  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.
