Dimer-monomer Model on the Towers of Hanoi Graphs

The number of dimer-monomers (matchings) of a graph $G$ is an important graph parameter in statistical physics. Following recent research, we study the asymptotic behavior of the number of dimer-monomers $m(G)$ on the Towers of Hanoi graphs and another variation of the Sierpi\'{n}ski graphs which is similar to the Towers of Hanoi graphs, and derive the recursion relations for the numbers of dimer-monomers. Upper and lower bounds for the entropy per site, defined as $\mu_{G}=\lim_{v(G)\rightarrow\infty}\frac{\ln m(G)}{v(G)}$, where $v(G)$ is the number of vertices in a graph $G$, on these Sierpi\'{n}ski graphs are derived in terms of the numbers at a certain stage. As the difference between these bounds converges quickly to zero as the calculated stage increases, the numerical value of the entropy can be evaluated with more than a hundred significant figures accuracy.