Dimer–monomer model on the Towers of Hanoi graphs

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

doi:10.1142/s0217979215501738

Cited by 10 publications

(11 citation statements)

References 25 publications

(30 reference statements)

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“…However, here we address the problem of perfect matching in the Hanoi graphs with one extreme vertex removed. Moreover, our technique is different from that in [46], but is partially similar to those in [47] and [48].…”

Section: Lemma 312mentioning

confidence: 85%

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Maximum matchings and minimum dominating sets in Apollonian networks and extended Tower of Hanoi graphs

Jin

Zhang

2017

Theoretical Computer Science

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The Apollonian networks display the remarkable power-law and small-world properties as observed in most realistic networked systems. Their dual graphs are extended Tower of Hanoi graphs, which are obtained from the Tower of Hanoi graphs by adding a special vertex linked to all its three extreme vertices. In this paper, we study analytically maximum matchings and minimum dominating sets in Apollonian networks and their dual graphs, both of which have found vast applications in various fields, e.g. structural controllability of complex networks. For both networks, we determine their matching number, domination number, the number of maximum matchings, as well as the number of minimum dominating sets.

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Section: Lemma 312mentioning

confidence: 85%

“…. ✷ It should be mentioned that the problem of all matchings in the Tower of Hanoi graphs has been previously studied [46]. However, here we address the problem of perfect matching in the Hanoi graphs with one extreme vertex removed.…”

Section: Lemma 312mentioning

confidence: 99%

Maximum matchings and minimum dominating sets in Apollonian networks and extended Tower of Hanoi graphs

Jin

Zhang

2017

Theoretical Computer Science

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“…Notice that although the lower and upper bounds given above are not exactly the same as those in [24], the convergent rate is equivalent. The lower and upper bounds given above apply to 2 ≤ d ≤ 4, and we conjecture that they are valid for any dimension d. We shall show that ω 3 (n) is an ascending function and α 3 (n) is a descending function here.…”

Section: )mentioning

confidence: 97%

“…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%

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Study of dimer–monomer on the generalized Hanoi graph

Chang

2020

Comp. Appl. Math.

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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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Silver Iodide‐mediated Insertion of Indium to Synthesize Water‐tolerant Organoindium Reagents and the Application in Cross‐coupling Reactions

Feng

et al. 2022

ChemistrySelect

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A direct synthesis method of efficient and mild water‐resistant organoindium reagents and their cross‐coupling reactions with aryl iodide by palladium‐catalyzed was described. At room temperature, organic indium reagents were mediated by AgI, and water‐tolerant alkyl indium reagents were directly and efficiently synthesized. The cross‐coupling reaction of organic indium reagents with various aryl halides could perform smoothly with palladium catalysis. The expected products were obtained in moderate to good yields, and the best yield can reach 99 %. These reactions are promising for their high reactivity, broad applicability, and functional group tolerance.

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Dimer–monomer model on the Towers of Hanoi graphs

Cited by 10 publications

References 25 publications

Maximum matchings and minimum dominating sets in Apollonian networks and extended Tower of Hanoi graphs

Maximum matchings and minimum dominating sets in Apollonian networks and extended Tower of Hanoi graphs

Study of dimer–monomer on the generalized Hanoi graph

Silver Iodide‐mediated Insertion of Indium to Synthesize Water‐tolerant Organoindium Reagents and the Application in Cross‐coupling Reactions

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