Báo cáo toán học: "Weighted spanning trees on some self-similar graphs" pot

Moro, 2 00185 Roma, Italia donno@mat.uniroma1.it Submitted: Aug 11, 2010; Accepted: Dec 20, 2010; Published: Jan 12, 2011 Mathematics Subject Classification: 05A15, 05C22, 20E08, 05C25 A

Weighted spanning trees on some self-similar graphs

Daniele D’Angeli

Departamento de Matem´aticas

Universidad de los Andes

Carrera 1, 18A - 70 Bogot´a, Colombia

dangeli@uniandes.edu.co

Alfredo Donno

Dipartimento di MatematicaSapienza Universit`a di RomaPiazzale A Moro, 2 00185 Roma, Italia

donno@mat.uniroma1.it

Submitted: Aug 11, 2010; Accepted: Dec 20, 2010; Published: Jan 12, 2011

Mathematics Subject Classification: 05A15, 05C22, 20E08, 05C25

Abstract

We compute the complexity of two infinite families of finite graphs: the Sierpi´nskigraphs, which are finite approximations of the well-known Sierpi´nski gasket, and theSchreier graphs of the Hanoi Towers group H(3) acting on the rooted ternary tree.For both of them, we study the weighted generating functions of the spanning trees,associated with several natural labellings of the edge sets

The enumeration of spanning trees in a finite graph is largely studied in the literature,and it has many applications in several areas of Mathematics as Algebra, Combinatorics,Probability and of Theoretical Computer Science

Given a connected finite graph Y = (V (Y ), E(Y )), where V (Y ) and E(Y ) denote thevertex set and the edge set of Y , respectively, a spanning tree of Y is a subgraph of Ywhich is a tree and whose vertex set coincides with V (Y )

The number of spanning trees of a graph Y is called the complexity of Y and is denoted

by τ (Y ) The famous Kirchhoff’s Matrix-Tree Theorem (1847) states that τ (Y ) is equal

to (the constant value of) any cofactor of the Laplace matrix of Y , which is obtained asthe difference between the degree matrix of Y and its adjacency matrix Equivalently,

τ (Y ) · |V (Y )| is given by the product of all nonzero eigenvalues of the Laplace matrix of

Y

It is interesting to study complexity when the system grows More precisely, given

a sequence {Yn}n≥1 of finite graphs with complexity τ (Yn), such that |V (Yn)| → ∞, thelimit

lim

|V (Y n )|→∞

log τ (Yn)

|V (Yn)| ,

when it exists, is called the asymptotic growth constant of the spanning trees of {Yn}n≥1

(see [13])

A spanning k-forest of Y is a subgraph of Y which is a k-forest, i.e., it is a forest with

k connected components, and its vertex set coincides with V (Y )

The enumeration of spanning subgraphs, in general, for a graph Y , is also strictlyrelated to the Tutte polynomial TY(x, y) of the graph: more precisely, it is known that

TY(1, 1) equals the complexity of Y , TY(2, 1) equals the number of spanning forests of

Y , and TY(1, 2) is the number of its connected spanning subgraphs (see [5, 9], wherethis analysis is developed for the finite Sierpi´nski graphs and for other examples of finitegraphs associated with the action of automorphisms groups of rooted regular trees)

A finer invariant of the graph Y is a finite abelian group Φ(Y ), whose order is exactlythe complexity of Y This group occurs in the literature under different names, depending

on the context It was introduced in [1] as the Picard group of Y (or the Jacobian of

Y ), whereas it is shown in [4] that the Picard group is isomorphic to the group of criticalconfigurations of the chip-firing game on Y As any finite abelian group, Φ(Y ) can bedecomposed into direct sum of invariant factors The dependence of this decomposition

on the properties of Y has been studied by several authors, (see, e.g., [12]), but not much

is known so far Explicit computations have been performed for certain families of graphs

In many optimization problems it is often useful to find a minimal spanning tree of aweighted graph Hence, it is interesting to study spanning trees when a weight function onE(Y ) is defined In order to do this, we introduce the formal variables we, with e ∈ E(Y ).These variables will be regarded as weights on the edges of the graph, so that we canassume that they take only positive real values Put w = {we}e∈E(Y ) and let T be the set

of all spanning trees of Y With each spanning tree t ∈ T , we can associate the weightfunction

X = {0, 1, , q − 1} Now let G < Aut(Tq) be a group acting on Tq by automorphisms

generated by a finite symmetric set of generators S Suppose, moreover, that the action

is transitive on each level of the tree

Definition 1.1 The n-th Schreier graph Σn of the action of G on Tq, with respect to thegenerating set S, is a graph whose vertex set coincides with the set of vertices of the n-thlevel of the tree, and two vertices u, v are adjacent if and only if there exists s ∈ S suchthat s(u) = v If this is the case, the edge joining u and v is labelled by s

The vertices of Σnare labelled by words of length n in X and the edges are labelled byelements of S The Schreier graph is thus a regular graph of degree |S| with qn vertices,and it is connected since the action of G is level-transitive

Definition 1.2 ([14]) A finitely generated group G < Aut(Tq) is self-similar if, for all

g ∈ G, x ∈ X, there exist h ∈ G, y ∈ X such that

g(xw) = yh(w),for all finite words w in the alphabet X

Self-similarity implies that G can be embedded into the wreath product Sym(q) ≀ G =Sym(q) ⋉ Gq, where Sym(q) denotes the symmetric group on q elements, so that anyautomorphism g ∈ G can be represented as

g = α(g0, , gq−1),where α ∈ Sym(q) describes the action of g on the first level of Tq and gi ∈ G, i =

0, , q − 1, is the restriction of g on the full subtree of Tq rooted at the vertex i of thefirst level of Tq (observe that any such subtree is isomorphic to Tq) Hence, if x ∈ X and

w is a finite word in X, we have

g(xw) = α(x)gx(w)

The class of self-similar groups contains many interesting examples of groups whichhave exotic properties: among them, we mention the first Grigorchuk group, which yieldsthe simplest solution of the Burnside problem (an infinite, finitely generated torsion group)and the first example of a group of intermediate growth (see [10] for a detailed accountand further references) In the last decades, automorphisms groups of rooted trees havebeen largely investigated: R Grigorchuk and a number of coauthors have developed anew exciting direction of research focusing on finitely generated groups acting by auto-morphisms on rooted trees [3] They proved that these groups have deep connectionswith the theory of profinite groups and with complex dynamics In particular, for manyexamples of groups belonging to this class, the property of self-similarity is reflected onfractalness of some limit objects associated with them [14]

Since the Schreier graphs are determined by group actions, their edges are naturally belled by the generators of the acting group and it takes sense to study weighted spanningtrees on them, with respect to this labelling

la-The paper is structured as follows In Section 2, we study weighted spanning trees onfinite approximations of the well-known Sierpi´nski gasket, endowed with three differentedge labellings:

• the “rotational-invariant”labelling, whose special symmetry allows to explicitly pute the generating function of the spanning trees (Theorem 2.2) and to perform

com-a stcom-atisticcom-al com-ancom-alysis com-about the number of edges, with com-a fixed lcom-abel, occurring in com-arandom spanning tree of the graph (Proposition 4.1);

• the “directional”labelling, where the weights depend on the direction of the edges;for this model, the weighted generating function of the spanning trees is describedvia the iteration of a polynomial map (Theorem 2.8);

• the “Schreier”labelling, strictly related to the labelling of the Schreier graphs of theHanoi Towers group H(3); also in this case, the weighted generating function of thespanning trees is described via the iteration of a polynomial map (Theorem 2.12)

In all these models we follow a combinatorial approach The self-similar structure ofthe graph (in the sense of [16]) allows to study both unweighted and weighted subgraphsrecursively More precisely, we introduce three different generating functions associatedwith spanning trees, 2-spanning forests, 3-spanning forests and, using self-similarity, weare able to establish recursive relations (Theorems 2.1, 2.6 and 2.10) and to give anexplicit description of them (Theorems 2.2, 2.8 and 2.12) More generally, the self-similarstructure of a graph turns out to be a powerful tool for investigating many combinatorialand statistical models on it: see, for instance, [7, 8, 15, 16, 17]

In Section 3, we consider the Schreier graphs of the Hanoi Towers group H(3), whoseaction on the ternary tree models the famous Hanoi Towers game on three pegs (see [11]),endowed with the natural edge labelling coming from the action of its generators Even ifthese graphs also have a self-similar structure, the combinatorial approach used in the case

of the Sierpi´nski graphs seems to be much harder here Therefore, our technique consists

in using a weighted version of the Kirchhoff’s Theorem: we construct the Laplace matrix

by using the self-similar presentation of the generators of the group, which is impossible

in the case of Sierpi´nski graphs, where there is no group structure In this case, thegenerating function is described in terms of iterations of a rational map (Theorem 3.5):this kind of approach already appears in [2, 11] (see also [8], where we use the samestrategy to compute the partition function of the dimer model on the Schreier graphs ofthe Hanoi Towers group)

The problem of enumeration of spanning trees in Sierpi´nski graphs was largely treated

in literature (see, for instance, [6, 15]) We consider here three different labellings of theedges of these graphs and write down the associated generating function of the spanningtrees In all the models, the self-similarity of the graphs plays a crucial role to study theproblem recursively The description of the generating function strongly depends on thesymmetry of the labelling of the graph: as we will see, in the first model that we consider,which is invariant under rotation, we are able to give an explicit formula for it; in the two

remaining models, where we do not have invariance under the action of any symmetrygroup, the generating function is described via the iteration of two polynomial maps.

Let Γ1 be the graph in the following picture

a

T T T

bc

T T T

to the rotation of 2π3 We represent in the following picture the graph Γ2

a

T T T

b

T T T

abab

b

T T T

a

T T T

• Tn(a, b, c) = weighted generating function of the spanning trees of Γn;

• Sn(a, b, c) = weighted generating function of the spanning 2-forests of Γn, wheretwo fixed outmost vertices belong to the same connected component and the thirdoutmost vertex belongs to the second connected component;

• Qn(a, b, c) = weighted generating function of the spanning 3-forests of Γn, wherethe three outmost vertices belong to three different connected components

Observe that, because of the rotational invariance of the labelling of the graph, the tion Sn(a, b, c) does not depend on the choice of the two outmost vertices In what follows,

func-we will often omit the argument (a, b, c) of the func-weighted generating functions

Theorem 2.1 For each n ≥ 1, the weighted generating functions Tn(a, b, c), Sn(a, b, c)and Qn(a, b, c) satisfy the following equations:

Sn+1 = 7TnSn2+ Tn2Qn (2)

Qn+1 = 12TnSnQn+ 14Sn3, (3)with initial conditions

T1(a, b, c) = 3(a + b)(ab + ac + bc)2

S1(a, b, c) = (a + b)(a + b + 3c)(ab + ac + bc) Q1(a, b, c) = (a + b)(a + b + 3c)2.Proof The graph Γn+1 can be represented as a triangle containing three copies of Γn







T T T

We will use the pictures

to denote, respectively, the case where in a copy of Γn:

• the three outmost vertices are in the same connected component;

• two outmost vertices are in the same connected component and the third one is in

a different connected component;

• the outmost vertices are in three different connected components

The only way to construct a spanning tree of Γn+1 is to choose a spanning tree in twocopies of Γn and a spanning 2-forest in the third one, as in the following picture



 T

This argument proves Equation (1), where the factor 6 is given by symmetry (we have

to take into account both reflections and rotations)

Next, we are going to prove Equation (2) (we analyze, for instance, the case where theleftmost and the rightmost vertices are in the same connected component) Consider thetwo following pictures

 T T



 T T

These possibilities, together with their symmetric, obtained by reflecting with respect

to the vertical axis, give a contribution to Sn+1 equal to 4TnS2

n Consider now the twofollowing configurations



 T T



 T T

Since the picture on the left has to be considered together with its symmetric, we get acontribution to Sn+1 equal to 3TnS2

n Finally, the contribution T2

nQn is described by thefollowing picture



 T T

This proves Equation (2)

We have now to prove Equation (3) about Qn+1 Consider the following situations

 T

 T

They provide, by symmetry, a contribution equal to 12TnSnQn The following picturesgive, by symmetry, a contribution of 12S3

 T

 T

Finally, the following two pictures give a contribution of 2S3



 T T



 T T

This completes the proof

Theorem 2.2 For each n ≥ 1, the weighted generating functions Tn(a, b, c), Sn(a, b, c)and Qn(a, b, c) satisfying Equations (1), (2) and (3), with the initial conditions given inTheorem 2.1, are:

Tn(a, b, c) = 23n−1−12 33n+2n−14 53n−1−2n+14 (a + b)3n−1(a + b + 3c)3n−1−12 (ab + ac + bc)3n+12 ;

Sn(a, b, c) = 2 2 3 4 5 4 (a + b)3n−1(a + b + 3c) 2 (ab + ac + bc) 2 ;

Qn(a, b, c) = 23n−1−12 33n−6n+34 53n−1+6n−74 (a + b)3n−1(a + b + 3c)3n−1+32 (ab + ac + bc)3n −32 Proof The proof is by induction on n It is easy to verify that, for n = 1, one gets theinitial conditions given in Theorem 2.1 Then, one can check that the functions given inthe claim satisfy Equations (1), (2) and (3) We omit here the explicit computations

It follows that Tn(1, 1, 1) = τ (Γn); similarly, s(Γn) := Sn(1, 1, 1) is the number ofspanning 2-forests of Γn, where two fixed outmost vertices belong to the same connectedcomponent and the third outmost vertex belongs to the second connected component;q(Γn) := Qn(1, 1, 1) is the number of spanning 3-forests of Γn, where the three outmostvertices belong to three different connected components

Corollary 2.3 For each n ≥ 1, one has:

2(3n+ 1) is the number of vertices of Γn, for each n ≥ 1

Remark 2.4 The same values of the complexity and of the asymptotic growth constanthave been found in [6] and [15], where the authors study unweighted spanning trees of

Γn

Consider now a new sequence of graphs {Γn}n≥1, which coincide, as unweighted graphs,with the graphs studied in Section 2.1, and whose edges are endowed with a new labelling,that we call directional labelling It is clear that an edge of Γn can point in three differentdirections: up (from left to right), down (from left to right) or horizontal Then, we label

by a each edge pointing up, by b each horizontal edge, and by c each edge pointing down,where, as usual, a, b, c ∈ R+ Here we draw the three first examples

T T T

c

T

Tcb

T T T

c

T

Tc

T T T

cbbbb

b

b b

T T

c

T T T

In this section, we study the weighted spanning trees of the graph Γn endowed withthe directional labelling For each n ≥ 1, we put:

• Tn(a, b, c) = weighted generating function of the spanning trees of Γn;

• Un(a, b, c) = weighted generating function of the spanning 2-forests of Γn, wherethe leftmost and the rightmost vertices belong to the same connected component,and the upmost vertex belongs to the second connected component Similarly, byrotation, we define Rn(a, b, c) (respectively Ln(a, b, c)) for the spanning 2-forests of

Γn, where the rightmost (respectively leftmost) vertex is not in the same connectedcomponent containing the two other outmost vertices;

• Qn(a, b, c) = weighted generating function of the spanning 3-forests of Γn, wherethe three outmost vertices belong to three different connected components

Observe that, in this model, we need to introduce three different functions Un(a, b, c),

Rn(a, b, c) and Ln(a, b, c), since the edge labelling is not invariant with respect to a rotation

of 2π3 as in the previous case On the other hand it is clear that, for each n ≥ 1, onehas Un(1, 1, 1) = Rn(1, 1, 1) = Ln(1, 1, 1) and this common value is equal to Sn−1(1, 1, 1),where Sn(a, b, c) is the generating function introduced in Section 2.1 In what follows, wewill often omit the argument (a, b, c) of the generating functions

Theorem 2.6 For each n ≥ 1, the weighted generating functions Tn(a, b, c), Un(a, b, c),

Rn(a, b, c), Ln(a, b, c) and Qn(a, b, c) satisfy the following equations:

Tn+1 = 2T2

Un+1= TnUn(2Rn+ 2Ln+ 3Un) + Tn2Qn (5)

Rn+1= TnRn(2Ln+ 2Un+ 3Rn) + Tn2Qn (6)

Ln+1= TnLn(2Rn+ 2Un+ 3Ln) + Tn2Qn (7)

Qn+1 = 4TnQn(Un+ Rn+ Ln) (8)

+ 2 Un2(Rn+ Ln) + R2n(Ln+ Un) + L2n(Rn+ Un)
+ 2UnRnLn,

with initial conditions

T1(a, b, c) = ab + ac + bc U1(a, b, c) = b R1(a, b, c) = a

F1(x, y, z) = 3x2+ 3xz + 3xy + yz F2(x, y, z) = 3y2+ 3xy + 3yz + xz

Theorem 2.8 The weighted generating functions Tn(a, b, c), Un(a, b, c), Rn(a, b, c),

Ln(a, b, c) and Qn(a, b, c) satisfying Equations (4), (5), (6), (7) and (8), with the initialconditions given in Theorem 2.6, are:

Proof The proof works by induction on n One can directly find:

T1(a, b, c) = φ1(a, b, c) U2(a, b, c) = φ1(a, b, c)F2(a, b, c)

R2(a, b, c) = φ1(a, b, c)F1(a, b, c) L2(a, b, c) = φ1(a, b, c)F3(a, b, c)

Q3(a, b, c) = 2φ31(a, b, c)f (F1(a, b, c), F2(a, b, c), F3(a, b, c)),and so the basis of induction holds We only prove the assertion for Tn(a, b, c), by showingthat Equation (4) is satisfied (the computations in the other cases are similar but morecomplicated) One has:

2.3 Third model: the “Schreier”labelling

Consider the graph Γ1 in the picture below and define by recurrence, for each n ≥ 1, thegraph Γn+1 as constituted by the union of three copies of Γn in the following way: foreach one of the outmost vertices of Γn+1, the corresponding copy is given by the graph

Γn, reflected with respect to the bisectrix of the corresponding angle

T T T

b

T T T

ab

T T T

b

T T T

a

T T T

cbac

b

T T T

a

T T T

Σn on the remaining edges (See Section 3.1.)

Define the generating functions Tn(a, b, c), Un(a, b, c), Rn(a, b, c), Ln(a, b, c) and Qn(a, b, c)

to have the same meaning as in the case of the directional labelling (Section 2.2) In whatfollows, we will often omit the argument (a, b, c) of the generating functions

Theorem 2.10 For each n ≥ 1, the weighted generating functions Tn(a, b, c), Un(a, b, c),

Rn(a, b, c), Ln(a, b, c) and Qn(a, b, c) satisfy the following equations:

Qn+1 = 4TnQn(Un+ Rn+ Ln) (13)

+ 2 Un2(Ln+ Rn) + R2n(Un+ Ln) + L2n(Rn+ Un)
+ 2UnRnLn,

with initial conditions

T1(a, b, c) = ab + ac + bc U1(a, b, c) = b R1(a, b, c) = a

G(x, y, z) = (G1(x, y, z), G2(x, y, z), G3(x, y, z)),with

Ln(a, b, c) and Qn(a, b, c) satisfying Equations (9), (10), (11), (12) and (13), with theinitial conditions given in Theorem 2.10, are:

k (a, b, c) for each n ≥ 1;

Proof The proof is by induction on n One can directly find:

T1(a, b, c) = ψ1(a, b, c) U2(a, b, c) = 2ψ1(a, b, c)G2(a, b, c)

R2(a, b, c) = 2ψ1(a, b, c)G1(a, b, c) L2(a, b, c) = 2ψ1(a, b, c)G3(a, b, c)

Q3(a, b, c) = 24ψ13(a, b, c)g(G1(a, b, c), G2(a, b, c), G3(a, b, c)),and so the basis on the induction holds We only prove the assertion for Tn(a, b, c), byshowing that Equation (9) is satisfied (the computations in the other cases are similarbut more complicated) One has:

k (a, b, c) = Tn+1

Remark 2.13 Note that the function g(a, b, c) coincides with the function f (a, b, c), troduced in Section 2.2 However, the functions G(x, y, z) and F (x, y, z) do not coincide,which implies that the functions ψi(a, b, c) and φi(a, b, c), factorizing the weighted gener-ating functions of the spanning trees in the directional model and in the Schreier model,are different

Towers group

In this section, we study spanning trees on the Schreier graphs of the Hanoi Towers group

H(3), which are very similar to the Sierpi´nski graphs studied in the previous sections Westart with a combinatorial recursive approach as in Section 2, but this leads us to some