Counting Spanning Trees on Fractal Graphs

Transcription

1 Jason Anema Cornell University 4th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals September 13, 2011

2 Spanning Trees Definition A Spanning Tree T = (V T, E T ) of a finite, connected, undirected graph G = (V G, E G ) is a tree such that V T = V G and E T E G. The number of spanning trees of such a graph G is denoted τ(g)

3 Spanning Trees Definition A Spanning Tree T = (V T, E T ) of a finite, connected, undirected graph G = (V G, E G ) is a tree such that V T = V G and E T E G. The number of spanning trees of such a graph G is denoted τ(g) Example By Cayley s formula it is know that τ(k n) = n n 2 where K n is the complete graph on n vertices.

4 Problem Statements Problem For a given fully symmetric self-similar structure on a finitely ramified fractal K, let V n denote its sequence of approximating graphs. What is τ(v n)?

5 Problem Statements Problem For a given fully symmetric self-similar structure on a finitely ramified fractal K, let V n denote its sequence of approximating graphs. What is τ(v n)? Answer It will later be show how to exactly compute τ(v n) for a given structure.

6 Problem Statements Problem For a given fully symmetric self-similar structure on a finitely ramified fractal K, with V n being it s sequence of approximating graphs. How does log(τ(v n)) grow asymptotically?

7 Problem Statements Problem For a given fully symmetric self-similar structure on a finitely ramified fractal K, with V n being it s sequence of approximating graphs. How does log(τ(v n)) grow asymptotically? Answer We could use the answer to the first problem to compute this, but there is a much more simple way. Information from the V 0 and V 1 graphs contain enough information to know the asymptotic growth of log(τ(v n)).

8 Kirchhoff s Matrix-tree theorem Theorem (Kirchhoff) Let G = (V, E) be a finite, connected, undirected, graph with n labeled vertices, let λ G 1,..., λ G n 1 be the non-zero eigenvalues of its graph Laplacian matrix and τ(g) denote the number of distinct spanning trees of G, then τ(g) = 1 n Y n 1 λ G i i=1

9 Kirchhoff s Matrix-tree theorem Theorem (Kirchhoff) Let G = (V, E) be a finite, connected, undirected, graph with n labeled vertices, let λ G 1,..., λ G n 1 be the non-zero eigenvalues of its graph Laplacian matrix and τ(g) denote the number of distinct spanning trees of G, then τ(g) = 1 n Y n 1 λ G i i=1 Note This graph Laplacian G is the (degree matrix) (adjacency matrix). It is not the probabilistic graph Laplacian P.

10 Relating G and P Proposition (A.) Fix G = (V, E) as before, let λ G 1,..., λ G n 1 be the non-zero eigenvalues of G and λ P 1,..., λ P n 1 be the non-zero eigenvalues of P, then n 1 Q Y n 1 λ G v V deg(v) i = n P v V deg(v) Y i=1 i=1 λ P i

11 Relating G and P Proposition (A.) Fix G = (V, E) as before, let λ G 1,..., λ G n 1 be the non-zero eigenvalues of G and λ P 1,..., λ P n 1 be the non-zero eigenvalues of P, then n 1 Q Y n 1 λ G v V deg(v) i = n P v V deg(v) Y i=1 i=1 λ P i Kirchhoff s Matrix-tree Theorem for Probabilistic Graph Laplacians (A.) Q n 1 v V deg(v) τ(g) = P v V deg(v) Y λ P i i=1

12 Products of Preiterates of Rational Functions Proposition: Rational Fact (A.) Let R(z) be a rational function such that R(0) = 0, deg(r(z)) = d, R(z) = P(z) Q(z), with deg(p(z)) > deg(q(z)). Let P d be the leading coefficient of P(z). Fix α C. Let {R ( n) (α)} be the set of n th preiterates of α under R(z). By convention, R (0) (α) := {α}. Then for n 0, Y z {R ( n) (α)} «d n 1 Q(0) d 1 z = α. P d

13 How to compute τ(v n ) Theorem: (A.) For a given fully symmetric self-similar structure on a finitely ramified fractal K, let V n be the sequence of approximating graphs. Arising naturally from the spectral decimation process, there is a rational function R(z), which satisfies the conditions of Proposition: Rational Fact, finite sets A, B R such that for all α A, β B, and integers n, k 0, there exist functions α n and β k n such that the number of spanning trees of V n is given by 0 V Y j=1 τ(v n) = 0 V X j=1 d j d j 1 A 1 A! 2 0 Y α αn 4 P n k=0 βn k Q(0) P d α A β B «P n k=0 β k n d k 1 d 1 «13 A5 where d is the degree of R(z), P d is the leading coefficient of the numerator of R(z), V n is the number of vertices of V n and d j is the degree of vertice j in V n.

14 How to use this Theorem 1 Carry out the Spectral Decimation Process as in [Vibration modes of 3n-gaskets and other fractals, N. Bajorin et. al] 2 A is the set α which satisfy items 2 or 8 of Prop. 3.1 of the same paper, and for α A, α n := mult n(α). 3 B := {β : for some n, mult n(β) 0 and mult n(r ( 1) (β)) 0} and for β B, β k n := mult n(r ( k) (β)). 4 Do the inductive calculations to find the degree of each vertex in V n 5 Solve for τ(v n)

15 Example: Sierpiński Gasket Theorem The number of spanning trees on the Sierpiński gasket at level n is given by τ(v n) = 2 fn 3 gn 5 hn, n 0 where f n = 1 2 g n = 1 4 `3n 1 3 n+1 + 2n + 1 h n = 1 4 `3n 2n 1. Note This was shown in [Spanning trees on the SierpińskiGasket, Chang, S. et.al.], however the previous Theorem gives a much more simple proof. Also, it applies to any Fractal which admits spectral decimation.

16 Diamond Fractal Theorem (A.) The number of spanning trees on the Diamond2.2fractal The at Fractal level n is given by τ(v n) = (4 n+1 +6n+5) n 0. The diamond self-similar hierarchical lattice appeared as an example in several physics works, including [Gefen V., Ahoranu A and Mandelbrot BB 1983, Phase transitions on fractals:...]. Figure [NEXT] shows the V1 and V2 networks for this. x2 x5 x6 x2 x9 x10 x1 x3 x1 x3 x12 x11 x8 x7 The V 0 and V 1 network of the Diamond Fractal x4 x4

17 Hexagasket Theorem (A.) The number of spanning trees on the Hexagasket at level n is given by τ(v n) = 2 fn 3 gn 7 hn n 0. where f n = n n 60n g n = n+1 + 5n h n = 1 25 `6n 5n 1. The V 1 network of the Hexagasket

18 Second Question Note 1 For the Sierpiński Gasket log(τ(v n)) θ(3 n ), meaning is bounded above and below by 3 n. 2 For the Diamond Fractal log(τ(v n)) θ(4 n ) 3 For the Hexagasket log(τ(v n)) θ(6 n ) Theorem: Asymptotic growth of log(τ(v n)) (A.) For a given fully symmetric self-similar structure on a finitely ramified fractal K. 1 If the V 1 network of K is tree then τ(v n) = 1 n 0 2 If not, then log(τ(v n)) θ(m n ), where m is the number of 1-cells in V 1.

19 Thank you! For a Preprint me at jaa72@cornell.edu