Counting spanning trees on fractal graphs and their asymptotic complexity
arXiv:1602.01996 · doi:10.1088/1751-8113/49/35/355101
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.
26 pages. arXiv admin note: substantial text overlap with arXiv:1211.7341