Study of dimer–monomer on the generalized Hanoi graph

Abstract

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.