Number of connected spanning subgraphs on the Sierpinski gasket

TL;DR

This paper investigates the enumeration of connected spanning subgraphs on generalized Sierpinski gaskets across various dimensions and layers, deriving bounds and numerical values for their asymptotic growth constants.

Contribution

It provides new bounds and numerical estimates for the growth constants of connected spanning subgraphs on generalized Sierpinski gaskets.

Findings

Derived upper and lower bounds for growth constants.

Numerical values of growth constants for specific gasket configurations.

Extended understanding of subgraph enumeration on fractal structures.

Abstract

We study the number of connected spanning subgraphs $f_{d,b}(n)$ 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 and four for $d=2$. The upper and lower bounds for the asymptotic growth constant, defined as $z_{SG_{d,b}}=\lim_{v \to \infty} \ln f_{d,b}(n)/v$ where $v$ is the number of vertices, on $SG_{2,b}(n)$ with $b=2,3,4$ are derived in terms of the results at a certain stage. The numerical values of $z_{SG_{d,b}}$ are obtained.