2009/11/10 , 台灣大學, D-bounded distance-regular graphs

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INDIA-TAIWAN CONFERENCE ON DISCRETE MATHEMATICS

D-bounded distance-regular graphs

Chih-wen Weng

(with Yu-pei Huang, Yeh-jong Pan)

Department of Applied Mathematics, National Chiao Tung University

November 10, 2009

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Notations

We always assume Γ = (X , R) is a connected graph with diameter D.

Chih-wen Weng (with Yu-pei Huang, Yeh-jong Pan) (Dep. of A. Math., NCTU)D-bounded distance-regular graphs November 10, 2009 2 / 37

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Notations

We always assume Γ = (X , R) is a connected graph with diameter D. For x ∈ X ,

Γ_i(x ) := {y ∈ X | ∂(x , y ) = i }.

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Γ = (X , R) is distance-regular if and only if for i ≤ D, ci := |C (x, y )|

a_i := |A(x, y )|, b_i := |B(x, y )|

are constantssubject to all vertices x , y with ∂(x , y ) = i , where C (x , y ) = Γ₁(x ) ∩ Γ_{i −1}(y ), A(x , y ) = Γ₁(x ) ∩ Γ_i(y ) and

B(x , y ) = Γ1(x ) ∩ Γi +1(y ).

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∂(x , y ) = i

d y

d x

c_i

ai

b_i

Note that a_i + b_i+ c_i = b0 and k := b0 is the valency of Γ.

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d y

d x

c_i

ai

b_i

Note that a_i + b_i+ c_i = b0 and k := b0 is thevalency of Γ.

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A distance-regular graph is also called a P-polynomial scheme which is an important and interesting mathematical object, and also plays the role as an underlying combinatorial structure of Coding Theory, Design Theory and Group Theory.

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Examples: Hamming Graphs H(D, 2)

Set F2 = {0, 1}, X = F₂^D, and

R = {uv | u, v ∈ X differ in exact one cordinate}.

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Examples: Hamming Graphs H(D, 2)

Set F2 = {0, 1}, X = F₂^D, and

Then Γ = (X , R) is a distance-regular graph of diameter D.

Γ is called the Hamming graph H(D, 2). H(2, 2) is a square and H(3, 2) is a cube.

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Examples: Hamming Graphs H(D, 2)

Set F2 = {0, 1}, X = F₂^D, and

Then Γ = (X , R) is a distance-regular graph of diameter D. Γ is called the Hamming graph H(D, 2).

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Examples: Hamming Graphs H(D, 2)

Set F2 = {0, 1}, X = F₂^D, and

Then Γ = (X , R) is a distance-regular graph of diameter D. Γ is called the Hamming graph H(D, 2). H(2, 2) is a square and H(3, 2) is a cube.

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Examples: Johnson Graphs J(n, D), 2D ≤ n

Set [n] = {1, 2, . . . , n}, X =

[n]

D

(the set of D-subsets of [n]) and

R = {uv | u, v ∈ X , |u ∩ v | = D − 1}.

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Examples: Johnson Graphs J(n, D), 2D ≤ n

Set [n] = {1, 2, . . . , n}, X =

[n]

D

R = {uv | u, v ∈ X , |u ∩ v | = D − 1}.

Then Γ = (X , R) is a distance-regular graph of diameter D.

Γ is called the Johnson graph J(n, D).

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Set [n] = {1, 2, . . . , n}, X =

[n]

D

R = {uv | u, v ∈ X , |u ∩ v | = D − 1}.

Then Γ = (X , R) is a distance-regular graph of diameter D. Γ is called the Johnson graph J(n, D).

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Recall that a sequence x , z, y of vertices of Γ is geodeticwhenever

∂(x , z) + ∂(z, y ) = ∂(x , y ), where ∂ is the distance function of Γ.

A sequence x , z, y of vertices of Γ is weak-geodetic whenever

∂(x , z) + ∂(z, y ) ≤ ∂(x , y ) + 1.

d x

dz

d d y

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∂(x , z) + ∂(z, y ) = ∂(x , y ),

where ∂ is the distance function of Γ. A sequence x , z, y of vertices of Γ is weak-geodetic whenever

∂(x , z) + ∂(z, y ) ≤ ∂(x , y ) + 1.

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∂(x , z) + ∂(z, y ) = ∂(x , y ),

where ∂ is the distance function of Γ. A sequence x , z, y of vertices of Γ is weak-geodetic whenever

∂(x , z) + ∂(z, y ) ≤ ∂(x , y ) + 1.

d x

dz

d d y

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Definition. A subset ∆ ⊆ X is weak-geodetically closed if for any weak-geodetic sequence x , z, y of Γ,

x , y ∈ ∆ =⇒ z ∈ ∆.

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Weak-geodetically closed subgraphs are called strongly closed subgraphsin some literature. If a weak-geodetically closed subgraph ∆ of diameter d is regular then it has valency a_d+ c_d = b₀− b_d, where a_d, c_d, b₀, b_d are intersection numbers of Γ. Furthermore ∆ is distance-regular with intersection numbers a_i(∆) = a_i(Γ) and c_i(∆) = c_i(Γ) for 1 ≤ i ≤ d . '

&

$

% d

y

d x

c_d

a_d

b_d

∆

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Definition. Γ is said to be i-bounded whenever for all x , y ∈ X with

∂(x , y ) ≤ i , there is a regular weak-geodetically closed subgraph of diameter ∂(x , y ) which contains x and y .

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Note that a (D − 1)-bounded distance-regular graph is clear to be D-bounded. The properties of D-bounded distance-regular graphs were studied in [—, D-bounded distance-regular graphs, European Journal of Combinatorics, 18(1997), 211-229], and these properties were used in the classification of classical distance-regular graphs of negative type [—, Classical distance-regular graphs of negative type, J. Combin. Theory Ser.

B, 76(1999), 93-116].

To state our main theorem we need more definitions.

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Note that a (D − 1)-bounded distance-regular graph is clear to be D-bounded. The properties of D-bounded distance-regular graphs were studied in [—, D-bounded distance-regular graphs, European Journal of Combinatorics, 18(1997), 211-229], and these properties were used in the classification of classical distance-regular graphs of negative type [—, Classical distance-regular graphs of negative type, J. Combin. Theory Ser.

B, 76(1999), 93-116].

To state our main theorem we need more definitions.

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A parallelogram of length i

A 4-tuple xywz of vertices in X is a parallelogram of length i if

∂(x , y ) = ∂(w , z) = 1, ∂(x , w ) = ∂(y , w ) = ∂(w , z) = i − 1 and

∂(x , z) = i .

i − 1

1

@

@@ i − 1

i − 1 1

d d

y z

x w

Note that if c_i = 1 then there is no parallelogram of length i .

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A parallelogram of length i

∂(x , y ) = ∂(w , z) = 1, ∂(x , w ) = ∂(y , w ) = ∂(w , z) = i − 1 and

∂(x , z) = i .

i − 1

1

@

@@ i − 1

i − 1 1

d d

y z

x w

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A parallelogram of length i

∂(x , y ) = ∂(w , z) = 1, ∂(x , w ) = ∂(y , w ) = ∂(w , z) = i − 1 and

∂(x , z) = i .

i − 1

1

@

@@ i − 1

i − 1 1

d d

y z

x w

Note that if c_i = 1 then there is no parallelogram of length i .

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A kite of length i

A 4-tuple xywz of vertices in X is a kite of length i if

∂(x , y ) = ∂(x , w ) = ∂(y , w ) = 1, ∂(w , z) = ∂(y , z) = i − 1 and

∂(x , z) = i .

wd HH

HHH

(( (( (( (( (( (( (( ((

1 i − 1

Note that if c_i = 1 or a1 = 0 then there is no kite of length i .

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A kite of length i

∂(x , y ) = ∂(x , w ) = ∂(y , w ) = 1, ∂(w , z) = ∂(y , z) = i − 1 and

∂(x , z) = i .

x d

yd

wd

HH

HHH

h d hh hh hh hh hh hh hh hh

(( (( (( (( (( (( (( (( (

z

1 i − 1

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∂(x , y ) = ∂(x , w ) = ∂(y , w ) = 1, ∂(w , z) = ∂(y , z) = i − 1 and

∂(x , z) = i .

x d

yd

wd

HH

HHH

(( (( (( (( (( (( (( (( (

z

1 i − 1

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A parallelogram of length 2 or a kite of length 2 (K

1,2,1

)

@

@@

d d

y z

x w

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@

@@

d d

y z

x w

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Main Theorem. Let Γ denote a distance-regular graph with diameter D ≥ 3. Suppose the intersection number a₂ 6= 0. Fix an integer

2 ≤ d ≤ D − 1. Then the following two conditions (i), (ii) are equivalent:

(i) Γ is d -bounded.

(ii) Γ contains no parallelograms of any length up to d + 1 and b₁ > b₂.

d d d

'

&

$

% '

&

$

%

∆

Ω

x

∆(x )

Use Ω(x ) ⊂ ∆(x ) to obtain b0− b₁= |Ω(x )| < |∆(x )| = b0− b₂.

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2 ≤ d ≤ D − 1. Then the following two conditions (i), (ii) are equivalent:

(i) Γ is d -bounded.

(ii) Γ contains no parallelograms of any length up to d + 1 and b₁ > b₂.

d d d

'

&

$

% '

&

$

%

∆

Ω

x

∆(x )

Use Ω(x ) ⊂ ∆(x ) to obtain b0− b₁= |Ω(x )| < |∆(x )| = b0− b₂.

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The Proof of d -bounded ⇒ no parallelogram

d

1

@

@@ d

d 1

d d

y z

x w '

&

$

%

∆(x , w )

If a parallelogram of length d + 1 exists as shown above, then x , y , z, w ∈ ∆(x , w ), but ∂(x , z) = d + 1 > d = diameter(∆(x , w )).

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To prove the other direction ”No parallelogram ⇒ d -bounded,” let’s try first to find the nonexistence of many other configurations from the nonexistence of parallelogram.

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Lemma

If Γ contains no parallelogram of any length up to d + 1 then Γ contains no kite of any length up to d + 1.

Proof. If there exists a kite xywz of smallest length 3 ≤ i ≤ d + 1.

x d

yd

wd

HH

HHH

(( (( (( (( (( (( (( (( (

z

1 i − 1

ud

Picku with ∂(u, z) = 1 and ∂(y , u) = i − 2. Then xwuz is a parallelogram of length i , a contradiction.

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Lemma

x d

yd

wd

HH

HHH

(( (( (( (( (( (( (( (( (

z

1 i − 1

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Lemma

x d

yd

wd

HH

HHH

(( (( (( (( (( (( (( (( (

z

1 i − 1

ud

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x d

yd

wd

HH

HHH

(( (( (( (( (( (( (( (( (

z

1 i − 1

ud

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Throughout the talk, we always assume that Γ does not contain parallelogram of any length up to d + 1.

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Non-existence configurations

x d

yd

wd

HH

HHH

d z

1 i − 1

du dv

sets have size c_{i −1}).

3 Similarly, Γ1(z) ∩ Γ_{i −2}(u) = Γ1(z) ∩ Γ_{i −2}(y ).

4 Show Γ1(z) ∩ Γi −2(v ) = Γ1(z) ∩ Γi −2(y ) to have a contradiction.

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Non-existence configurations

x d

yd

wd

HH

HHH

d z

1 i − 1

du dv

Sketch of Proof. Find minimal i that the above configuration exists.

1 Show Γ1(z) ∩ Γ_{i −2}(v ) ⊆ Γ1(z) ∩ Γ_{i −2}(w ).

2 Show Γ1(z) ∩ Γi −2(v ) = Γ1(z) ∩ Γi −2(w ) by finiteness theorem (both sets have size c_{i −1}).

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Non-existence configurations

x d

yd

wd

HH

HHH

d z

1 i − 1

du dv

1 Show Γ₁(z) ∩ Γ_{i −2}(v ) ⊆ Γ₁(z) ∩ Γ_{i −2}(w ).

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Non-existence configurations

x d

yd

wd

HH

HHH

d z

1 i − 1

du dv

1 Show Γ₁(z) ∩ Γ_{i −2}(v ) ⊆ Γ₁(z) ∩ Γ_{i −2}(w ).

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Non-existence configurations

x d

yd

wd

HH

HHH

d z

1 i − 1

du dv

1 Show Γ₁(z) ∩ Γ_{i −2}(v ) ⊆ Γ₁(z) ∩ Γ_{i −2}(w ).

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Non-existence configurations

x d

yd

wd

HH

HHH

d z

1 i − 1

du dv

1 Show Γ₁(z) ∩ Γ_{i −2}(v ) ⊆ Γ₁(z) ∩ Γ_{i −2}(w ).

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Non-existence configurations

x d

yd

wd

HH

HHH

d z

1 i − 1

du

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Non-existence configurations

x d

yd

wd

HH

HHH

d z

1 i − 1

du

Proof.

Skip

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x d

yd

wd

HH

HHH

d z

1 i − 1

du

Proof. Skip

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Non-existence configurations

q q

q q q

q q q q

q q q

q p p p p p p

p p p p p p

x v₂

y

v w y4

z

w4

w3

y₃

u

p₄

p₃ distance to x

0 i − 1 i i + 1 i + 2

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We need a theory to reduce the load of the proof.

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Definition. Assume x ∈ ∆ ⊆ X .. The subset ∆ is weak-geodetically closed with respect to x if for any weak-geodetic sequence x , z, y of Γ,

x , y ∈ ∆ =⇒ z ∈ ∆.

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Theorem

Let Γ be a distance-regular graph with diameter D ≥ 3. Let Ω be a regular subgraph of Γ with valency γ and set d := min{i | γ ≤ c_i + a_i}. Then the following (i),(ii) are equivalent.

(i) Ω is weak-geodetically closed with respect to at least one vertex x ∈ Ω.

(ii) Ω is weak-geodetically closed with diameter d .

In this case γ = c_d+ a_d.

([Theorem 4.6 in —, Weak-geodetically closed subgraphs in

distance-regular graphs, Graphs and Combinatorics, 14(1998), 275-304])

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Theorem

Let Γ be a distance-regular graph with diameter D ≥ 3. Let Ω be a regular subgraph of Γ with valency γ and set d := min{i | γ ≤ c_i + a_i}. Then the following (i),(ii) are equivalent.

(i) Ω is weak-geodetically closed with respect to at least one vertex x ∈ Ω.

(ii) Ω is weak-geodetically closed with diameter d .

In this case γ = c_d+ a_d.

([Theorem 4.6 in —, Weak-geodetically closed subgraphs in

distance-regular graphs, Graphs and Combinatorics, 14(1998), 275-304])

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The construction

Definition

For any vertex x ∈ X and any subset Π ⊆ X , define [x , Π] to be the set {v ∈ X | there exists y⁰ ∈ Π, such that the sequence x, v , y⁰is geodetic }.

For any x , y ∈ X with ∂(x , y ) = d , set

Πxy := {y⁰∈ Γ_d(x ) | B(x , y ) = B(x , y⁰)}

and

∆(x , y ) = [x , Πxy].

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We shall prove that for any vertices x , y ∈ X with ∂(x , y ) = d the following statements Wd, Rd hold.

(W_d) ∆(x , y ) is weak-geodetically closed with respect to x , and (R_d) the subgraph induced on ∆(x , y ) is regular with valency a_d+ c_d.

To prove W_d in the case c2> 1, we use induction on d and induction on d − ∂(x , z) to show v ∈ ∆(x , y ) for any z ∈ ∆(x , y ) and v ∈ A(z, x ) in the following picture.

x d

zd d v

du d w

A A

d w

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To prove W_d in the case c₂ > 1, we use induction on d and induction on d − ∂(x , z) to show v ∈ ∆(x , y ) for any z ∈ ∆(x , y ) and v ∈ A(z, x ) in the following picture.

x d

zd d v

du A

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x d

zd d v

du

d w

A A

d w

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x d

zd d v

du d w

A A

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x d

zd d v

du d w

A A

d w

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In the case c2= 1 to prove Wd is more difficult with the following diagram involved.

x d

zd d v

du d w

A A

d s

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In the case c2= 1 to prove Wd is more difficult with the following diagram involved.

x d

zd d v

du d w

A A

d s

The idea is to show B(x , s) = B(x , u) and use this to show s ∈ ∆(x , y ).

Then v ∈ ∆(x , y ) by the construction.

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Precisely, we need to show the following.

Proposition

For any vertices x , y ∈ X with ∂(x , y ) = d and for any vertex z ∈ ∆(x , y ) ∩ Γi(x ), where 1 ≤ i ≤ d , we have the following (i), (ii).

(i) A(z, x ) ⊆ ∆(x , y ).

(ii) For any vertex w ∈ Γ_i(x ) ∩ Γ₂(z) with B(x , w ) = B(x , z), we have w ∈ ∆(x , y ).

In particular (Wd) holds.

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Precisely, we need to show the following.

Proposition

For any vertices x , y ∈ X with ∂(x , y ) = d and for any vertex z ∈ ∆(x , y ) ∩ Γi(x ), where 1 ≤ i ≤ d , we have the following (i), (ii).

(i) A(z, x ) ⊆ ∆(x , y ).

(ii) For any vertex w ∈ Γ_i(x ) ∩ Γ₂(z) with B(x , w ) = B(x , z), we have w ∈ ∆(x , y ).

In particular (Wd) holds.

The proof is quite technical. To prove (i) we need (ii) to help. The nonexistence of many configurations listed before is used in the proof of (ii).

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The following proves R_d. Proposition

For any vertices x , y ∈ X with ∂(x , y ) = d , ∆(x , y ) is regular with valency a_d+ c_d.

1 ≤ j ≤ d , and xd ∈ Π_xy, it suffices to show

|Γ₁(x_i) ∩ ∆| = a_d+ c_d (1) for 1 ≤ i ≤ d − 1. We show (1) holds for i = 0, d , and for each integer 1 ≤ i ≤ d , we use W_d to show

|Γ₁(x_{i −1}) \ ∆| ≤ |Γ₁(x_i) \ ∆|.

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The following proves R_d. Proposition

For any vertices x , y ∈ X with ∂(x , y ) = d , ∆(x , y ) is regular with valency a_d+ c_d.

Proof.

(Sketch) Since each vertex in ∆ = ∆(x , y ) appears in a sequence of vertices x = x₀, x₁, . . . , x_d in ∆, where ∂(x , x_j) = j , ∂(x_{j −1}, x_j) = 1 for 1 ≤ j ≤ d , and xd ∈ Π_xy, it suffices to show

|Γ₁(x_i) ∩ ∆| = a_d+ c_d (1) for 1 ≤ i ≤ d − 1. We show (1) holds for i = 0, d , and for each integer 1 ≤ i ≤ d , we use W_d to show

|Γ₁(x_{i −1}) \ ∆| ≤ |Γ₁(x_i) \ ∆|.

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Thank you for your attention.

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http://jupiter.math.nctu.edu.tw/∼weng/papers/Dbounded 10 26 2009.pdf