On the spectral distributions of distance-k graph of free product graphs

1.1 MB·

by  Octavio Arizmendi

ON THE SPECTRAL DISTRIBUTIONS OF DISTANCE-k GRAPH
OF FREE PRODUCT GRAPHS.
OCTAVIOARIZMENDIANDTULIOGAXIOLA
6
1
0
2

n Abstract. We calculate the distribution with respect to the vacuum state
a of the distance-k graph of a d-regular tree. From this result we show that
J thedistance-kgraphofad-regulargraphsconvergestothedistributionofthe

8 distance-kgraphofaregulartree. Finally,weprovethat,properlynormalized,
1 the asymptotic distributions of distance-k graphs of the d-fold free product
graph,asdtendstoinfinity,isgivenbythedistributionofPk(s),wheresisa

] semicirclerandomvariableandPk isthek-thChebychevpolynomial.
O
C 1. Introduction
.
h In this paper we consider three problems on the distance-k graphs, which gener-
t alize results of Kesten [11] (on random walks on free groups), McKay [13] (on the
a
m asymptoticdistributionofd-regulargraphs)andthefreecentrallimitofVoiculescu
[15]. Thefirstoneisfinding,forfixedd,thedistributionw.r.t. thevacuumstateof
[
the distance-k graphs of a d-regular tree. Then we consider two related problems

1 which are in the asymptotic regime. On one hand, we show that the asymptotic
v
distributions of distance-k graphs of d-fold free product graphs, as d tends to infin-
2
ity, are given by the distribution of P (s), where s is a semicircle distribution and
5 k
7 Pk is the k-th Chebychev polynomial. On the other hand, we find the asymptotic
4 spectral distribution of the distance-k graph of a random d-regular graph of size n,
0 as n tends to infinity.
.
1 More precisely our first result is the following.
0
Theorem 1.1. For d ≥ 2, k ≥ 1, let A[k] be the adjacency matrix of distance-k
6 d
1 graph of the d-regular tree. Then the distribution with respect to the vacuum state
: of A[k] is given by the probability distribution of
v d
i (cid:114) (cid:18) (cid:19) (cid:18) (cid:19)
X d−1 b 1 b
T (b)= P √ − P √ ,
r k d k 2 d−1 (cid:112)d(d−1) k−2 2 d−1
a
where P is the Chebyshev polynomial of order k and b is a random variable with
k
Kesten-McKay distribution, µ .
d
Thespectrumofthedistance-kgraphoftheCartesianproductofgraphswasfirst
studied by Kurihara and Hibino [10] where they consider the distance-2 graph of
K ×···×K (the n-dimensional hypercube). More recently, in a series of papers
2 2
[7, 8, 9, 10, 12, 14] the asymptotic spectral distribution of the distance-k graph
of the N-fold power of the Cartesian product was studied. These investigations,
finallyleadtothefollowingtheoremwhichgeneralizesthecentrallimittheoremfor
Cartesian products of graphs.
Date:January20,2016.
1
2 OCTAVIOARIZMENDIANDTULIOGAXIOLA
Theorem 1.2 (Hibino, Lee and Obata [8]). Let G = (V,E) be a finite connected
graph with |V| ≥ 2. For N ≥ 1 and k ≥ 1 let G[N,k] be the distance-k graph of
GN =G×···×G(N-foldCartesianpower)and A[N,k] itsadjacencymatrix. Then,
for a fixed k ≥1, the eigenvalue distribution of N−k/2A[N,k] converges in moments
as N →∞ to the probability distribution of
(cid:18)2|E|(cid:19)k/2 1
(1.1) H˜ (g),
|V| k! k
where H˜ is the monic Hermite polynomial of degree k and g is a random variable
k
obeying the standard normal distribution N(0,1).
Inthesamespirit,in[2],weconsidertheanalogofTheorem1.2bychangingthe
Cartesian product by the star product.
Theorem 1.3 (Arizmendi and Gaxiola [2]). Let G = (V,E,e) be a locally finite
connected graph and let k ∈ N be such that G[k] is not trivial. For N ≥ 1 and
k ≥1 let G[(cid:63)N,k] be the distance-k graph of G(cid:63)N =G(cid:63)···(cid:63)G (N-fold star power)
and A[(cid:63)N,k] its adjacency matrix. Furthermore, let σ = V[k] be the number of
e
neighbors of e in the distance-k graph of G, then the distribution with respect to the
vacuum state of (Nσ)−1/2A[(cid:63)N,k] converges in distribution as N →∞ to a centered
Bernoulli distribution. That is,
A[(cid:63)N,k] 1 1
√ −→ δ + δ ,
Nσ 2 −1 2 1
weakly.
Our second theorem is the free counterpart of the theorems above.
Theorem 1.4. Let G = (V,E,e) be a finite connected graph and let k ∈ N. For
N ≥1 and k ≥1 let G[∗N,k] be the distance-k graph of G∗N =G∗···∗G (N-fold
free power) and A[∗N,k] its adjacency matrix. Furthermore, let σ be the number of
neighbors of e in the graph G. Then the distribution with respect to the vacuum
state of (Nσ)−k/2A[∗N,k] converges in moments (and then weakly) as N → ∞ to
the probability distribution of
(1.2) P (s),
k
whereP istheChebychevpolynomialoforderk andsisarandomvariableobeying
k
the semicircle law.
Finally, our third theorem considers the asymptotic spectral distribution of the
distance-k graph of d-regular random graphs.
Theorem 1.5. Let d, k be fixed integers and, for each n, let F (x) be the expected
n
eigenvaluedistributionofthedistance-k graphofarandomregulargraphwithdegree
d and order 2n. Then, as n tends to infinity, F (x) converges to the distribution
n
of A[k] with respect to the vacuum state, described in Theorem 1.1.
d
Apart from this introduction the paper is organized as follows. In Section
2 we give the basic preliminaries on graphs, orthogonal polynomials and Non-
CommutativeProbabilityandKesten-McKaydistributions. Section3isdevotedto
proveTheorem1.3. WeproveTheorem1.4inSections4and5. Section4considers
the case k = 2, while Section 5 considers the case k ≥ 3. Finally, in Section 6 we
use the results of Section 3 to prove Theorem 1.5.
DISTANCE-k GRAPHS OF FREE PRODUCT 3
2. Preliminaries
In this section we give very basic preliminaries on graphs, free product graphs,
orthogonal polynomials, Jacobi parameters and non-commutative probability. The
reader familiar with these objects may skip this section.
2.1. Graphs. By a rooted graph we understand a pair (G,e), where G = (V,E),
is a undirected graph with set of vertices V = V(G), and the set of edges E =
E(G) ⊆ {(x,x(cid:48)) : x, x(cid:48) ∈ V, x (cid:54)= x(cid:48)} and e ∈ V is a distinguished vertex called
the root. For rooted graphs we will use the notation V0 = V\{e}. Two vertices
x,x(cid:48) ∈ V are called adjacent if (x,x(cid:48)) ∈ E, i.e. vertices x,x(cid:48) are connected with
an edge. Then we write x ∼ x(cid:48). Simple graphs have no loops, i.e. (x,x) ∈/ E for
all x ∈ V. A graph is called finite if |V| < ∞. The degree of x ∈ V is defined by
κ(x) = |{x(cid:48) ∈ V : x(cid:48) ∼ x}|, where |I| stands for the cardinality of I. A graph is
called locally finite if κ(x)<∞ for every x∈V. It is called uniformly locally finite
if sup{κ(x):x∈V}<∞.
We define the free product of the rooted vertex sets (V ,e ), i ∈ I, where I is a
i i
countable set, by the rooted set (∗ V ,e), where
i∈I i
∗ V ={e}∪{v v ···v :v ∈V0, and i (cid:54)=i (cid:54)=···(cid:54)=i , m∈N},
i∈I i 1 2 m k ik 1 2 m
and e is the empty word.
Definition 2.1. The free product of rooted graph (G ,e ), i ∈ I, is defined by the
i i
rooted graph (∗ G ,e) with vertex set ∗ V and edge set ∗ E , defined by
i∈I i i∈I i i∈I i
(cid:91)
∗ E :={(vu,v(cid:48)u):(v,v(cid:48))∈ E and u, vu, v(cid:48)u∈∗ V }.
i∈I i i i∈I i
i∈I
We denote this product by ∗ (G ,e ) or ∗ G if no confusion arises. If I = [n],
i∈I i i i∈I
we denote by G∗n =(∗ G,e).
i∈I
Noticethatforafixedwordu=v v ···v withj ∈I withv ∈/ V thesubgraph
1 2 m 1 j
of (∗ G ,e) induced by the vertex set {wu : w ∈ V } is isomorphic to G . This
i∈I (cid:105)i j j
motivates the following definition
Definition 2.2. If x,y ∈ ∗ V , we say that x and y are in the same copy of G
i∈I i i
if x=vu and y =v(cid:48)u for some u∈∗ V and v,v(cid:48) ∈V0 for some j ∈I.
i∈I i j
For a given graph G=(V,E), its distance-k graph G[k] =(V,E[k]) is defined by
E[k] ={(x,y):x,y ∈V, ∂ (x,y)=k}.
G
For x ∈ V, let δ(x) be the indicator function of the one-element set {x}. Then
{δ(x), x∈V}isanorthonormalbasisoftheHilbertspacel2(V)ofsquareintegrable
functions on the set V, with the usual inner product.
The adjacency matrix A=A(G) of G is a 0-1 matrix defined by
(cid:26) 1 if x∼x(cid:48)
(2.1) A =
x,x(cid:48) 0 otherwise.
We identify A with the densely defined symmetric operator on l2(V) defined by
(cid:88)
(2.2) Aδ(x)= δ(x(cid:48))
x∼x(cid:48)
for x ∈ V. Notice that the sum on the right-hand-side is finite since our graph is
assumed to be locally finite. It is known that A(G) is bounded if and only if G is
4 OCTAVIOARIZMENDIANDTULIOGAXIOLA
uniformly locally finite. If A(G) is essentially self-adjoint, its closure is called the
adjacency operator of G and its spectrum is called the spectrum of G.
TheunitalalgebrageneratedbyA,i.e. thealgebraofpolynomialsinA,iscalled
the adjacency algebra of G and is denoted by A(G) or simply A.
2.2. Orthogonal Polynomials and The Jacobi Parameters. Letµbeaprob-
ability measure with all moments, that is m (µ):=(cid:82) |xn|µ(dx)<∞. The Jacobi
n R
parameters γ =γ (µ)≥0,β =β (µ)∈R, are defined by the recursion
m m m m
xQ (x)=Q (x)+β Q (x)+γ Q (x),
m m+1 m m m−1 m−1
where the polynomials Q (x) = 0, Q (x) = 1 and (Q ) is a sequence of
−1 0 m m≥0
orthogonal monic polynomials with respect to µ, that is,
(cid:90)
Q (x)Q (x)µ(dx)=0 if m(cid:54)=n.
m n
R
Example 2.3. The Chebyshev polynomials of the second kind are defined by the
recurrence relation
P (x)=1, P (x)=x,
0 1
and
(2.3) xP (x)=P (x)+P (x) ∀n≥1.
n n+1 n−1
These polynomials are orthogonal with respect to the semicircular law, which is
defined by the density
1 (cid:112)
dµ= 4−x2dx.
2π
The Jacobi parameters of µ are β =0 and γ =1 for all m≥0.
m m
2.3. Non-Commutative Probability Spaces. A C∗-probability space is a pair
(A,ϕ), where A is a unital C∗-algebra and ϕ : A → C is a positive unital linear
functional. The elements of A are called (non-commutative) random variables. An
element a∈A such that a=a∗ is called self-adjoint.
Thefunctionalϕshouldbeunderstoodastheexpectationinclassicalprobability.
For a ,...,a ∈ A, we will refer to the values of ϕ(a ···a ), 1 ≤ i ,...i ≤ k,
1 k i1 in 1 n
n ≥ 1, as the joint moments of a ,...,a . If there exists 1 ≤ m,l,≤ n with
1 k
i(m)(cid:54)=i(l) we call it a mixed moment.
For any self-adjoint element a∈A there exists a unique probability measure µ
a
(its spectral distribution) with the same moments as a, that is,
(cid:90)
xkµ (dx)=ϕ(ak), ∀k ∈N.
a
R
We say that a sequence a ∈ A converges in distribution to a ∈ A if µ con-
n n an
verges in distribution to µ . In this setting convergence in distribution is replaced
a
by convergence in moments. Let (φ ,A ) be a sequence of C∗-probability spaces
n n
and let a ∈ (A,ϕ) be a selfadjoint random variable. We say that the sequence
a ∈(φ ,A ) of selfadjoint random variables converges to a in moments if
n n n
lim φ (ak)=φ(ak) for all k ∈N.
n n
n→∞
If a is bounded then convergence in moments implies convergence in distribution.
The following proposition is straightforward and will be used frequently in the
paper. A sequence of polynomials {P = (cid:80)l c xi} of degree at most l ≥ k
n i=0 n,i n>0
DISTANCE-k GRAPHS OF FREE PRODUCT 5
is said to converge to a polynomial P = (cid:80)k c xi of degree k if c → c for
i=0 i i,n i
0≤i≤k and c →0 for k <i≤l.
i,n
Proposition 2.4. Suppose that the the sequence of random variables {a } con-
n n>0
verges in moments to a and the sequence of polynomials {P } converges to P.
n n>0
Then the random variables P (a ) converges to P(a).
n n
In this work we will only consider the C∗-probability spaces (M ,ϕ ), where
n 1
M is the set of matrices of size n×n and for a matrix M ∈ M the functional
n n
ϕ evaluated in M is given by
1
ϕ (M)=M .
1 11
Let G = (V,E,1) be a finite rooted graph with vertex set {1,...,n} and let A
G
betheadjacencymatrix. WedenotebyA(G)⊂M betheadjacencyalgebra, i.e.,
n
the ∗-algebra generated by A .
G
It is easy to see that the k-th moment of A with respect to the ϕ is given the
1
the number of walks in G of size k starting and ending at the vertex 1. That is,
ϕ (Ak)=|{(v ,...,v ):v =v =1 and (v ,v )∈E}|.
1 1 k 1 k i i+1
Thus one can get combinatorial information of G from the values of ϕ in ele-
1
ments of A(G) and vice versa.
Let us recall the free central limit theorem for free product of graphs (see, e.g.
[1]) which follows from the usual free central limit theorem for random variables
[15].
Theorem 2.5 (Free Central Limit Theorem for Graphs). Let G = (V,E,e) be a
finite connected graph. Let A be the adjacency matrix of the N-fold free power
N
G∗N,andletσ bethenumberofneighborsofeinthegraphG. Thenthedistribution
with respect to the vacuum state of (Nσ)−1/2A converges in moments (and thus
N
weakly) as N →∞ to the semicircular law.
For the rest of the paper we define an order which will become handy when
estimating vanishing terms in Sections 4 and 5.
Definition 2.6. Let A and B be matrices (possibly infinite), we define the order
A(cid:23)B if A ≥B for all entries ij.
ij ij
Remark 2.7. 1) ϕ (Ak)≥ϕ (Bk) if A(cid:23)B.
1 1
2) For G and G graphs with n vertices, G is a subgraph of G iff A (cid:23)A .
1 2 2 1 G1 G2
3) If A(cid:23)B and C (cid:23)D implies AC (cid:23)BD.
2.4. Kesten-McKay Distribution. As we know, by the free central limit theo-
rem,ifwehaveasequenceofd-regulartrees,thenthelimitingspectraldistribution
of the sequence, as d→∞, converges to a semicircular law. However, if d is fixed,
and we consider a sequence of d-regular graphs, such that the number of vertices
tends to infinity, then the limiting spectral distribution is not semicircular. These
limiting spectral distributions, which are known as the Kesten-McKay distribu-
tions, were found by McKay [13] while studying properties of d-regular graphs and
by Kesten [11] in his works on random walk on (free) groups.
Letd≥2beaninteger,wedefineKesten-McKaydistribution,µ ,bythedensity
d
(cid:112)
d 4(d−1)−x2
(2.4) dµ = dx.
d 2π(d2−x2)
6 OCTAVIOARIZMENDIANDTULIOGAXIOLA
The orthogonal polynomials and the Jacobi parameters of these distributions are
well known. More precisely, for d≥2, the polynomials defined by
T (x)=1, T (x)=x,
0 1
and the recurrence formula
(2.5) xT (x)=T (x)+(d−1)T (x),
k k+1 k−1
are orthogonal with respect to the distribution µ . Thus, it follows that the Jacobi
d
parameters of µ are given by
d
β =0, ∀m≥0 and γ =d, γ =d−1 ∀n≥1.
m 0 n
Remark 2.8. If we define the following polynomials
(cid:40) 1, k =0
T˜ (x)= (cid:113)
k d−1P (x)− √ 1 P (x), k =1,2,3,...,
d k d(d−1) k−2
√
then, T (x)=T˜ (x/2 d−1).
k k
In Section 6 we will generalize the following theorem due to McKay [13] which
givesaconnectionbetweenlarged-regulargraphsandKesten-McKaydistributions.
Theorem 2.9. Let X , X , ... be a sequence of regular graphs with degree d≥2
1 2
such that n(X ) → ∞ and c (X )/n(X ) → 0 as i → ∞ for each k ≥ 3, where
i k i i
n(X ) is the order of X and c (X ) is the number of k-cycles in X . Then, the
i i k i i
limiting distribution for the eigenvalues X as i→∞ is given by µ .
i d
3. Distance-k graph of d-regular trees
The d-regular tree is the d-fold free product graph of K , the complete graph
2
withtwovertices. Beforeweconsiderasymptoticbehaviorofthegeneralcaseofthe
free product of graphs, we study the distance-k graph of a d-regular tree for fixed
d and k. This is an example where we can find the distribution with respect to the
vacuum state in a closed form. Moreover, this example sheds light on the general
case of the d-fold free product of graphs, in the same way as the d-dimensional
cube was the leading example for investigations of the distance-k graph of the
d-fold Cartesian product of graphs (Kurihara [9]).
As a warm up and base case, we calculate the distribution of the distance-2
graph with respect to the vacuum state.
Ford≥2, letA[k] betheadjacencymatrixofdistance-k graphofd-regulartree.
d
We will sometimes omit the subindex d in the notation and write A[1] =A . Then
wehavethefollowingequality, whichexpressesA2 intermsofthedistance-2graph
and the identity matrix (see Figure 1). :
(3.1) A2 =A[2]+dI.
d
Since A[2] =A2−dI then the distribution is given by the law of x2−d, where x is
d
a random variable obeying the Kesten-McKay distribution of parameter d, µ .
d
For k ≥2 we have the following recurrence formula.
Lemma 3.1. Let d≥1 fixed, then A[1] =A, A[2] =A2−dI, and
(3.2) AA[k] =A[k+1]+(d−1)A[k−1] k =2, ...,d−1.
DISTANCE-k GRAPHS OF FREE PRODUCT 7
Figure 1. Graph of A2 split in two parts A2 =A[2]+dI.
d
Proof. Letiandj beverticesofthed-regulartree,Y . Wehavethefollowingthree
d
cases.
Case 1. If ∂(i,j)=k+1 then (A[k]A) =1, that is because, in this case, there is
ij
only one way to get from vertex j to vertex i. Indeed, since this Y is a tree there
d
is only one walk from i to j of size k+1 in Y . Thus, there is exactly one neighbor
d
l of j at distance k from i and thus the only way to go across the distance-k graph
and after across Y to reach j is trough l.
d
Case 2. When we have ∂(i,j) = k−1, then (A[k]A) = d−1. In fact, for the
ij
vertex i there are d−1 ways to arrive to j from a neighbor of j at distance k from
i. Thus, if we are in vertex i, there are d−1 ways to travel across the distance-k
graph and finally go down one level in the d-regular tree to vertex j, .
Case 3. Suppose |∂(i,j)−k| (cid:54)= 1, then (A[k]A) = 0. To see this, we just note
ij
that, in the d-regular tree we can go up one-level or go down one-level, after going
across the distances-k graph, this means that the distance between i and j would
be k−1 or k+1, which is a contradiction. Therefore if |∂(i,j)−k| =(cid:54) 1, there is
no way to go from the vertex i to the vertex j, going across the distance-k graph
and after, across the d-regular tree in one step.
Thanks to the above, we obtain the following recurrence formula
(3.3) A[k]A=A[k+1]+(d−1)A[k−1].
From the equations (3.1) and (3.3) we can see that A[k] is a polynomial in A for
k ≥1, andthuscommuteswithA. Thenwecanrewriteequation(3.3)inthemore
convenient way as follows
AA[k] =A[k+1]+(d−1)A[k−1].
(cid:3)
Now we can calculate the distribution of the distance-k graph of the d-regular
tree, for d fixed, which is exactly Theorem 1.1.
Proof of Theorem 1.1. From equation (3.2) we can see that A[k] fulfills the same
d
recurrence formula than T in (2.5). Since A is distributed as the Kesten-McKay
k
distribution µ , we arrive to the conclusion. (cid:3)
d
8 OCTAVIOARIZMENDIANDTULIOGAXIOLA
To end this section we observe that from the considerations above, by letting
d approach infinity, we may find the asymptotic behavior of the distribution of
the distance-k graph of the d-regular tree. The same behavior is expected when
changing the d-regular tree with the d-fold free product of any finite graph. We
will prove this in Section 5 of the paper.
Theorem 3.2. For d≥2, let A[k] be the adjacency matrix of the distance-k graph
d
of the d-regular tree. Then the distribution with respect to the vacuum state of
d−k/2A[k] converges in moments as d→∞ to the probability distribution of
d
(3.4) P (s),
k
where P (s) is the Chebychev polynomial of degree k and s is a random variable
k
obeying the semicircle law.
Proof. If we divide the equation (3.2) by d(k+1)/2 we obtain
A A[k] A[k+1] A[k−1] 1 A[k−1]
d d = d + d − d
d1/2dk/2 d(k+1)/2 d(k−1)/2 dd(k−1)/2
We write X = Ad , then we have
d1/2
P(1)(X)=X, P(2)(X)=X2−I,
and the recurrence
1
XP(k)(X)=P(k+1)(X)+P(k−1)(X)− P(k−1)(X),
d
which when d→∞ becomes the recurrence formula
XP(k)(X)=P(k+1)(X)+P(k−1)(X).
Thus P(k)(x) and P (x) satisfy the same recurrence formula asymptotically and
k
thanks to the free central limit theorem for graphs (Theorem 2.5) we have the
m
convergence, X −→ s. Consequently, combining these two observations and using
Lemma 2.4 we obtain the proof. (cid:3)
4. Distance-2 graph of free products
In this section we derive the asymptotic spectral distribution of the distance-2
graph of the n-free power of a graph when n goes to infinity.
In order to analyze the distance-2 graphs we give a simple, but useful, decom-
position of the square of the adjacency matrix.
Lemma 4.1. Let G be a simple graph with adjacency matrix A, we have the fol-
lowing decomposition of A2:
(4.1) A2 =A˜[2]+D+∆,
whereDisdiagonalwith(D) =deg(i),(∆) =|triangles in G with one side (i,j)|
ii ij
and (A˜[2]) = |paths of size 2 from i to j|, whenever (A[2]) = 1 and (A˜[2]) = 0
ij ij ij
if (A[2]) =0.
ij
Proof. Indeed (A2) is zero if the distance between i and j is greater than 2.
ij
So (A2) > 0 implies that ∂(i,j) = 0,1 or 2. If ∂(i,j) = 0 then i = j and
ij
since (A2) = deg(i) we get D, a diagonal matrix with (D) = deg(i). Next, if
ii ii
∂(i,j) = 1 then each path of size 2 which forms a triangle with side (i,j) will
contribute to (A2) = (∆) where (∆) = |triangles in G with one side (i,j)|.
ij ij ij
DISTANCE-k GRAPHS OF FREE PRODUCT 9
Finally if ∂(i,j) = 2 then (A2) equals the number of paths of size 2 from i to j,
ij
which is non-zero exactly when (A˜[2]) >0. (cid:3)
ij
Remark 4.2. NoticeinLemma4.1,thatwhenGisatreethen∆=0,A˜[2] =A[2],
therefore A[2] =A2−D.
Let G=(V,E,e) be a rooted graph, A =A and define D and ∆ by the
n G∗N n n
decomposition (4.1) applied to G∗N =G∗···∗G, i.e.
(4.2) A2 =A˜[2]+D +∆ .
n n n n
We will describe the asymptotic behavior of each of these matrices. First, we
consider the diagonal matrix D .
n
Lemma 4.3. D /n → Ideg(e) entrywise and in distribution w.r.t. the vacuum
n
state.
Proof. For any i ∈ G (D ) = deg (i) = c +(n−1)deg(e) for some 0 < c <
n n ii Gn i i
maxdeg(G). Thus,
(D ) c (n−1)deg(e)
n ii = i + →deg(e).
n n n
(cid:3)
In order to consider the other matrices in the decomposition we will use the
order (cid:23) from Definition 2.6.
Lemma 4.4. The mixed moments of A2/n and ∆ /n asymptotically vanish.
n n
Proof. Note that the free product does not generate new triangles other than the
ones in copies of the original graph. Thus, for c=max deg(G) the relation cA (cid:23)
n
∆ holds. Hence,form , m , ..., m , l , l , ..., l ∈Nandl >0,fromRemark
n 1 2 s 1 2 s 1
2.7, we have that
(cid:34)(cid:18)A2(cid:19)m1(cid:18)∆ (cid:19)l1 (cid:18)A2(cid:19)ms(cid:18)∆ (cid:19)ls(cid:35)
ϕ n n ··· n
1 n n n n
≤c(cid:80)iliϕ1(cid:34)(cid:18)An2n(cid:19)m1(cid:18)An(cid:19)l1···(cid:18)An2(cid:19)ms(cid:18)An(cid:19)ls(cid:35).
From Theorem 2.5 we have that A2/n and A/n1/2 converge, then the right hand
side of the preceding inequality converges to zero as n goes to infinity. (cid:3)
Since A˜[2] and D are subgraphs of A2 we have the following.
n n n
Corollary4.5. Themixedmomentsofthepairs(A˜[2]/n,∆/n,)and(D /n,∆/n,)
n n
asymptotically vanish.
Finally, we consider the matrix A˜[2].
Lemma 4.6. A˜[2] converges to A[2] as n goes to infinity.
n n
Proof. Observe that we can write A[2] as
n
A˜[2] =A[2]+(cid:3) ,
n n n
where for (i,j) at distance 2 in G∗n, the entry ((cid:3) ) exceeds in one the number of
n ij
vertices k such that i∼k and k ∼j.
10 OCTAVIOARIZMENDIANDTULIOGAXIOLA
WewillextendGinthefollowingway. Foreach(i,j)suchthat(cid:3) ispositivewe
ij
put the edge ij and call this new graph G(ext). Now notice that, by construction,
∆ (cid:23) (cid:3) and A (cid:23) A . Finally, by the previous lemma the mixed
G(ext)∗n G(ext)∗n G∗n
moments of ∆ and A2 asymptotically vanish. But A2 (cid:23) A[2],
G(ext)∗n G(ext)∗n G(ext)∗n n
sothemixedmomentsofA[2] and(cid:3) alsovanishinthelimit. Thisofcoursemeans
n n
that A˜[2] and A[2] are asymptotically equal in distribution.
n n
(cid:3)
We have shown that asymptotically D /n approximates I, A˜[2] approximates
n n
A[2] and that the joint moments between A˜[2] or D and ∆ vanish. Thus, we
n n n n
arrive to the following theorem.
Theorem 4.7. The asymptotic distributions of distance-2 graph of the n-fold free
product graph, as n tends to infinity, is given by the distribution of s2−1, where s
is a semicircle.
Proof. From the equation (4.2), and thanks to Lemmas 2.4, 4.3, 4.6, Corolary 4.5
and Theorem 2.5 we have
A[2] −D→A˜[2] −D→A2 −D −∆ −D→A2 −I −D→s2−1.
n n n n n n
(cid:3)
5. Distance-k graphs of free products
This section contains the proof of Theorem 1.4 which describes the asymptotic
behavior of the distance-k graph of the d-fold free power of graphs. We will show
that the adjacency matrix satisfies in the limit the recurrence formula (2.3). We
start by showing a decomposition similar to the one seen above for d-regular trees
which plays the role of Lemma 4.1 in the last section.
Theorem 5.1. Let G be a simple finite graph, let N,k ∈N with N ≥2 and k ≥3
and let A = A denote the adjacency matrix of G∗N. Then, we have de following
N
recurrence formula
(5.1) A[k]A=A˜[k+1]+(N −1)deg(e)A[k−1]+D[k−1]+∆[k],
N N
where (A˜[k+1]) =|{l∼j :∂(i,l)=k}| whenever ∂(i,j)=k+1,
ij
(D[k−1]) =|{l∼j :∂(i,l)=k,and j and l are in the same copy of G}|if∂(i,j)=
N ij
k−1 and (∆[k]) =|{l∼j :∂(i,l)=k}| when ∂(i,j)=k.
N ij
Proof. It’s easy to see that (A[k]A) is zero if |∂(i,j)−k| ≥ 2. So (A[k]A) > 0
ij ij
implies that ∂(i,j)=k−1, k or k+1.
Notice that for each neighbor l of j at distance k from i, there is one edge from
i to l in A[k] and one from l to j in A. Thus each of these neighbors adds 1 to
(A[k]A) and there is no further contribution.
ij
First, if ∂(i,j)=k−1 there are two types of neighbors l at distance k in G∗N.
The first ones come from the (N −1) copies of G in G∗N which have j as a root
andcontributetothematrixA[k−1] by(N−1)deg(e)andthesecondonesinwhich
j is in the same copy that l, which contribute to D[k−1].
N
Secondly, if ∂(i,j)=k and (A[k]A) >0 is the number of neighbors of j which
ij
are at distance k from i, then we get ∆[k].
N