Spanning trees on the Sierpinski gasket

TL;DR

This paper calculates the number of spanning trees on Sierpinski gaskets of various dimensions and generalizations, providing explicit formulas and conjectures for arbitrary dimensions.

Contribution

It provides exact counts for spanning trees on Sierpinski gaskets of dimensions 2, 3, and 4, and conjectures a general formula for any dimension.

Findings

Exact number of spanning trees for $SG_2(n)$, $SG_3(n)$, and $SG_4(n)$.

Explicit counts for generalized Sierpinski gaskets with $b=3,4$.

A conjectured general expression for arbitrary dimension $d$.

Abstract

We obtain the numbers of spanning trees on the Sierpinski gasket $SG_d(n)$ with dimension $d$ equal to two, three and four. The general expression for the number of spanning trees on $SG_d(n)$ with arbitrary $d$ is conjectured. The numbers of spanning trees on the generalized Sierpinski gasket $SG_{d,b}(n)$ with $d=2$ and $b=3,4$ are also obtained.