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Structure of spanning trees on the two-dimensional Sierpinski gasket

Abstract : Consider spanning trees on the two-dimensional Sierpinski gasket SG(n) where stage n is a non-negative integer. For any given vertex x of SG(n), we derive rigorously the probability distribution of the degree j ∈{1,2,3,4} at the vertex and its value in the infinite n limit. Adding up such probabilities of all the vertices divided by the number of vertices, we obtain the average probability distribution of the degree j. The corresponding limiting distribution φj gives the average probability that a vertex is connected by 1, 2, 3 or 4 bond(s) among all the spanning tree configurations. They are rational numbers given as φ1=10957/40464, φ2=6626035/13636368, φ3=2943139/13636368, φ4=124895/4545456.
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Shu-Chiuan Chang, Lung-Chi Chen. Structure of spanning trees on the two-dimensional Sierpinski gasket. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2011, special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity, 12 (3), pp.151-176. ⟨hal-00993549⟩

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