Graph partitioning and characteristic polynomials of Laplacian matrics of Roach-type graphs

Graph partitioning and eigen polynomials ofLaplacian matrices of Roach-type graphs

Yoshihiro Mizoguchi

Institute of Mathematics for Industry,Kyushu University　

[email protected]

Algebraic Graph Theory,Spectral Graph Theory and Related Topics

5th Jan. 2013 at Nagoya University

Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 1 / 32

Table of contents

...1 Introduction

...2 Chebyshev polynomials

...3 Tridiagonal matrices

...4 Laplacian Matrix

...5 Mcut, Lcut and spectral clustering

...6 Conclusion


Biogeography-Based Optimization (1)

Let Ps be the probability that the habitat contains exactly S species. Wecan arrange Ps equations into the single matrix equation

P0

P1

P2...Pn−1

Pn

=

−(λ0 + µ0) µ1 0 · · · 0

λ0 −(λ1 + µ1) µ2. . .

......

. . .. . .

. . ....

.... . . λn−2 −(λn−1 + µn−1) µn

0 . . . 0 λn−1 −(λn + µn)

P0P1P2...Pn−1Pn

where λs and µs are the immigration and emigration rates when there are

S species in the habitat.Generally λ0 > λ1 > · · · > λn and µ0 < µ1 < · · · < µn hold and weassume λs =

n−sn and µs =

sn in this talk.

[Sim08] D.Simon, Biogeography-Based Optimization,IEEE Trans. on evolutionary computation, 2008.


Biogeography-Based Optimization (2)

.Theorem..

......

The (n + 1) eigenvalues of the biogeography matrix

A =

−1 1/n 0 · · · 0

n/n −1 2/n . . ....

.... . .

. . .. . .

......

. . . 2/n −1 n/n0 . . . 0 1/n −1

are {0,−2/n,−4/n, . . . ,−2}.

[IS11] B.Igelnik, D. Simon, The eigenvalues of a tridiagonal matrix inbiogeography, Appl. Mathematics and Computation, 2011.


Heat equation (Crank-Nicolson method) (1)

ut = uxx x ∈ [0, 1] and t > 0

with initial and Dirichlet boundary condition given by:

u(x, 0) = f (x), u(0, t) = g(t) and u(1, t) = h(t)

The finite difference discretization can be expressed as:

Aun+1 = Bun + c

where

A =

1 + α −r/2 0−r/2 1 + r −r/2

. . .. . .

. . .

−r/2 1 + r −r/20 −r/2 1 + α


Heat equation (Crank-Nicolson method) (2)

and

B =

1 − β r/2 0r/2 1 − r r/2

. . .. . .

. . .

r/2 1 − r r/20 r/2 1 − β

.

We note un = (un1, un

2, . . . , un

m)T. The parapmeters α and β are given by:

α = β = 3r/2 for the implicit boundary conditions;

α = r and β = 2r for the explicit boundary conditions

The iteration matrix M(r) = A−1B controls the stability of the numericalmethod to compute Aun+1 = Bun + c.

[CM10] J.A. Cuminato, S. McKee, A note on the eigenvalues of a specialclass of matrices, J. of Computational and Applied Mathematics, 2010.


Tridiagonal Matrix (1)

An =

−α + b c 0 0 · · · 0 0a b c 0 · · · 0 00 a b c · · · 0 0· · · · · · · · · · · · · · · · · · · · ·0 0 0 0 · · · b c0 0 0 0 · · · a −β + b

n×n

.Theorem..

......

Suppose α = β =√

ac , 0. Then the eigenvalues λk of An are given by

λk = b + 2√

ac coskπn

and the corresponding eigenvectors u(k) = (u(k)j

)

are given by u(k)j= ρ j−1 sin

k(2 j − 1)π2n

for k = 1, 2, · · · , n − 1 and

u(n)j= (−ρ) j−1 where ρ =

√a/c.

[Yue05] W-C. Yueh, Eigenvalues of several tridiagonal matrices, AppliedMathematics E-Notes, 2005.


Tridiagonal Matrix (2)

Consider the n × n matrix C = (min{ai − b, a j − b})i, j=1,...,n..Proposition..

......

For a > 0 and a , b, the tridiagonal matrix of order n

Tn =

1 + aa−b −1−1 2 −1

......

...−1 2 −1

−1 1

is the inverse of (1/a)C.

[dF07] C.M. da Fonseca, On the eigenvalues of some tridiagonal matrices,J. of Computational and Applied Mathematics, 2007.


Chebyshev polynomials

For n ∈ N and x ∈ R, we define functions Tn(x) and Un(x) as follows.

T0(x) = 1, T1(x) = x,U0(x) = 1, U1(x) = 2x,

Tn+1(x) = 2xTn(x) − Tn−1(x), and

Un+1(x) = 2xUn(x) − Un−1(x).

We note cos nθ = Tn(cos θ), and sin(n + 1)θ = Un(cos θ) sin θ for θ ∈ R..Proposition..

......

Let x = cos θ. Then

Tn(x) = 0 ⇔ x = cos((2k + 1)π

2n) (k = 0, · · · , n − 1).

Un(x) = 0 ⇔ x = cos(kπ

n + 1) (k = 1, · · · , n).


Tridiagonal matrix An(a, b)We define a n × n matrix An(a, b) as follows:

An(a, b) =

a b 0 · · · · · · · · · 0

b a b 0...

0 b a b 0...

.... . .

. . .. . .

. . .. . .

...... 0 b a b 0... 0 b a b0 · · · · · · · · · 0 b a

.

We put |A0(a, 1)| = 1, then |An(a, 1)| = a|An−1(a, 1)| − |An−2(a, 1)|,|A1(a, 1)| = a and An(a, b) = bn · An (a/b, 1)..Proposition..

......|An(a, b)| = bn ·

sin(n + 1)θsin θ

where cos θ =a

2b.


Tridiagonal matrix Bn and Cn (1)Let n ≥ 3.

Bn(a0, b0, a, b) =

a0 b0 0 · · · 0b00 An−1(a, b)...0

Cn(a, b, a0, b0) =

0

An−1(a, b)...0b0

0 · · · 0 b0 a0

We note that

|Bn(a0, b0, a, b)| = a0|An−1(a, b)| − b20|An−2(a, b)|, and

||Cn(a, b, a0, b0)|| = |a0|An−1(a, b)| − b20|An−2(a, b)||.


Tridiagonal matrix Bn and Cn (2)We define functions

gn(β) = 2 sin((n + 1)β) + sin nβ − sin((n − 1)β), and

hn(β) = 2 sin((n + 1)β) − sin nβ − sin((n − 1)β)

before introducing the next Lemma..Proposition..

......

Let n ≥ 3.∣∣∣∣∣∣∣Bn(λ − 1,1√

2, λ − 1,

12

)

∣∣∣∣∣∣∣ = 12n−1

cos nα, (λ = 1 + cos α),∣∣∣∣∣∣∣Cn(η − 23,

13, η − 1

2,

1√

6)

∣∣∣∣∣∣∣ = 12 · 3n · sin β

gn(β), (η =23

(1 + cos β)),∣∣∣∣∣∣∣Cn(µ − 43,

13, µ − 3

2,

1√

6)

∣∣∣∣∣∣∣ = 12 · 3n · sin β

hn(β), (µ =23

(2 + cos β)).


Tridiagonal matrix Qn(a0, b0, a, b)

Let n ≥ 4. We define n × n matrix Qn(a0, b0, a, b) as follows:

Qn(a0, b0, a, b) =

a0 b0 0 · · · 0

b0...

0 An−2(a, b) 0... b00 · · · 0 b0 a0

We note that

|Qn(a0, b0, a, b)| = a0|Cn−1(a, b, a0, b0)| − b20|Cn−2(a, b, a0, b0)|.


Laplacian matrix of a graph.Definition (Weighted normalized Laplacian)..

......

The weighted normalized Laplacian L(G) = (ℓi j) is defined as

ℓi j =

1 − w j j

d jif i = j,

− wi j√di d j

if vi and v j are adjacent and i , j,

0 otherwise.

The adjacency matrix A(P5) and the normalized Laplacian matrix L(P5) ofa path graph P5.

A(P5) =

0 1 0 0 01 0 1 0 00 1 0 1 00 0 1 0 10 0 0 1 0

L(P5) =

1 − 1√2

0 0 0− 1√

21 − 1

2 0 00 − 1

2 1 − 12 0

0 0 − 12 1 − 1√

20 0 0 − 1√

21


Characteristic polynomial of L(Pn).Proposition..

......

Let n ≥ 4.

|λIn − L(Pn)| = −(

12

)n−2

(sin α sin((n − 1)α))

where λ = 1 + cos α. That is λ = 1 − cos(kπ

n − 1) (k = 0, . . . , n − 1).

We note

L(Pn) = Qn

1,− 1√

2, 1,−1

2

, and

λIn − L(Pn) = Qn

λ − 1,1√

2, λ − 1,

12

.Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 15 / 32

Characteristic polynomial of L(Pn,k) (1)

Let n ≥ 3 and k ≥ 3. Then

L(Pn,k) =

Bn(1,− 1√2, 1,− 1

2 ) Xn,k

X tn,k

Ck( 23 ,−

13 ,

12 ,−

1√6)

where Xn,k is the n × k matrix defined by

Xn,k =

0 · · · · · · 0...

...

0 0...

− 1√6

0 · · · 0

.


Characteristic polynomial of L(Pn,k) (2)

.Proposition..

......

Let

pn,k(λ) =1

2n3k sin β(gk(β) cos(nα)) − gk−1(β) cos((n − 1)α)).

Then ∣∣∣λIn+k − L(Pn,k)∣∣∣ = pn,k(λ),

where λ = 1 + cos α and λ =23

(1 + cos β).

∣∣∣λIn+k − L(Pn,k)∣∣∣ =

∣∣∣∣∣∣∣∣Bn(λ − 1, 1√

2, λ − 1, 1

2 ) Xn,k

X tn,k

Ck(λ − 23 ,

13 , λ −

12 ,

1√6)

∣∣∣∣∣∣∣∣Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 17 / 32

Characteristic polynomial of L(Rn,k) (1)

1 2 3 4 5 6

7 8 9 10 11 12

Let n ≥ 3 and k ≥ 3. Then L(Rn,k) =Bn(1,− 1√

2, 1,− 1

2 ) Xn,k O OX t

n,kCk(1,− 1

3 , 1,−1√6) O Ck(− 1

3 , 0,−12 , 0)

O O Bn(1,− 1√2, 1,− 1

2 ) Xn,k

O Ck(− 13 , 0,−

12 , 0) X t

n,kCk(1,− 1

3 , 1,−1√6)

.


Characteristic polynomial of L(Rn,k) (2).Proposition..

......

Let n ≥ 3, k ≥ 3 and

pn,k(λ) =1

2n3k sin β(gk(β) cos(nα)) − gk−1(β) cos((n − 1)α)), and

qn,k(λ) =1

2n3k sin γ(hk(γ) cos(nα) − hk−1(γ) cos((n − 1)α)). Then

|λIn+k − L(Rn,k)| = pn,k(λ) · qn,k(λ).

where λ = 1 + cos α =23

(1 + cos β) =23

(2 + cos γ).

∣∣∣λIn+k − L(Rn,k)∣∣∣ =

∣∣∣∣∣∣∣∣Bn(λ − 1, 1√

2, λ − 1, 1

2 ) Xn,k

X tn,k

Ck(λ − 23 ,

13 , λ −

12 ,

1√6)

∣∣∣∣∣∣∣∣×

∣∣∣∣∣∣∣∣Bn(λ − 1, 1√

2, λ − 1, 1

2 ) Xn,k

X tn,k

Ck(λ − 43 ,

13 , λ −

32 ,

1√6)

∣∣∣∣∣∣∣∣= pn,k(λ) × qn,k(λ).


∣∣∣∣Bn(λ − 1, 1√2, λ − 1, 1

2 )∣∣∣∣ (Calculation)

Let λ = 1 + cos α. Then we have∣∣∣∣∣∣∣Bn(λ − 1,1√

2, λ − 1,

12

)

∣∣∣∣∣∣∣ = (λ − 1)∣∣∣∣∣An−1(λ − 1,

12

)∣∣∣∣∣ − 1

2

∣∣∣∣∣An−2(λ − 1,12

)∣∣∣∣∣

= (λ − 1)(

12

)n−1 sin nαsin α

− 12

(12

)n−2 sin(n − 1)αsin α

=

(12

)n−1

· 1sin α

((λ − 1) sin nα − sin(n − 1)α)

=

(12

)n−1

· 1sin α

(cos α sin nα − sin(nα − α))

=

(12

)n−1

· 1sin α

(cos nα sin α)

=

(12

)n−1

cos nα.


∣∣∣∣Bn(λ − 1, 1√2, λ − 1, 1

2 )∣∣∣∣ (Mathematica)


∣∣∣∣Cn(η − 23 ,

13 , η −

12 ,

1√6)∣∣∣∣ (Calculation)

Let η =23

(1 + cos β) and gn(β) = 2 sin(n + 1)β + sin nβ − sin(n − 1)β. Then we have∣∣∣∣∣∣∣Cn(η − 23,

13, η − 1

2,

1√

6)

∣∣∣∣∣∣∣ =

(η − 1

2

) ∣∣∣∣∣∣An−1

(η − 2

3,

13

)∣∣∣∣∣∣ − 16

∣∣∣∣∣∣An−2

(η − 2

3,

13

)∣∣∣∣∣∣=

(13

)n−1 ((η − 1

2

) sin nβsin β

− 12

sin(n − 1)βsin β

)=

(13

)n−1 ((16+

23

cos β) sin nβ

sin β− 1

2sin(n − 1)β

sin β

)=

(13

)n−1 16 sin β

(sin nβ + 4 cos β sin nβ − 3 sin(nβ − β))

=

(13

)n−1 16 sin β

(2 sin(n + 1)β + sin nβ − sin(n − 1)β)

=

(13

)n 12 sin β

gn(β).


∣∣∣∣Cn(η − 23 ,

13 , η −

12 ,

1√6)∣∣∣∣ (Mathematica)

Some manual computations for gn(θ) (× sin).Y.Mizoguchi (Kyushu University) Roach-type Graph Laplacian Matrices 2013/01/05 23 / 32

pn,k(λ) (Mathematica)


qn,k(λ) (Mathematica)


Minimum Normalized Cut Mcut(G)

.Definition (Normalized cut)..

......

Let G = (V, E) be a connected graph. Let A, B ⊂ V, A , ∅, B , ∅ andA ∩ B = ∅. Then the normalized cut Ncut(A, B) of G is defined by

Ncut(A, B) = cut(A, B)(

1vol(A)

+1

vol(B)

).

.Definition (Mcut(G))..

......

Let G = (V, E) be a connected graph. The Mcut(G) is defined by

Mcut(G) = min{Mcut j(G) | j = 1, 2, . . . }.

Where,

Mcut j(G) = min{Ncut(A,V \ A) | cut(A,V \ A) = j, A ⊂ V}.


Mcut(R6,3)

G0 (even) G1 (odd)1 2 3 4 5 6

7 8 9 10 11 12

1 2 3 4 5 6

7 8 9 10 11 12

Ncut(A0, B0) = 2 × (116+

110

) =1340= 0.325

Ncut(A1, B1) = 3 × (113+

113

) =6

13≈ 0.462


Spectral Clustering.Definition (Lcut(G))..

......

Let G = (V, E) be a connected graph, λ2 the second smallest eigenvalueof L(G), U2 = ((U2)i) (1 ≤ i ≤ |V|) a second eigenvector of L(G) with λ2.We assume that λ2 is simple. Then Lcut(G) is defined asLcut(G) = Ncut(V+(U2) ∪ V0(U2),V−(U2)).

1

2

3

4

-

-

-

-

5

6

7

+

+

+

1 2 3 4 5 6 7 8 9 10 11

12 13 14 15 16 17 18 19 20 21 22

Lcut(G) = Mcut(G) Lcut(R4,7) = Mcut(R4,7)

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

Mcut(R6,4) Lcut(R6,4)


Roach Graph Observation

.Proposition..

......

Let Rn,k be a roach-type graph. If Lcut(Rn,k) = Mcut(Rn,k) then a secondeigen vector of L(Rn,k) is an even vector.

.Proposition..

......

Let R2k,k be a roach-type graph, P2k,k a weighted path and P4k a pathgraph.

1. λ2(L(P4k)) = 1 − π4k−1 .

2. λ2(L(R2k,k)) < λ2(L(P4k)).3. λ2(L(P4k)) < λ2(L(P2k,k)).4. A second eigenvector of L(R2k,k) is an odd vector.

5. Mcut(R2k,k) < Lcut(R2k,k).


Conclusion

We give followings in this talks:

Tridiagonal matrices, Laplacian of graphs and spectral clusteringmethod.

Concrete formulae of characteristic polynomials of tridiagonalmatrices.

Mathematica computations for characteristic polynomials.

Concrete formulae of eigen-polynomials of (P2k,k) and L(R2k,k).Proof of Lcut does not always give an optimal cut.

We are not able to decide the simpleness of the second eigenvalue forPn,k and Rn,k.


Reference I

A. Behn, K. R. Driessel, and I. R. Hentzel.The eigen-problem for some special near-toeplitz centro-skewtridiagonal matrices.arXiv:1101.5788v1 [math.SP], Jan 2011.

H-W. Chang, S-E. Liu, and R. Burridge.Exact eigensystems for some matrices arising from discretizations.Linear Algebra and its Applications, 430:999–1006, 2009.

J. A. Cuminato and S. McKee.A note on the eigenvalues of a special class of matrices.Journal of Computational and Applied Mathematics, 234:2724–2731,2010.

C. M. da Fonseca.On the eigenvalues of some tridiagonal matrices.Journal of Computational and Applied Mathematics, 200:283–286,2007.


Reference II

B. Igelnik and D. Simon.The eigenvalues of a tridiagonal matrix in biogeography.Applied Mathematics and Computation, 218:195–201, 2011.

S. Kouachi.Eigenvalues and eigenvectors of tridiagonal matrices.Electronic Journal of Linear Algebra, 15:115–133, 2006.

D. Simon.Biogeography-based optimization.IEEE Transactions on Evolutionary Computation, 12(6):702–713,2008.

W. Yueh.Eigenvalues of several tridiagonal matrices.Applied Mathematics E-Notes, 5:66–74, 2005.


Graph partitioning and characteristic polynomials of Laplacian matrics of Roach-type graphs

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