Asymptotic enumeration of independent sets on the Sierpinski gasket

Chang, Shu Chiuan; Chen, Lung Chi; Yan, Weigen

doi:10.2298/fil1301023c

Cited by 3 publications

(1 citation statement)

References 48 publications

(45 reference statements)

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“…It is of interest to consider independent sets on self-similar fractal lattices which have scaling invariance rather than translational invariance [35]. The recursion relations for the numbers of independent sets on the Sierpiński gasket were derived by Chang, Chen and Yan [6]. A special type of self-similar graph that has been of interest is the Hanoi graph, which has been extensively studied in several contexts [5,7,8,12,17,18,19,20,22,25,28,31].…”

Section: Introductionmentioning

confidence: 99%

Independent sets on the Towers of Hanoi graphs

Chen

Huang

et al. 2016

Ars Math. Contemp.

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The number of independent sets is equivalent to the partition function of the hardcore lattice gas model with nearest-neighbor exclusion and unit activity. In this article, we mainly study the number of independent sets i(H n) on the Tower of Hanoi graph H n at stage n, and derive the recursion relations for the numbers of independent sets. Upper and lower bounds for the asymptotic growth constant µ on the Towers of Hanoi graphs are derived in terms of the numbers at a certain stage, where µ = lim v→∞ ln i(G) v(G) and v(G) is the number of vertices in a graph G. Furthermore, we also consider the number of independent sets on the Sierpiński graphs which contain the Towers of Hanoi graphs as a special case.

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Section: Introductionmentioning

confidence: 99%

Independent sets on the Towers of Hanoi graphs

Chen

Huang

et al. 2016

Ars Math. Contemp.

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Asymptotic behavior of a generalized independent sets model on the two-dimensional Sierpinski gasket

Chang,

Chen

2024

Journal of Mathematical Physics

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We consider a generalized independent set model on the two-dimensional Sierpinski gasket SGn. Let A be a set of symbols and each vertex of SGn should take a symbol from A. Given a subset S⊂A such that the symbols of nearest-neighbor vertices are not allowed to be the forbidden blocks F={ij:i,j∈S}. The probability of each vertex of SGn to take a symbol in S is denoted by p ∈ (0, 1), and the probability of each vertex to take a symbol not in S is given by 1 − p. When A={0,1}, S={0}, and p = 1/2, this reduces to independent set model or golden mean shift in symbolic dynamics. We investigate the asymptotic behavior of this generalized model on the two-dimensional Sierpinski gasket and obtain upper and lower bounds of the entropy per site for p ∈ (0, 1).

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Independence polynomial and matching polynomial of the Koch network

Liao

Xie

2015

Int. J. Mod. Phys. B

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The lattice gas model and the monomer-dimer model are two classical models in statistical mechanics. It is well known that the partition functions of these two models are associated with the independence polynomial and the matching polynomial in graph theory, respectively. Both polynomials have been shown to belong to the “[Formula: see text]-complete” class, which indicate the problems are computationally “intractable”. We consider these two polynomials of the Koch networks which are scale-free with small-world effects. Explicit recurrences are derived, and explicit formulae are presented for the number of independent sets of a certain type.

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Asymptotic enumeration of independent sets on the Sierpinski gasket

Cited by 3 publications

References 48 publications

Independent sets on the Towers of Hanoi graphs

Independent sets on the Towers of Hanoi graphs

Asymptotic behavior of a generalized independent sets model on the two-dimensional Sierpinski gasket

Independence polynomial and matching polynomial of the Koch network

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