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Study of dimer-monomer on the generalized Hanoi graph. (English) Zbl 1449.05032

Summary: We study the number of dimer-monomers \(M_d(n)\) on the Hanoi graphs \(H_d(n)\) at stage \(n\) with dimension \(d\) equal to 3 and 4. The entropy per site is defined as \(z_{H_d}=\lim_{v \rightarrow \infty}\ln M_d(n)/v\), where \(v\) is the number of vertices on \(H_d(n)\). 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 values of \(z_{H_d}\) for \(d=3, 4\) are 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 \(z_{H_d}\) with arbitrary \(d\).

MSC:

05A20 Combinatorial inequalities
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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