Independent sets on the Towers of Hanoi graphs

Chen, Hanlin; Wu, Renfang; Huang, Gui‐Xiang; Deng, Hanyuan

doi:10.26493/1855-3974.783.9b5

“…In [17], S. Wagner and I. Gutman wrote a survey of results and techniques on the Hosoya index and Merrifield-Simmons index. Other recent results on the number of independent sets can be found in [2,4,3,9].…”

Section: Introductionmentioning

confidence: 93%

The number of independent sets in a connected graph and its complement

Wei

¹

,

Hu

²

2018

Art Discrete Appl. Math.

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For a connected graph G, the total number of independent vertex sets (including the empty vertex set) is denoted by i(G). In this paper, we consider Nordhaus-Gaddum-type inequalities for the number of independent sets of a connected graph with connected complement. First we define a transformation on a graph that increases i(G) and i(G). Next, we obtain the minimum and maximum value of i(G)+i(G), where graph G is a tree T with connected complement and a unicyclic graph G with connected complement, respectively. In each case, we characterize the extremal graphs. Finally, we establish an upper bound on the i(G) in terms of the Wiener polarity index.

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“…In this study, we shall consider the ice model on the 3D Hanoi graph. Several models have been considered on the Hanoi graph [12][13][14][15][16][17]. The ice model on the 3D Hanoi graph contains many configurations and looks much more complex than that on the 2D Sierpinski gasket.…”

Section: Introductionmentioning

confidence: 99%

The ice model on the three-dimensional Hanoi graph

Chang,

Chen,

Yang

2023

J. Stat. Mech.

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We study the ice model (with Boltzmann factors equal to one) on the 3D Hanoi graph H(n) at stage n. We derive recursive equations for functions that provide the number of configurations for each type of restricted arrow diagram. The total number of configurations I(n) can be computed exactly using these functions. The entropy per site in the thermodynamic limit is S H = 2 3 ln 2 = 0.462 098 120 373 … for H(n). This is comparable to the entropy per site S s q = 3 2 ln 4 3 = 0.431 523 108 677 … for the square lattice (Lieb 1967 Phys. Rev. Lett. 18 692–4 and Lieb E H 1967 Phys. Rev. 162 162–72).

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“…The fractals with noninteger Hausdorff dimension can be constructed from certain basic shape [19,20]. A famous fractal is the Tower of Hanoi graph, and it has been discussed in different contexts [21][22][23].…”

Section: Introductionmentioning

confidence: 99%

“…The fractals with noninteger Hausdorff dimension can be constructed from certain basic shape [19,20]. A famous fractal is the Tower of Hanoi graph, and it has been discussed in different contexts [21][22][23].The dimer-monomer problem on the Tower of Hanoi graph with dimension d = 2 was discussed in [24]. In this article, we shall first recall some basic definitions in section II.…”

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confidence: 99%

Study of dimer–monomer on the generalized Hanoi graph

Li

¹

,

Chang

²

2020

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We study the number of dimer-monomers M d (n) on the Tower of Hanoi graphs T H d (n) at stage n with dimension d equal to 3 and 4. The entropy per site is defined aswhere v is the number of vertices on T H d (n). We obtain the lower and upper bounds of the entropy per site, and the convergence of these bounds approaches to zero rapidly when the calculated stage increases. The numerical value of z T H d is evaluated to more than a hundred digits correct. Using the results with d less than or equal to 4, we predict the general form of the lower and upper bounds for z T H d with arbitrary d.The dimer-monomer model is an interesting but elusive model in statistical mechanics [1][2][3]. In this model, a dimer is realized by a diatomic molecule with two neighboring sites attaching to a surface or lattice. For the sites that are not occupied by any dimers, they could be regarded as covered by monomers. Let us define N DM (G) to be the number of dimer-monomers on a graph G.The computation of the general dimer-monomer model remains to be a difficult problem [4], in contrast to the closed-packed dimer problem on planar lattices that had been discussed and solved more than fifty years ago [5][6][7]. Recent computation of close-packed dimers, dimers with a single monomer, and general dimer-monomer models on regular lattices are given in Refs. [8][9][10][11][12][13][14][15][16][17][18]. It is also interesting to discuss the dimer-monomer problem on fractals with scaling invariance but not translational invariance. The fractals with noninteger Hausdorff dimension can be constructed from certain basic shape [19,20]. A famous fractal is the Tower of Hanoi graph, and it has been discussed in different contexts [21][22][23].The dimer-monomer problem on the Tower of Hanoi graph with dimension d = 2 was discussed in [24]. In this article, we shall first recall some basic definitions in section II. In section III, we present the recursion relations for the number of dimer-monomers on T H d (n) with dimension d = 3, then enumerate the entropy per site using lower and upper bounds in details. The calculation for T H d (n) with dimension d = 4 will be given in section IV. In the last section, we shall predict the general form of the lower and upper bounds of the entropy per site for dimer-monomers on the Tower of Hanoi graph with arbitrary dimension. II. PRELIMINARIESIn this section, let us review some basic terminology. A graph G = (V, E) that is connected and has no loops is defined by the vertex (site) set V and edge (bond) set E [25,26].Denote v(G) = |V | as the number of vertices in G and e(G) = |E| as the number of edges.Two vertices a and b are neighboring if the edge ab is included in E. A matching of a graph G is an independent edge subset where the edges have no common vertices. The number of matching in G is denoted as N DM (G), which corresponds to the number of dimer-monomers in statistical mechanics. Although monomer and dimer weights can be associated to each

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Independent sets on the Towers of Hanoi graphs

Cited by 4 publications

References 36 publications

The number of independent sets in a connected graph and its complement

The number of independent sets in a connected graph and its complement

The ice model on the three-dimensional Hanoi graph

Study of dimer–monomer on the generalized Hanoi graph

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