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The Number of Spanning Trees in Self-Similar Graphs
Elmar Teufl1 and Stephan Wagner2
1Fachbereich Mathematik, Eberhard Karls Universit¨at T¨ubingen, Auf der Morgenstelle 10, 72076 T¨ubingen, Germany
elmar.teufl@uni-tuebingen.de
2Department of Mathematical Sciences, Stellenbosch University, Private Bag X1, Matieland 7602, South Africa
swagner@sun.ac.za
Annals of Combinatorics 15 (2) pp.355-380 April, 2011
AMS Subject Classification: 05C30; 05C05, 34B45
Abstract:
The number of spanning trees of a graph, also known as the complexity, is computed for graphs constructed by a replacement procedure yielding a self-similar structure. It is shown that under certain symmetry conditions exact formulas for the complexity can be given. These formulas indicate interesting connections to the theory of electrical networks. Examples include the well-known Sierpi´nski graphs and their higher-dimensional analogues. Several auxiliary results are provided on the way — for instance, a property of the number of rooted spanning forests is proven for graphs with a high degree of symmetry.
Keywords: spanning trees, self-similar graphs

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