Study of dimer–monomer on the generalized Hanoi graph

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 zHd=limv→∞lnMd(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 zHd 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 zHd with arbitrary d.