Dimer-monomer Model on the Towers of Hanoi Graphs

TL;DR

This paper investigates the asymptotic behavior of dimer-monomers on Towers of Hanoi and related Sierpiński graphs, deriving recursion relations and tight bounds for the entropy per site with high numerical precision.

Contribution

It introduces recursion relations and tight bounds for the entropy of dimer-monomers on these fractal graphs, enabling highly accurate numerical estimates.

Findings

Derived recursion relations for dimer-monomers

Established upper and lower bounds for entropy per site

Achieved numerical entropy values with over a hundred significant figures

Abstract

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.