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Laplacian and RandomWalks on Graphs
Linyuan Lu
University of South Carolina
Selected Topics on Spectral Graph Theory (II)Nankai University, Tianjin, May 22, 2014
Five talks
Laplacian and Random Walks on Graphs Linyuan Lu – 2 / 59
Selected Topics on Spectral Graph Theory
1. Graphs with Small Spectral RadiusTime: Friday (May 16) 4pm.-5:30p.m.
2. Laplacian and Random Walks on GraphsTime: Thursday (May 22) 4pm.-5:30p.m.
3. Spectra of Random GraphsTime: Thursday (May 29) 4pm.-5:30p.m.
4. Hypergraphs with Small Spectral RadiusTime: Friday (June 6) 4pm.-5:30p.m.
5. Lapalacian of Random HypergraphsTime: Thursday (June 12) 4pm.-5:30p.m.
Backgrounds
Laplacian and Random Walks on Graphs Linyuan Lu – 3 / 59
Linear Algebra
I
Graph Theory
II
Probability Theory
III
I: Spectral Graph Theory II: Random Graph TheoryIII: Random Matrix Theory
Outline
Laplacian and Random Walks on Graphs Linyuan Lu – 4 / 59
■ Combinatorial Laplacian
■ Normalized Laplacian
■ An application
Graphs and Matrices
Laplacian and Random Walks on Graphs Linyuan Lu – 5 / 59
There are several ways to associate a matrix to a graph G.
■ Adjacency matrix
Graphs and Matrices
Laplacian and Random Walks on Graphs Linyuan Lu – 5 / 59
There are several ways to associate a matrix to a graph G.
■ Adjacency matrix
■ Combinatorial Laplacian
Graphs and Matrices
Laplacian and Random Walks on Graphs Linyuan Lu – 5 / 59
There are several ways to associate a matrix to a graph G.
■ Adjacency matrix
■ Combinatorial Laplacian
■ Normalized Laplacian
...
Basic Graph Notation
Laplacian and Random Walks on Graphs Linyuan Lu – 6 / 59
■ G = (V,E): a simple connected graph on n vertices
Basic Graph Notation
Laplacian and Random Walks on Graphs Linyuan Lu – 6 / 59
■ G = (V,E): a simple connected graph on n vertices■ A(G): the adjacency matrix
Basic Graph Notation
Laplacian and Random Walks on Graphs Linyuan Lu – 6 / 59
■ G = (V,E): a simple connected graph on n vertices■ A(G): the adjacency matrix■ D(G) = diag(d1, d2, . . . , dn): the diagonal degree matrix
Basic Graph Notation
Laplacian and Random Walks on Graphs Linyuan Lu – 6 / 59
■ G = (V,E): a simple connected graph on n vertices■ A(G): the adjacency matrix■ D(G) = diag(d1, d2, . . . , dn): the diagonal degree matrix■ L = D − A: the combinatorial Laplacian
Basic Graph Notation
Laplacian and Random Walks on Graphs Linyuan Lu – 6 / 59
■ G = (V,E): a simple connected graph on n vertices■ A(G): the adjacency matrix■ D(G) = diag(d1, d2, . . . , dn): the diagonal degree matrix■ L = D − A: the combinatorial Laplacian■ L is semi-definite and 1 is always an eigenvector for the
eigenvalue 0.
Basic Graph Notation
Laplacian and Random Walks on Graphs Linyuan Lu – 6 / 59
■ G = (V,E): a simple connected graph on n vertices■ A(G): the adjacency matrix■ D(G) = diag(d1, d2, . . . , dn): the diagonal degree matrix■ L = D − A: the combinatorial Laplacian■ L is semi-definite and 1 is always an eigenvector for the
eigenvalue 0.
① ① ①
①
S4
L(S4) =
3 −1 −1 −1−1 1 0 0−1 0 1 0−1 0 0 1
Basic Graph Notation
Laplacian and Random Walks on Graphs Linyuan Lu – 6 / 59
■ G = (V,E): a simple connected graph on n vertices■ A(G): the adjacency matrix■ D(G) = diag(d1, d2, . . . , dn): the diagonal degree matrix■ L = D − A: the combinatorial Laplacian■ L is semi-definite and 1 is always an eigenvector for the
eigenvalue 0.
① ① ①
①
S4
L(S4) =
3 −1 −1 −1−1 1 0 0−1 0 1 0−1 0 0 1
Combinatorial Laplacian eigenvalues of S4: 0, 1, 1, 4.
Matrix-tree Theorem
Laplacian and Random Walks on Graphs Linyuan Lu – 7 / 59
Kirchhoff’s Matrix-tree Theorem: The (i, j)-cofactor ofD − A equals (−1)i+jt(G), where t(G) is the number ofspanning trees in G.
Matrix-tree Theorem
Laplacian and Random Walks on Graphs Linyuan Lu – 7 / 59
Kirchhoff’s Matrix-tree Theorem: The (i, j)-cofactor ofD − A equals (−1)i+jt(G), where t(G) is the number ofspanning trees in G.
Proof: Fix an orientation of G, let B be the incidencematrix of the orientation, i.e., bve = 1 if v is the head of thearc e, bve = −1 if i is the tail of e, and bve = 0 otherwise.Let L11 be the sub-matrix obtained from L by deleting thefirst row and first column, and B1 be the matrix obtainedfrom B by deleting the first row. Then L11 = B1B
′1.
det(L11) = det(B1B′1)
=∑
S
det(BS)2 By Cauchy-Binet formula
= the number of Spanning Trees. �
An application
Laplacian and Random Walks on Graphs Linyuan Lu – 8 / 59
Corollary: If G is connected, and λ1, . . . , λn−1 be thenon-zero eigenvalues of L. Then the number of spanningtree is
1
nλ1λ2 · · ·λn−1.
An application
Laplacian and Random Walks on Graphs Linyuan Lu – 8 / 59
Corollary: If G is connected, and λ1, . . . , λn−1 be thenon-zero eigenvalues of L. Then the number of spanningtree is
1
nλ1λ2 · · ·λn−1.
Chung-Yau [1999]: The number of spanning trees in anyd-regular graph on n vertices is at most
(1 + o(1))2 log n
dn log d
(
(d− 1)d−1
(d2 − 2d)d/2−1
)n
.
This is best possible within a constant factor.
Normalized Laplacian
Laplacian and Random Walks on Graphs Linyuan Lu – 9 / 59
Normalized Laplacian: L = I −D−1/2AD−1/2.
Normalized Laplacian
Laplacian and Random Walks on Graphs Linyuan Lu – 9 / 59
Normalized Laplacian: L = I −D−1/2AD−1/2.
■ L is always semi-definite.■ 0 is always an eigenvalue of L with eigenvector
(√d1, . . . ,
√dn)
′.
Normalized Laplacian
Laplacian and Random Walks on Graphs Linyuan Lu – 9 / 59
Normalized Laplacian: L = I −D−1/2AD−1/2.
■ L is always semi-definite.■ 0 is always an eigenvalue of L with eigenvector
(√d1, . . . ,
√dn)
′.
① ① ①
①
L(S4) =
1 − 1√
3− 1
√
3− 1
√
3
− 1√
31 0 0
− 1√
30 1 0
− 1√
30 0 1
(Normalized) Laplacian eigenvalues of S4:λ0 = 0, λ1 = λ2 = 1, λ3 = 2.
Facts
Laplacian and Random Walks on Graphs Linyuan Lu – 10 / 59
General properties:
■ The multiplicity of 0 is the number of connectedcomponents.
Facts
Laplacian and Random Walks on Graphs Linyuan Lu – 10 / 59
General properties:
■ The multiplicity of 0 is the number of connectedcomponents.
■ Laplacian eigenvalues: λ0, . . . , λn−1
0 = λ0 ≤ λ1 ≤ · · · ≤ λn−1 ≤ 2.
Facts
Laplacian and Random Walks on Graphs Linyuan Lu – 10 / 59
General properties:
■ The multiplicity of 0 is the number of connectedcomponents.
■ Laplacian eigenvalues: λ0, . . . , λn−1
0 = λ0 ≤ λ1 ≤ · · · ≤ λn−1 ≤ 2.
■ λn−1 = 2 if and only if G is bipartite.
Facts
Laplacian and Random Walks on Graphs Linyuan Lu – 10 / 59
General properties:
■ The multiplicity of 0 is the number of connectedcomponents.
■ Laplacian eigenvalues: λ0, . . . , λn−1
0 = λ0 ≤ λ1 ≤ · · · ≤ λn−1 ≤ 2.
■ λn−1 = 2 if and only if G is bipartite.
■ λ1 > 1 if and only if G is the complete graph.
Rayleigh quotients
Laplacian and Random Walks on Graphs Linyuan Lu – 11 / 59
The Laplacian eigenvalues can also be computed by Rayleighquotients: for 0 ≤ i ≤ n− 1,
λi = supdim(M)=n−i
inff∈M
∑
x∼y(f(x)− f(y))2∑
x f(x)2dx
.
Rayleigh quotients
Laplacian and Random Walks on Graphs Linyuan Lu – 11 / 59
The Laplacian eigenvalues can also be computed by Rayleighquotients: for 0 ≤ i ≤ n− 1,
λi = supdim(M)=n−i
inff∈M
∑
x∼y(f(x)− f(y))2∑
x f(x)2dx
.
In particular, λ1 can be evaluated by
λ1 = inff⊥D1
∑
x∼y(f(x)− f(y))2∑
x f(x)2dx
.
Rayleigh quotients
Laplacian and Random Walks on Graphs Linyuan Lu – 11 / 59
The Laplacian eigenvalues can also be computed by Rayleighquotients: for 0 ≤ i ≤ n− 1,
λi = supdim(M)=n−i
inff∈M
∑
x∼y(f(x)− f(y))2∑
x f(x)2dx
.
In particular, λ1 can be evaluated by
λ1 = inff⊥D1
∑
x∼y(f(x)− f(y))2∑
x f(x)2dx
.
An important parameter
Laplacian and Random Walks on Graphs Linyuan Lu – 12 / 59
λ1 is related to
■ the mixing rate of random walks
An important parameter
Laplacian and Random Walks on Graphs Linyuan Lu – 12 / 59
λ1 is related to
■ the mixing rate of random walks
■ diameter
An important parameter
Laplacian and Random Walks on Graphs Linyuan Lu – 12 / 59
λ1 is related to
■ the mixing rate of random walks
■ diameter
■ neighborhood/edge expansion
An important parameter
Laplacian and Random Walks on Graphs Linyuan Lu – 12 / 59
λ1 is related to
■ the mixing rate of random walks
■ diameter
■ neighborhood/edge expansion
■ Cheeger constant
An important parameter
Laplacian and Random Walks on Graphs Linyuan Lu – 12 / 59
λ1 is related to
■ the mixing rate of random walks
■ diameter
■ neighborhood/edge expansion
■ Cheeger constant
■ quasi-randomness
An important parameter
Laplacian and Random Walks on Graphs Linyuan Lu – 12 / 59
λ1 is related to
■ the mixing rate of random walks
■ diameter
■ neighborhood/edge expansion
■ Cheeger constant
■ quasi-randomness
■ many other applications.
Random walks
Laplacian and Random Walks on Graphs Linyuan Lu – 13 / 59
A walk on a graph is a sequence of vertices together asequence of edges:
v0, v1, v2, v3, . . . , vk, vk+1, . . .
v0v1, v1v2, v2v3, . . . , vkvk+1, . . .
Random walks
Laplacian and Random Walks on Graphs Linyuan Lu – 13 / 59
A walk on a graph is a sequence of vertices together asequence of edges:
v0, v1, v2, v3, . . . , vk, vk+1, . . .
v0v1, v1v2, v2v3, . . . , vkvk+1, . . .
Random walks on a graph G:
fk+1 = fkD−1A.
D−1A ∼ D−1/2AD−1/2 = I−L.λ determines the mixing rate ofrandom walks. ✍✌
✎☞v ✍✌
✎☞
✍✌✎☞
✍✌✎☞
✲��������✒✻
1dv
1dv1
dv
Convergence
Laplacian and Random Walks on Graphs Linyuan Lu – 14 / 59
■ row vector fk: the vertex probability distribution at timek.
fk = f0(D−1A)k.
Convergence
Laplacian and Random Walks on Graphs Linyuan Lu – 14 / 59
■ row vector fk: the vertex probability distribution at timek.
fk = f0(D−1A)k.
■ Stationary distribution π = 1vol(G)(d1, d2, . . . , dn).
π(D−1A) = π.
Convergence
Laplacian and Random Walks on Graphs Linyuan Lu – 14 / 59
■ row vector fk: the vertex probability distribution at timek.
fk = f0(D−1A)k.
■ Stationary distribution π = 1vol(G)(d1, d2, . . . , dn).
π(D−1A) = π.
■ Mixing:
‖(fk − π)D−1/2‖ ≤ λk‖(f0 − π)D−1/2‖.
Convergence
Laplacian and Random Walks on Graphs Linyuan Lu – 14 / 59
■ row vector fk: the vertex probability distribution at timek.
fk = f0(D−1A)k.
■ Stationary distribution π = 1vol(G)(d1, d2, . . . , dn).
π(D−1A) = π.
■ Mixing:
‖(fk − π)D−1/2‖ ≤ λk‖(f0 − π)D−1/2‖.
If G is bipartite, then the random walk does not mix. Inthis case, we will use the α-lazy random walk with thetransition matrix αI + (1− α)D−1A.
Diameter
Laplacian and Random Walks on Graphs Linyuan Lu – 15 / 59
Suppose that G is not a complete graph. Then the diameterof G satisfies
diam(G) ≤⌈
log(vol(G)/δ)
log λn−1+λ1
λn−1−λ1
⌉
,
where δ is the minimum degree of G.
Edge discrepancy
Laplacian and Random Walks on Graphs Linyuan Lu – 16 / 59
Let vol(X) =∑
x∈X dx and λ = max{1− λ1, λn−1 − 1}.Then
∣
∣
∣|E(X, Y )| − vol(X)vol(Y )
vol(G)
∣
∣
∣≤ λ
√vol(X)vol(Y )vol(X)vol(Y )
vol(G) .
X Y
E(X,Y)
Proof
Laplacian and Random Walks on Graphs Linyuan Lu – 17 / 59
Let 1X and 1Y be the indicated vector of X and Yrespectively. Let α0, α1, . . . , αn−1 be orthogonal uniteigenvectors of L. Write D1/2
1X =∑n−1
i=0 xiαi and
D1/21Y =
∑n−1i=0 yiαi. Then
|E(X, Y )| = 1′XA1Y
= (D1/21X)
′(I − L)D1/21Y
=n−1∑
i=0
(1− λi)xiyi.
continue
Laplacian and Random Walks on Graphs Linyuan Lu – 18 / 59
Note x0 =vol(X)√vol(G)
and y0 =vol(Y )√vol(G)
. Hence,
∣
∣
∣
∣
|E(X, Y )| − vol(X)vol(Y )
vol(G)
∣
∣
∣
∣
=n
∑
i=1
(1− λi)xiyi
≤ λ
√
√
√
√
n−1∑
i=1
x2i
√
√
√
√
n−1∑
i=1
y2i
= λ
√
vol(X)vol(Y )vol(X)vol(Y )
vol(G). �
Cheeger Constant
Laplacian and Random Walks on Graphs Linyuan Lu – 19 / 59
For a subset S ⊂ V , we define
hG(S) =|E(S, S)|
min(vol(S), vol(S)).
The Cheeger constant hG of a graph G is defined to behG = minS hG(S).
Cheeger Constant
Laplacian and Random Walks on Graphs Linyuan Lu – 19 / 59
For a subset S ⊂ V , we define
hG(S) =|E(S, S)|
min(vol(S), vol(S)).
The Cheeger constant hG of a graph G is defined to behG = minS hG(S).
Cheeger’s inequality:
2hG ≥ λ1 ≥h2G
2.
d-regular graph
Laplacian and Random Walks on Graphs Linyuan Lu – 20 / 59
If G is d-regular graph, then adjacency matrix, combinatorialLaplacian, and normalize Laplacian are all equivalent.
d-regular graph
Laplacian and Random Walks on Graphs Linyuan Lu – 20 / 59
If G is d-regular graph, then adjacency matrix, combinatorialLaplacian, and normalize Laplacian are all equivalent.
Suppose A has eigenvalues µ1, . . . , µn. Then
■ D − A has eigenvalues d− µ1, . . . , d− µn.
■ I −D−1/2AD−1/2 has eigenvalues1− µ1/d, . . . , 1− µn/d.
d-regular graph
Laplacian and Random Walks on Graphs Linyuan Lu – 20 / 59
If G is d-regular graph, then adjacency matrix, combinatorialLaplacian, and normalize Laplacian are all equivalent.
Suppose A has eigenvalues µ1, . . . , µn. Then
■ D − A has eigenvalues d− µ1, . . . , d− µn.
■ I −D−1/2AD−1/2 has eigenvalues1− µ1/d, . . . , 1− µn/d.
The theories of three matrices apply to the d-regular graphs.
An application
Laplacian and Random Walks on Graphs Linyuan Lu – 21 / 59
Constructing Small Folkman Graphs.
Ramsey’s theorem
Laplacian and Random Walks on Graphs Linyuan Lu – 22 / 59
For integers k, l ≥ 2, there exists a least positive integerR(k, l) such that no matter how the complete graph KR(k,l)
is two-colored, it will contain a blue subgraph Kk or a redsubgraph Kl.
Ramsey’s theorem
Laplacian and Random Walks on Graphs Linyuan Lu – 22 / 59
For integers k, l ≥ 2, there exists a least positive integerR(k, l) such that no matter how the complete graph KR(k,l)
is two-colored, it will contain a blue subgraph Kk or a redsubgraph Kl.
R(3, 3) = 6
R(4, 4) = 18
43 ≤ R(5, 5) ≤ 49
102 ≤ R(6, 6) ≤ 165
...
(1+o(1))
√2
en2n/2 ≤ R(n, n) ≤ (n−1)−C log(n−1)
log log(n−1)
(
2(n− 1)
n− 1
)
.
Spencer[1975] Colon [2009]
Ramsey number R(3, 3) = 6
Laplacian and Random Walks on Graphs Linyuan Lu – 23 / 59
■ If edges of K6 are 2-colored then there exists amonochromatic triangle.
Ramsey number R(3, 3) = 6
Laplacian and Random Walks on Graphs Linyuan Lu – 23 / 59
■ If edges of K6 are 2-colored then there exists amonochromatic triangle.
■ There exists a 2-coloring of edges of K5 with nomonochromatic triangle.
Rado’s arrow notation
Laplacian and Random Walks on Graphs Linyuan Lu – 24 / 59
G → (H): if the edges of G are 2-colored then there exists amonochromatic subgraph of G isomorphic to H.
K6 → (K3)K5 6→ (K3)
Rado’s arrow notation
Laplacian and Random Walks on Graphs Linyuan Lu – 24 / 59
G → (H): if the edges of G are 2-colored then there exists amonochromatic subgraph of G isomorphic to H.
K6 → (K3)K5 6→ (K3)
Fact: If K6 ⊂ G, then G → (K3).
An Erdos-Hajnal Question
Laplacian and Random Walks on Graphs Linyuan Lu – 25 / 59
Is there a K6-free graph G with G → (K3)?
An Erdos-Hajnal Question
Laplacian and Random Walks on Graphs Linyuan Lu – 25 / 59
Is there a K6-free graph G with G → (K3)?
Graham (1968): Yes!
K8 \ C5
Graham’s graph K8 \ C5
Laplacian and Random Walks on Graphs Linyuan Lu – 26 / 59
Suppose G has no monochromatic triangle.
Graham’s graph K8 \ C5
Laplacian and Random Walks on Graphs Linyuan Lu – 27 / 59
Graham’s graph K8 \ C5
Laplacian and Random Walks on Graphs Linyuan Lu – 28 / 59
Graham’s graph K8 \ C5
Laplacian and Random Walks on Graphs Linyuan Lu – 29 / 59
Graham’s graph K8 \ C5
Laplacian and Random Walks on Graphs Linyuan Lu – 30 / 59
Graham’s graph K8 \ C5
Laplacian and Random Walks on Graphs Linyuan Lu – 31 / 59
(b, r)
(r, b)
Label the vertices of C5 by either (r, b) or (b, r).
Graham’s graph K8 \ C5
Laplacian and Random Walks on Graphs Linyuan Lu – 32 / 59
(b, r)
(b, r)
Label the vertices of C5 by either (r, b) or (b, r).A red triangle is unavoidable since χ(C5) = 3.
K5-free G with G → (K3)
Laplacian and Random Walks on Graphs Linyuan Lu – 33 / 59
Year Authors |G|1969 Schauble 421971 Graham, Spencer 231973 Irving 181979 Hadziivanov, Nenov 161981 Nenov 15
K5-free G with G → (K3)
Laplacian and Random Walks on Graphs Linyuan Lu – 33 / 59
Year Authors |G|1969 Schauble 421971 Graham, Spencer 231973 Irving 181979 Hadziivanov, Nenov 161981 Nenov 15
In 1998, Piwakowski, Radziszowski and Urbanski used acomputer-aided exhaustive search to rule out all possiblegraphs on less than 15 vertices.
General results
Laplacian and Random Walks on Graphs Linyuan Lu – 34 / 59
Folkman’s theorem (1970): For any k2 > k1 ≥ 3, thereexists a Kk2-free graph G with G → (Kk1).
General results
Laplacian and Random Walks on Graphs Linyuan Lu – 34 / 59
Folkman’s theorem (1970): For any k2 > k1 ≥ 3, thereexists a Kk2-free graph G with G → (Kk1).
These graphs are called Folkman Graphs.
General results
Laplacian and Random Walks on Graphs Linyuan Lu – 34 / 59
Folkman’s theorem (1970): For any k2 > k1 ≥ 3, thereexists a Kk2-free graph G with G → (Kk1).
These graphs are called Folkman Graphs.
Nesetril-Rodl’s theorem (1976): For p ≥ 2 and any
k2 > k1 ≥ 3, there exists a Kk2-free graph G with
G → (Kk1)p.
Here G → (H)p: if the edges of G are p-colored then thereexists a monochromatic subgraph of G isomorphic to H.
f (p, k1, k2)
Laplacian and Random Walks on Graphs Linyuan Lu – 35 / 59
Let f(p, k1, k2) denote the smallest integer n such that thereexists a Kk2-free graph G on n vertices with G → (Kk1)p.
■ Grahamf(2, 3, 6) = 8.
f (p, k1, k2)
Laplacian and Random Walks on Graphs Linyuan Lu – 35 / 59
Let f(p, k1, k2) denote the smallest integer n such that thereexists a Kk2-free graph G on n vertices with G → (Kk1)p.
■ Grahamf(2, 3, 6) = 8.
■ Nenov, Piwakowski, Radziszowski and Urbanski
f(2, 3, 5) = 15.
f (p, k1, k2)
Laplacian and Random Walks on Graphs Linyuan Lu – 35 / 59
Let f(p, k1, k2) denote the smallest integer n such that thereexists a Kk2-free graph G on n vertices with G → (Kk1)p.
■ Grahamf(2, 3, 6) = 8.
■ Nenov, Piwakowski, Radziszowski and Urbanski
f(2, 3, 5) = 15.
■ What about f(2, 3, 4)?
Upper bound of f (2, 3, 4)
Laplacian and Random Walks on Graphs Linyuan Lu – 36 / 59
■ Folkman, Nesetril-Rodl ’s upper bound is huge.■ Frankl and Rodl (1986)
f(2, 3, 4) ≤ 7× 1011.
Upper bound of f (2, 3, 4)
Laplacian and Random Walks on Graphs Linyuan Lu – 36 / 59
■ Folkman, Nesetril-Rodl ’s upper bound is huge.■ Frankl and Rodl (1986)
f(2, 3, 4) ≤ 7× 1011.
■ Erdos set a prize of $100 for the challenge
f(2, 3, 4) ≤ 1010.
Upper bound of f (2, 3, 4)
Laplacian and Random Walks on Graphs Linyuan Lu – 36 / 59
■ Folkman, Nesetril-Rodl ’s upper bound is huge.■ Frankl and Rodl (1986)
f(2, 3, 4) ≤ 7× 1011.
■ Erdos set a prize of $100 for the challenge
f(2, 3, 4) ≤ 1010.
■ Spencer (1988) claimed the prize.
f(2, 3, 4) ≤ 3× 109.
Upper bound of f (2, 3, 4)
Laplacian and Random Walks on Graphs Linyuan Lu – 36 / 59
■ Folkman, Nesetril-Rodl ’s upper bound is huge.■ Frankl and Rodl (1986)
f(2, 3, 4) ≤ 7× 1011.
■ Erdos set a prize of $100 for the challenge
f(2, 3, 4) ≤ 1010.
■ Spencer (1988) claimed the prize.
f(2, 3, 4) ≤ 3× 109.
Most wanted Folkman Graph
Laplacian and Random Walks on Graphs Linyuan Lu – 37 / 59
Most wanted Folkman Graph
Laplacian and Random Walks on Graphs Linyuan Lu – 38 / 59
Difficulty
Laplacian and Random Walks on Graphs Linyuan Lu – 39 / 59
■ There is no efficient algorithm to test whetherG → (K3).
Difficulty
Laplacian and Random Walks on Graphs Linyuan Lu – 39 / 59
■ There is no efficient algorithm to test whetherG → (K3).
■ For moderate n, Folkman graphs are very rare among allK4-free graphs on n vertices.
Difficulty
Laplacian and Random Walks on Graphs Linyuan Lu – 39 / 59
■ There is no efficient algorithm to test whetherG → (K3).
■ For moderate n, Folkman graphs are very rare among allK4-free graphs on n vertices.
■ Probabilistic methods are generally good choices forasymptotic results. However, it is not good for moderatesize n.
Our approach
Laplacian and Random Walks on Graphs Linyuan Lu – 40 / 59
■ Find a simple and sufficient condition for G → (K3), andan efficient algorithm to verify this condition.
Our approach
Laplacian and Random Walks on Graphs Linyuan Lu – 40 / 59
■ Find a simple and sufficient condition for G → (K3), andan efficient algorithm to verify this condition.
■ Search a special class of graphs so that we have a betterchance of finding a Folkman graph.
Our approach
Laplacian and Random Walks on Graphs Linyuan Lu – 40 / 59
■ Find a simple and sufficient condition for G → (K3), andan efficient algorithm to verify this condition.
■ Search a special class of graphs so that we have a betterchance of finding a Folkman graph.
■ Use spectral analysis instead of probabilistic methods.
Our approach
Laplacian and Random Walks on Graphs Linyuan Lu – 40 / 59
■ Find a simple and sufficient condition for G → (K3), andan efficient algorithm to verify this condition.
■ Search a special class of graphs so that we have a betterchance of finding a Folkman graph.
■ Use spectral analysis instead of probabilistic methods.
■ Localization and δ-fairness.
Our approach
Laplacian and Random Walks on Graphs Linyuan Lu – 40 / 59
■ Find a simple and sufficient condition for G → (K3), andan efficient algorithm to verify this condition.
■ Search a special class of graphs so that we have a betterchance of finding a Folkman graph.
■ Use spectral analysis instead of probabilistic methods.
■ Localization and δ-fairness.
■ Circulant graphs and L(m, s).
My result
Laplacian and Random Walks on Graphs Linyuan Lu – 41 / 59
I received $100-award by provingTheorem [Lu, 2007]: f(2, 3, 4) ≤ 9697.
My result
Laplacian and Random Walks on Graphs Linyuan Lu – 41 / 59
I received $100-award by provingTheorem [Lu, 2007]: f(2, 3, 4) ≤ 9697.
Shortly Dudek and Rodl improved it to f(2, 3, 4) ≤ 941.They also received a $50-award.
My result
Laplacian and Random Walks on Graphs Linyuan Lu – 41 / 59
I received $100-award by provingTheorem [Lu, 2007]: f(2, 3, 4) ≤ 9697.
Shortly Dudek and Rodl improved it to f(2, 3, 4) ≤ 941.They also received a $50-award.
Lange-Radziszowski-Xu [2012+]: f(2, 3, 4) ≤ 786.
Spencer’s Lemma
Laplacian and Random Walks on Graphs Linyuan Lu – 42 / 59
Notations:
- Gv: the induced graph on the the neighborhood of v.- b(H): the maximum size of edge-cuts for H.
Spencer’s Lemma
Laplacian and Random Walks on Graphs Linyuan Lu – 42 / 59
Notations:
- Gv: the induced graph on the the neighborhood of v.- b(H): the maximum size of edge-cuts for H.
Lemma (Spencer) If∑
v b(Gv) <23
∑
v |E(Gv)|, thenG → (K3).
Spencer’s Lemma
Laplacian and Random Walks on Graphs Linyuan Lu – 42 / 59
Notations:
- Gv: the induced graph on the the neighborhood of v.- b(H): the maximum size of edge-cuts for H.
Lemma (Spencer) If∑
v b(Gv) <23
∑
v |E(Gv)|, thenG → (K3).
Localization
Laplacian and Random Walks on Graphs Linyuan Lu – 43 / 59
For 0 < δ < 12 , a graph H is δ-fair if
b(H) < (1
2+ δ)|E(H)|.
Localization
Laplacian and Random Walks on Graphs Linyuan Lu – 43 / 59
For 0 < δ < 12 , a graph H is δ-fair if
b(H) < (1
2+ δ)|E(H)|.
G is a Folkman graph if for each v
- Gv is 16-fair.
- Gv is K3-free.
Localization
Laplacian and Random Walks on Graphs Linyuan Lu – 43 / 59
For 0 < δ < 12 , a graph H is δ-fair if
b(H) < (1
2+ δ)|E(H)|.
G is a Folkman graph if for each v
- Gv is 16-fair.
- Gv is K3-free.
For vertex transitive graph G, all Gv’s are isomorphic.
Spectral lemma
Laplacian and Random Walks on Graphs Linyuan Lu – 44 / 59
- H: a graph on n vertices- A: the adjacency matrix of H- d = (d1, d2, . . . , dn): degrees of H- Vol(S) =
∑
v∈S dv: the volume of S
- d = Vol(H)n : the average degree
Spectral lemma
Laplacian and Random Walks on Graphs Linyuan Lu – 44 / 59
- H: a graph on n vertices- A: the adjacency matrix of H- d = (d1, d2, . . . , dn): degrees of H- Vol(S) =
∑
v∈S dv: the volume of S
- d = Vol(H)n : the average degree
Lemma (Lu) If the smallest eigenvalue of
M = A− 1Vol(H)d ·d′ is greater than −2δd, then H is δ-fair.
Spectral lemma
Laplacian and Random Walks on Graphs Linyuan Lu – 44 / 59
- H: a graph on n vertices- A: the adjacency matrix of H- d = (d1, d2, . . . , dn): degrees of H- Vol(S) =
∑
v∈S dv: the volume of S
- d = Vol(H)n : the average degree
Lemma (Lu) If the smallest eigenvalue of
M = A− 1Vol(H)d ·d′ is greater than −2δd, then H is δ-fair.
Similar results hold for A and L. However, they are weakerthan using M in experiments.
Corollary
Laplacian and Random Walks on Graphs Linyuan Lu – 45 / 59
Corollary Suppose H is a d-regular graph and the smallest
eigenvalue of its adjacency matrix A is greater than −2δd.Then H is δ-fair.
Corollary
Laplacian and Random Walks on Graphs Linyuan Lu – 45 / 59
Corollary Suppose H is a d-regular graph and the smallest
eigenvalue of its adjacency matrix A is greater than −2δd.Then H is δ-fair.
Proof: We can replace M by A in the previous lemma.
- 1 is an eigenvector of A with respect to d.- M is the projection of A to the hyperspace 1
⊥.- M and A have the same smallest eigenvalues.
The proof of the Lemma
Laplacian and Random Walks on Graphs Linyuan Lu – 46 / 59
■ V (H) = X ∪ Y : a partition of the vertex-set.
The proof of the Lemma
Laplacian and Random Walks on Graphs Linyuan Lu – 46 / 59
■ V (H) = X ∪ Y : a partition of the vertex-set.
■ 1X , 1Y : indicated functions of X and Y .
1X + 1Y = 1.
The proof of the Lemma
Laplacian and Random Walks on Graphs Linyuan Lu – 46 / 59
■ V (H) = X ∪ Y : a partition of the vertex-set.
■ 1X , 1Y : indicated functions of X and Y .
1X + 1Y = 1.
■ We observe M1 = 0.
The proof of the Lemma
Laplacian and Random Walks on Graphs Linyuan Lu – 46 / 59
■ V (H) = X ∪ Y : a partition of the vertex-set.
■ 1X , 1Y : indicated functions of X and Y .
1X + 1Y = 1.
■ We observe M1 = 0.
■ For each t ∈ (0, 1), let α(t) = (1− t)1X − t1Y . We have
α(t)′ ·M · α(t) = −e(X, Y ) +1
Vol(H)Vol(X)Vol(Y ).
The proof of the Lemma
Laplacian and Random Walks on Graphs Linyuan Lu – 47 / 59
Let ρ be the smallest eigenvalue of M . We have
e(X, Y )− Vol(X)Vol(Y )
Vol(H)≤ −α(t)′ ·M · α(t) ≤ −ρ‖αt‖2.
The proof of the Lemma
Laplacian and Random Walks on Graphs Linyuan Lu – 47 / 59
Let ρ be the smallest eigenvalue of M . We have
e(X, Y )− Vol(X)Vol(Y )
Vol(H)≤ −α(t)′ ·M · α(t) ≤ −ρ‖αt‖2.
Choose t = |X|n so that ‖α(t)‖2 reaches its minimum |X||Y |
n .
The proof of the Lemma
Laplacian and Random Walks on Graphs Linyuan Lu – 47 / 59
Let ρ be the smallest eigenvalue of M . We have
e(X, Y )− Vol(X)Vol(Y )
Vol(H)≤ −α(t)′ ·M · α(t) ≤ −ρ‖αt‖2.
Choose t = |X|n so that ‖α(t)‖2 reaches its minimum |X||Y |
n .We have
e(X, Y ) ≤ Vol(X)Vol(Y )
Vol(H)+ ρ
|X||Y |n
.
≤ Vol(H)
4− ρ
n
4
< (1
2+ δ)|E(H)|, since ρ > −2δd. �
Circulant graphs
Laplacian and Random Walks on Graphs Linyuan Lu – 48 / 59
- Zn = Z/nZ- S: a subset of Zn satisfying −S = S and 0 6∈ S.
We define a circulant graph H by
- V (H) = Zn
- E(H) = {xy | x− y ∈ S}.
Example: A circulant graphwith n = 8 and S = {±1,±3}.
Spectrum of circulant graphs
Laplacian and Random Walks on Graphs Linyuan Lu – 49 / 59
Lemma: The eigenvalues of the adjacency matrix for the
circulant graph generated by S ⊂ Zn are
∑
s∈Scos
2πis
n
for i = 0, . . . , n− 1.
Spectrum of circulant graphs
Laplacian and Random Walks on Graphs Linyuan Lu – 49 / 59
Lemma: The eigenvalues of the adjacency matrix for the
circulant graph generated by S ⊂ Zn are
∑
s∈Scos
2πis
n
for i = 0, . . . , n− 1.
Proof: Note A =g(J), where
g(x) =∑
s∈Sxs.
J =
0 1 0 · · · 0 00 0 1 · · · 0 00 0 0 · · · 0 0...
...... . . . . . . ...
0 0 0 · · · 0 11 0 0 · · · 0 0
Proof continues...
Laplacian and Random Walks on Graphs Linyuan Lu – 50 / 59
Let φ = e2π
√
−1n denote the primitive n-th unit root.
J has eigenvalues
1, φ, φ2, . . . , φn−1.
Proof continues...
Laplacian and Random Walks on Graphs Linyuan Lu – 50 / 59
Let φ = e2π
√
−1n denote the primitive n-th unit root.
J has eigenvalues
1, φ, φ2, . . . , φn−1.
Thus, the eigenvalues of A = g(J) are
g(1), g(φ), . . . , g(φn−1).
For i = 0, 1, 2, . . . , n− 1, we have
g(φi) = ℜ(g(φi)) =∑
s∈Scos
2πis
n. �
Graph L(m, s)
Laplacian and Random Walks on Graphs Linyuan Lu – 51 / 59
Suppose s and m are relatively prime to each other. Let nbe the least positive integer satisfying
sn ≡ 1 mod m.
Graph L(m, s)
Laplacian and Random Walks on Graphs Linyuan Lu – 51 / 59
Suppose s and m are relatively prime to each other. Let nbe the least positive integer satisfying
sn ≡ 1 mod m.
We define the graph L(m, s) to be a circulant graph on mvertices with
S = {si mod m | i = 0, 1, 2, . . . , n− 1}.
Graph L(m, s)
Laplacian and Random Walks on Graphs Linyuan Lu – 51 / 59
Suppose s and m are relatively prime to each other. Let nbe the least positive integer satisfying
sn ≡ 1 mod m.
We define the graph L(m, s) to be a circulant graph on mvertices with
S = {si mod m | i = 0, 1, 2, . . . , n− 1}.
Proposition: The local graph Gv of L(m, s) is also acirculant graph.
Algorithm
Laplacian and Random Walks on Graphs Linyuan Lu – 52 / 59
■ For each L(m, s), compute the local graph Gv.■ If Gv is not triangle-free, reject it and try a new graph
L(m, s).■ If the ratio the smallest eigenvalue verse the largest
eigenvalue of Gv is less than −13 , reject it and try a new
graph L(m, s).■ Output a Folkman graph L(m, s).
Computational results
Laplacian and Random Walks on Graphs Linyuan Lu – 53 / 59
L(m, s) σL(127, 5) −0.6363 · · ·L(761, 3) −0.5613 · · ·L(785, 53) −0.5404 · · ·L(941, 12) −0.5376 · · ·L(1777, 53) −0.5216 · · ·L(1801, 125) −0.4912 · · ·L(2641, 2) −0.4275 · · ·L(9697, 4) −0.3307 · · ·L(30193, 53) −0.3094 · · ·L(33121, 2) −0.2665 · · ·L(57401, 7) −0.3289 · · ·
■ σ is the ratioof the smallesteigenvalue to thelargest eigenvaluein the local graph.
■ All graphs on theleft are K4-free.
■ Graphs in red areFolkman graphs.
■ Graphs in blackare good candi-dates.
Improvements
Laplacian and Random Walks on Graphs Linyuan Lu – 54 / 59
Our method has inspired two improvements.
■ Dudek-Rodl [2008]: f(2, 3, 4) ≤ 941.
Improvements
Laplacian and Random Walks on Graphs Linyuan Lu – 54 / 59
Our method has inspired two improvements.
■ Dudek-Rodl [2008]: f(2, 3, 4) ≤ 941.
■ Lange-Radziszowski-Xu [2012+]: f(2, 3, 4) ≤ 786.
Dudek and Rodl
Laplacian and Random Walks on Graphs Linyuan Lu – 55 / 59
Given a graph G, a triangle graph HG is defined as
■ V (HG) = E(G)■ e1 ∼ e2 in HG if e1 and e2 belong to the same triangle of
G.
Dudek and Rodl
Laplacian and Random Walks on Graphs Linyuan Lu – 55 / 59
Given a graph G, a triangle graph HG is defined as
■ V (HG) = E(G)■ e1 ∼ e2 in HG if e1 and e2 belong to the same triangle of
G.
Dudek, Rodl, 2008
■ If b(HG) <23 |E(HG)|, then G → (K3).
■ If HG is 16-fair, then G → (K3).
Dudek and Rodl
Laplacian and Random Walks on Graphs Linyuan Lu – 56 / 59
Theorem [Dudek, Rodl, 2008]
f(2, 3, 4) ≤ 941.
Dudek and Rodl
Laplacian and Random Walks on Graphs Linyuan Lu – 56 / 59
Theorem [Dudek, Rodl, 2008]
f(2, 3, 4) ≤ 941.
Proof: Let G = L(941, 12). Then G is 188 regular. Thetriangle graph H has 941 ∗ 188/2 = 88454 vertices and2122896 edges.
Dudek and Rodl
Laplacian and Random Walks on Graphs Linyuan Lu – 56 / 59
Theorem [Dudek, Rodl, 2008]
f(2, 3, 4) ≤ 941.
Proof: Let G = L(941, 12). Then G is 188 regular. Thetriangle graph H has 941 ∗ 188/2 = 88454 vertices and2122896 edges.
Using Matlab, they calculate the least eigenvalue
µn ≥ −15.196 > −(1
2+
1
6)24.
So H is 16-fair. Done.
Lange-Radziszowski-Xu
Laplacian and Random Walks on Graphs Linyuan Lu – 57 / 59
Instead of spectral methods, they use semi-definite program(SDP) to approximate the MAX-CUT problem.
■ First they try the graph G1 obtained from L(941, 12) bydeleting 81 vertices. They showed
3b(HG1) < 1084985 < 1085028 = 2|E(HG1
)|.
This implies f(2, 3, 4) ≤ 860.
Lange-Radziszowski-Xu
Laplacian and Random Walks on Graphs Linyuan Lu – 57 / 59
Instead of spectral methods, they use semi-definite program(SDP) to approximate the MAX-CUT problem.
■ First they try the graph G1 obtained from L(941, 12) bydeleting 81 vertices. They showed
3b(HG1) < 1084985 < 1085028 = 2|E(HG1
)|.
This implies f(2, 3, 4) ≤ 860.
■ Second they try the graph G2 obtained from L(785, 53)by one vertex and some 60 edges. They showed
3b(HG2) < 857750 < 857762 = 2|E(HG2
)|.
This implies f(2, 3, 4) ≤ 786.
Open questions
Laplacian and Random Walks on Graphs Linyuan Lu – 58 / 59
■ Exoo conjectured L(127, 5) is a Folkman graph.
■ In 2012 SIAMDM, Ronald Graham announced a $100award for determining if f(2, 3, 4) < 100.
■ A open problem on 3-colors: prove or disprove
f(3, 3, 4) ≤ 334
.
References
Laplacian and Random Walks on Graphs Linyuan Lu – 59 / 59
1. Linyuan Lu, Explicit Construction of Small FolkmanGraphs. SIAM Journal on Discrete Mathematics,21(4):1053-1060, January 2008.
2. Andrzej Dudek and Vojtech Rodl. On the FolkmanNumber f(2, 3, 4), Experimental Mathematics,17(1):63-67, 2008.
3. A. Lange, S. Radziszowski, and X. Xu, Use ofMAX-CUT for Ramsey Arrowing of Triangles,http://arxiv.org/pdf/1207.3750.pdf.
Homepage: http://www.math.sc.edu/∼ lu/
Thank You