Counting spanning trees on fractal graphs and their asymptotic complexity

TL;DR

This paper derives a closed-form solution for counting spanning trees on fractal graphs using spectral decimation and Kirchhoff's theorem, analyzing their asymptotic complexity across various fractals.

Contribution

It introduces a novel method combining spectral decimation with Kirchhoff's theorem to compute spanning trees on fractal graphs, providing explicit formulas and asymptotic analysis.

Findings

Closed-form solutions for spanning trees on several fractals

Asymptotic complexity constants for each fractal example

Bounds established for the complexity constants

Abstract

Using the method of spectral decimation and a modified version of Kirchhoff's Matrix-Tree Theorem, a closed form solution to the number of spanning trees on approximating graphs to a fully symmetric self-similar structure on a finitely ramified fractal is given in Theorem \ref{thm:maintheoremfull}. We show how spectral decimation implies the existence of the asymptotic complexity constant and obtain some bounds for it. Examples calculated include the Sierpinski Gasket, a non post critically finite analog of the Sierpinski Gasket, the Diamond fractal, and the Hexagasket. For each example, the asymptotic complexity constant is found.