is defined as a univariate polynomial Z G (w) in which w x is set equal to the same value w for all vertices x [4,41,42,56,51,52,53,63,27,28,64,77]. But one of our main contentions in this paper is that Z G is more naturally understood as a multivariate polynomial; this allows us, in particular, to exploit the fact that Z G is multiaffine.…”

Section: The Repulsive Lattice Gasmentioning

confidence: 99%

“…In the special case of a hard-core self-repulsion and hard-core nearest-neighbor exclusion (i.e. no site can be multiply occupied and no pair of adjacent sites can be simultaneously occupied), the partition function of the lattice gas coincides with the independent-set polynomial in combinatorics (also known as the independence polynomial or the stable-set polynomial) [4,41,42,56,51,52,53,63,27,28,64,77,35]. Moreover, the hard-core lattice gas is the universal statistical-mechanical model in the sense that any statistical-mechanical model living on a vertex set V 0 can be mapped onto a gas of nonoverlapping "polymers" on V 0 , i.e.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Repulsive Lattice Gas, the Independent-Set Polynomial, and the Lov�sz Local Lemma

2005

View full text Add to dashboard Cite

We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovász local lemma in probabilistic combinatorics. We show that the conclusion of the Lovász local lemma holds for dependency graph G and probabilities {p x } if and only if the independent-set polynomial for G is nonvanishing in the polydisc of radii {p x }. Furthermore, we show that the usual proof of the Lovász local lemma -which provides a sufficient condition for this to occur -corresponds to a simple inductive argument for the nonvanishing of the independent-set polynomial in a polydisc, which was discovered implicitly by Shearer [98] and explicitly by Dobrushin [37,38]. We also present some refinements and extensions of both arguments, including a generalization of the Lovász local lemma that allows for "soft" dependencies. In addition, we prove some general properties of the partition function of a repulsive lattice gas, most of which are consequences of the alternating-sign property for the Mayer coefficients. We conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs.

show abstract

Section: The Repulsive Lattice Gasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Repulsive Lattice Gas, the Independent-Set Polynomial, and the Lov�sz Local Lemma

2005

View full text Add to dashboard Cite

show abstract

“…Such type of graph polynomial was ever introduced under the name "independent polynomial" for various utilities, see [25,24] with cited references therein. So we now recall some properties on the IS polynomial of a graph.…”

Section: Sextet Polynomialmentioning

confidence: 99%

“…So we now recall some properties on the IS polynomial of a graph. Following are three well-known lemmas [25,24].…”

Section: Sextet Polynomialmentioning

confidence: 99%

See 1 more Smart Citation

Clar and sextet polynomials of buckminsterfullerene

Shiu

Lam

Zhang

2003

Journal of Molecular Structure: THEOCHEM

View full text Add to dashboard Cite

Study of special types of boundary layer natural convection flow problems through the clique polynomial method

Kumbinarasaiah

Raghunatha

2021

Heat Trans

View full text Add to dashboard Cite

This study proposed a new novel approach called the clique polynomial method (CPM) and applied it to the system of highly nonlinear ordinary differential equations (ODEs) associated with different cases of boundary layer natural convection fluid flow problems. Here, three specific problems are considered: (1) laminar film condensation of a fluid-saturated flow with isothermal vertical plate; (2) natural convection boundary layer fluid flow; and (3) heat transfer boundary layer flow due to a stretching sheet. CPM results, which do not need small parameters, are compared with numerical results, and the outcomes are established to be in good agreement. The CPM is hugely proficient and a sensible choice for finding outcomes to this type of coupled nonlinear ODEs. The work is in outstanding accord for coupled nonlinear ODEs in engineering applications.

show abstract

Clique polynomials and independent set polynomials of graphs

Abstract: This paper introduces two kinds of graph polynomials, clique polynomial and independent set polynomial.The paper focuses on expansions of these polynomials. Some open problems are mentioned.

Cited by 128 publications

References 10 publications

The Repulsive Lattice Gas, the Independent-Set Polynomial, and the Lov�sz Local Lemma

The Repulsive Lattice Gas, the Independent-Set Polynomial, and the Lov�sz Local Lemma

Clar and sextet polynomials of buckminsterfullerene

Study of special types of boundary layer natural convection flow problems through the clique polynomial method

Contact Info

Product

Resources

About