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arXiv · 1602.01996

Counting spanning trees on fractal graphs and their asymptotic complexity

Jason A. Anema·Konstantinos Tsougkas
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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.

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Jason A. Anema, Konstantinos Tsougkas. 2016-02-05. Counting spanning trees on fractal graphs and their asymptotic complexity. https://doi.org/10.1088/1751-8113%2F49%2F35%2F355101

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