Study of dimer–monomer on the generalized Hanoi graph

Abstract

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

Solving the dimer problem on Apollonian gasket

“…When the considered dimer configurations are not close-packed, i.e., when one deals with monomer-dimer configurations, all quantites of interest become much more difficult to determine; see, for example, a recent paper concerned with generalized Tower of Hanoi graphs [7]. It would be interesting to examine whether some progress could be made by using the approach of the present paper.…”

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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.…”