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Counting spanning trees on fractal graphs and their asymptotic complexity
Univ Illinois, Dept Math, Urbana, IL 61801 USA..
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
2016 (English)In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 49, no 35, article id 355101Article in journal (Refereed) Published
Abstract [en]

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 3.4. 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.

Place, publisher, year, edition, pages
2016. Vol. 49, no 35, article id 355101
Keywords [en]
fractal graphs, spanning trees, spectral decimation, asymptotic complexity
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-304155DOI: 10.1088/1751-8113/49/35/355101ISI: 000381302500005OAI: oai:DiVA.org:uu-304155DiVA, id: diva2:1014934
Available from: 2016-10-03 Created: 2016-10-03 Last updated: 2018-12-17Bibliographically approved
In thesis
1. Combinatorial and analytical problems for fractals and their graph approximations
Open this publication in new window or tab >>Combinatorial and analytical problems for fractals and their graph approximations
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The recent field of analysis on fractals has been studied under a probabilistic and analytic point of view. In this present work, we will focus on the analytic part developed by Kigami. The fractals we will be studying are finitely ramified self-similar sets, with emphasis on the post-critically finite ones. A prototype of the theory is the Sierpinski gasket. We can approximate the finitely ramified self-similar sets via a sequence of approximating graphs which allows us to use notions from discrete mathematics such as the combinatorial and probabilistic graph Laplacian on finite graphs. Through that approach or via Dirichlet forms, we can define the Laplace operator on the continuous fractal object itself via either a weak definition or as a renormalized limit of the discrete graph Laplacians on the graphs.

The aim of this present work is to study the graphs approximating the fractal and determine connections between the Laplace operator on the discrete graphs and the continuous object, the fractal itself.

In paper I, we study the number of spanning trees on the sequence of graphs approximating a self-similar set admitting spectral decimation.

In paper II, we study harmonic functions on p.c.f. self-similar sets. Unlike the standard Dirichlet problem and harmonic functions in Euclidean space, harmonic functions on these sets may be locally constant without being constant in their entire domain. In that case we say that the fractal has a degenerate harmonic structure. We prove that for a family of variants of the Sierpinski gasket the harmonic structure is non-degenerate.

In paper III, we investigate properties of the Kusuoka measure and the corresponding energy Laplacian on the Sierpinski gaskets of level k.

In papers IV and V, we establish a connection between the discrete combinatorial graph Laplacian determinant and the regularized determinant of the fractal itself. We establish that for a certain class of p.c.f. fractals the logarithm of the regularized determinant appears as a constant in the logarithm of the discrete combinatorial Laplacian.

Place, publisher, year, edition, pages
Uppsala: Department of Mathematics, 2019. p. 37
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 112
Keywords
Fractal graphs, energy Laplacian, Kusuoka measure
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:uu:diva-369918 (URN)978-91-506-2739-8 (ISBN)
Public defence
2019-02-15, Polhemsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
Opponent
Supervisors
Available from: 2019-01-23 Created: 2018-12-17 Last updated: 2019-01-23

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