Asymptotic enumeration of independent sets on the Sierpinski gasket

TL;DR

This paper investigates the asymptotic number of independent sets on generalized Sierpinski gaskets across various dimensions and layers, providing precise bounds and conjectures for their growth constants.

Contribution

It derives bounds for the asymptotic growth constants of independent sets on generalized Sierpinski gaskets and offers highly accurate numerical evaluations.

Findings

Derived bounds for growth constants on specific Sierpinski gaskets.

Numerical values of growth constants computed with over a hundred significant figures.

Conjectured bounds for growth constants in general dimensions.

Abstract

The number of independent sets is equivalent to the partition function of the hard-core lattice gas model with nearest-neighbor exclusion and unit activity. We study the number of independent sets $m_{d,b}(n)$ on the generalized Sierpinski gasket $SG_{d,b}(n)$ at stage $n$ with dimension $d$ equal to two, three and four for $b=2$, and layer $b$ equal to three for $d=2$. The upper and lower bounds for the asymptotic growth constant, defined as $z_{SG_{d,b}}=\lim_{v \to \infty} \ln m_{d,b}(n)/v$ where $v$ is the number of vertices, on these Sierpinski gaskets are derived in terms of the results at a certain stage. The numerical values of these $z_{SG_{d,b}}$ are evaluated with more than a hundred significant figures accurate. We also conjecture the upper and lower bounds for the asymptotic growth constant $z_{SG_{d,2}}$ with general $d$.