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

Graph partitioning and eigen polynomials of
Laplacian matrices of Roach-type graphs
Yoshihiro Mizoguchi
Institute of Mathematics for Industry,
Kyushu University
ym@imi.kyushu-u.ac.jp
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. We
can 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)




P0
P1
P2
...
Pn−1
Pn


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 we
assume λs = n−s
n and µs = s
n 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/n
0 . . . 0 1/n −1


are {0, −2/n, −4/n, . . . , −2}.
[IS11] B.Igelnik, D. Simon, The eigenvalues of a tridiagonal matrix in
biogeography, 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/2
0 −r/2 1 + α



Heat equation (Crank-Nicolson method) (2)
and
B =


1 − β r/2 0
r/2 1 − r r/2
...
...
...
r/2 1 − r r/2
0 r/2 1 − β


.
We note un = (un
1
, 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 numerical
method to compute Aun+1 = Bun + c.
[CM10] J.A. Cuminato, S. McKee, A note on the eigenvalues of a special
class of matrices, J. of Computational and Applied Mathematics, 2010.

Tridiagonal Matrix (1)
An =


−α + b c 0 0 · · · 0 0
a b c 0 · · · 0 0
0 a b c · · · 0 0
· · · · · · · · · · · · · · · · · · · · ·
0 0 0 0 · · · b c
0 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 cos
kπ
n
and the corresponding eigenvectors u(k) = (u(k)
j
)
are given by u(k)
j
= ρj−1
sin
k(2j − 1)π
2n
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, Applied
Mathematics E-Notes, 2005.

Tridiagonal Matrix (2)
Consider the n × n matrix C = (min{ai − b, aj − b})i,j=1,...,n.
.
Proposition
..
......
For a > 0 and a b, the tridiagonal matrix of order n
Tn =


1 + a
a−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 b
0 · · · · · · · · · 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)θ
sin θ
where cos θ =
a
2b
.

Tridiagonal matrix Bn and Cn (1)
Let n ≥ 3.
Bn(a0, b0, a, b) =


a0 b0 0 · · · 0
b0
0 An−1(a, b)
...
0


Cn(a, b, a0, b0) =


0
An−1(a, b)
...
0
b0
0 · · · 0 b0 a0


We note that
|Bn(a0, b0, a, b)| = a0|An−1(a, b)| − b2
0
|An−2(a, b)|, and
||Cn(a, b, a0, b0)|| = |a0|An−1(a, b)| − b2
0
|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,
1
2
) =
1
2n−1
cos nα, (λ = 1 + cos α),
Cn(η −
2
3
,
1
3
, η −
1
2
,
1
√
6
) =
1
2 · 3n · sin β
gn(β), (η =
2
3
(1 + cos β)),
Cn(µ −
4
3
,
1
3
, µ −
3
2
,
1
√
6
) =
1
2 · 3n · sin β
hn(β), (µ =
2
3
(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
... b0
0 · · · 0 b0 a0


We note that
|Qn(a0, b0, a, b)| = a0|Cn−1(a, b, a0, b0)| − b2
0
|Cn−2(a, b, a0, b0)|.

Laplacian matrix of a graph
.
Deﬁnition (Weighted normalized Laplacian)
..
......
The weighted normalized Laplacian L(G) = (ℓij) is deﬁned as
ℓij =



1 −
wj j
dj
if i = j,
−
wi j
√
didj
if vi and vj are adjacent and i j,
0 otherwise.
The adjacency matrix A(P5) and the normalized Laplacian matrix L(P5) of
a path graph P5.
A(P5) =


0 1 0 0 0
1 0 1 0 0
0 1 0 1 0
0 0 1 0 1
0 0 0 1 0


L(P5) =


1 − 1
√
2
0 0 0
− 1
√
2
1 −1
2
0 0
0 −1
2
1 −1
2
0
0 0 −1
2
1 − 1
√
2
0 0 0 − 1
√
2
1



Characteristic polynomial of L(Pn)
.
Proposition
..
......
Let n ≥ 4.
|λIn − L(Pn)| = −
(
1
2
)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,
1
2

 .

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
Xt
n,k
Ck(2
3
, −1
3
, 1
2
, − 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 =
2
3
(1 + cos β).
λIn+k − L(Pn,k) =
Bn(λ − 1, 1
√
2
, λ − 1, 1
2
) Xn,k
Xt
n,k
Ck(λ − 2
3
, 1
3
, λ − 1
2
, 1
√
6
)

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 O
Xt
n,k
Ck(1, −1
3
, 1, − 1
√
6
) O Ck(−1
3
, 0, −1
2
, 0)
O O Bn(1, − 1
√
2
, 1, −1
2
) Xn,k
O Ck(−1
3
, 0, −1
2
, 0) Xt
n,k
Ck(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 =
2
3
(1 + cos β) =
2
3
(2 + cos γ).
λIn+k − L(Rn,k) =
Bn(λ − 1, 1
√
2
, λ − 1, 1
2
) Xn,k
Xt
n,k
Ck(λ − 2
3
, 1
3
, λ − 1
2
, 1
√
6
)
×
Bn(λ − 1, 1
√
2
, λ − 1, 1
2
) Xn,k
Xt
n,k
Ck(λ − 4
3
, 1
3
, λ − 3
2
, 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,
1
2
) = (λ − 1) An−1(λ − 1,
1
2
) −
1
2
An−2(λ − 1,
1
2
)
= (λ − 1)
(
1
2
)n−1
sin nα
sin α
−
1
2
(
1
2
)n−2
sin(n − 1)α
sin α
=
(
1
2
)n−1
·
1
sin α
((λ − 1) sin nα − sin(n − 1)α)
=
(
1
2
)n−1
·
1
sin α
(cos α sin nα − sin(nα − α))
=
(
1
2
)n−1
·
1
sin α
(cos nα sin α)
=
(
1
2
)n−1
cos nα.

Bn(λ − 1, 1
√
2
, λ − 1, 1
2
) (Mathematica)

Cn(η − 2
3
, 1
3
, η − 1
2
, 1
√
6
) (Calculation)
Let η =
2
3
(1 + cos β) and gn(β) = 2 sin(n + 1)β + sin nβ − sin(n − 1)β. Then we have
Cn(η −
2
3
,
1
3
, η −
1
2
,
1
√
6
) =
(
η −
1
2
)
An−1
(
η −
2
3
,
1
3
)
−
1
6
An−2
(
η −
2
3
,
1
3
)
=
(
1
3
)n−1 ((
η −
1
2
)
sin nβ
sin β
−
1
2
sin(n − 1)β
sin β
)
=
(
1
3
)n−1 ((
1
6
+
2
3
cos β
)
sin nβ
sin β
−
1
2
sin(n − 1)β
sin β
)
=
(
1
3
)n−1
1
6 sin β
(sin nβ + 4 cos β sin nβ − 3 sin(nβ − β))
=
(
1
3
)n−1
1
6 sin β
(2 sin(n + 1)β + sin nβ − sin(n − 1)β)
=
(
1
3
)n
1
2 sin β
gn(β).

Cn(η − 2
3
, 1
3
, η − 1
2
, 1
√
6
) (Mathematica)
Some manual computations for gn(θ) (× sin).

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 ∅ and
A ∩ B = ∅. Then the normalized cut Ncut(A, B) of G is defined by
Ncut(A, B) = cut(A, B)
(
1
vol(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 × (
1
16
+
1
10
) =
13
40
= 0.325
Ncut(A1, B1) = 3 × (
1
13
+
1
13
) =
6
13
≈ 0.462

Spectral Clustering
.
Deﬁnition (Lcut(G))
..
......
Let G = (V, E) be a connected graph, λ2 the second smallest eigenvalue
of 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 as
Lcut(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 second
eigen 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 path
graph.
1. λ2(L(P4k)) = 1 − π
4k−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 clustering
method.
Concrete formulae of characteristic polynomials of tridiagonal
matrices.
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 for
Pn,k and Rn,k.

Reference I
A. Behn, K. R. Driessel, and I. R. Hentzel.
The eigen-problem for some special near-toeplitz centro-skew
tridiagonal 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.

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