Laplacian and Random Walks on Graphs

Laplacian and Random Walks on Graphs Linyuan Lu University of South Carolina Selected Topics on Spectral Graph Theory (II) Nankai University, Tianjin, May 22, 2014

Five talks Selected Topics on Spectral Graph Theory 1. Graphs with Small Spectral Radius Time: Friday (May 16) 4pm.-5:30p.m. 2. Laplacian and Random Walks on Graphs Time: Thursday (May 22) 4pm.-5:30p.m. 3. Spectra of Random Graphs Time: Thursday (May 29) 4pm.-5:30p.m. 4. Hypergraphs with Small Spectral Radius Time: Friday (June 6) 4pm.-5:30p.m. 5. Lapalacian of Random Hypergraphs Time: Thursday (June 12) 4pm.-5:30p.m. Laplacian and Random Walks on Graphs Linyuan Lu 2 / 59

Backgrounds Linear Algebra III I II Graph Theory Probability Theory I: Spectral Graph Theory II: Random Graph Theory III: Random Matrix Theory Laplacian and Random Walks on Graphs Linyuan Lu 3 / 59

Outline Combinatorial Laplacian Normalized Laplacian An application Laplacian and Random Walks on Graphs Linyuan Lu 4 / 59

Graphs and Matrices There are several ways to associate a matrix to a graph G. Adjacency matrix Laplacian and Random Walks on Graphs Linyuan Lu 5 / 59

Graphs and Matrices There are several ways to associate a matrix to a graph G. Adjacency matrix Combinatorial Laplacian Laplacian and Random Walks on Graphs Linyuan Lu 5 / 59

Graphs and Matrices There are several ways to associate a matrix to a graph G. Adjacency matrix Combinatorial Laplacian Normalized Laplacian. Laplacian and Random Walks on Graphs Linyuan Lu 5 / 59

Basic Graph Notation G = (V,E): a simple connected graph on n vertices Laplacian and Random Walks on Graphs Linyuan Lu 6 / 59

Basic Graph Notation G = (V,E): a simple connected graph on n vertices A(G): the adjacency matrix Laplacian and Random Walks on Graphs Linyuan Lu 6 / 59

Basic Graph Notation G = (V,E): a simple connected graph on n vertices A(G): the adjacency matrix D(G) = diag(d 1,d 2,...,d n ): the diagonal degree matrix Laplacian and Random Walks on Graphs Linyuan Lu 6 / 59

Basic Graph Notation G = (V,E): a simple connected graph on n vertices A(G): the adjacency matrix D(G) = diag(d 1,d 2,...,d n ): the diagonal degree matrix L = D A: the combinatorial Laplacian L is semi-definite and 1 is always an eigenvector for the eigenvalue 0. Laplacian and Random Walks on Graphs Linyuan Lu 6 / 59

Basic Graph Notation G = (V,E): a simple connected graph on n vertices A(G): the adjacency matrix D(G) = diag(d 1,d 2,...,d n ): 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 S 4 L(S 4 ) = 3 1 1 1 1 1 0 0 1 0 1 0 1 0 0 1 Laplacian and Random Walks on Graphs Linyuan Lu 6 / 59

Basic Graph Notation G = (V,E): a simple connected graph on n vertices A(G): the adjacency matrix D(G) = diag(d 1,d 2,...,d n ): 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 S 4 L(S 4 ) = 3 1 1 1 1 1 0 0 1 0 1 0 1 0 0 1 Combinatorial Laplacian eigenvalues of S 4 : 0,1,1,4. Laplacian and Random Walks on Graphs Linyuan Lu 6 / 59

Matrix-tree Theorem Kirchhoff s Matrix-tree Theorem: The (i, j)-cofactor of D A equals ( 1) i+j t(g), where t(g) is the number of spanning trees in G. Laplacian and Random Walks on Graphs Linyuan Lu 7 / 59

Matrix-tree Theorem Kirchhoff s Matrix-tree Theorem: The (i, j)-cofactor of D A equals ( 1) i+j t(g), where t(g) is the number of spanning trees in G. Proof: Fix an orientation of G, let B be the incidence matrix of the orientation, i.e., b ve = 1 if v is the head of the arc e, b ve = 1 if i is the tail of e, and b ve = 0 otherwise. Let L 11 be the sub-matrix obtained from L by deleting the first row and first column, and B 1 be the matrix obtained from B by deleting the first row. Then L 11 = B 1 B 1. det(l 11 ) = det(b 1 B 1) = det(b S ) 2 By Cauchy-Binet formula S = the number of Spanning Trees. Laplacian and Random Walks on Graphs Linyuan Lu 7 / 59

An application Corollary: If G is connected, and λ 1,...,λ n 1 be the non-zero eigenvalues of L. Then the number of spanning tree is 1 n λ 1λ 2 λ n 1. Laplacian and Random Walks on Graphs Linyuan Lu 8 / 59

An application Corollary: If G is connected, and λ 1,...,λ n 1 be the non-zero eigenvalues of L. Then the number of spanning tree is 1 n λ 1λ 2 λ n 1. Chung-Yau [1999]: The number of spanning trees in any d-regular graph on n vertices is at most (1+o(1)) 2logn dnlogd ( (d 1) d 1 (d 2 2d) d/2 1 ) n. This is best possible within a constant factor. Laplacian and Random Walks on Graphs Linyuan Lu 8 / 59

Normalized Laplacian Normalized Laplacian: L = I D 1/2 AD 1/2. Laplacian and Random Walks on Graphs Linyuan Lu 9 / 59

Normalized Laplacian Normalized Laplacian: L = I D 1/2 AD 1/2. L is always semi-definite. 0 is always an eigenvalue of L with eigenvector ( d 1,..., d n ). Laplacian and Random Walks on Graphs Linyuan Lu 9 / 59

Normalized Laplacian Normalized Laplacian: L = I D 1/2 AD 1/2. L is always semi-definite. 0 is always an eigenvalue of L with eigenvector ( d 1,..., d n ). 1 1 1 1 L(S 4 ) = 1 1 3 1 3 1 3 3 1 0 0 3 0 1 0 3 0 0 1 1 1 1 (Normalized) Laplacian eigenvalues of S 4 : λ 0 = 0, λ 1 = λ 2 = 1, λ 3 = 2. Laplacian and Random Walks on Graphs Linyuan Lu 9 / 59

Facts General properties: The multiplicity of 0 is the number of connected components. Laplacian and Random Walks on Graphs Linyuan Lu 10 / 59

Facts General properties: The multiplicity of 0 is the number of connected components. Laplacian eigenvalues: λ 0,...,λ n 1 0 = λ 0 λ 1 λ n 1 2. Laplacian and Random Walks on Graphs Linyuan Lu 10 / 59

Facts General properties: The multiplicity of 0 is the number of connected components. Laplacian eigenvalues: λ 0,...,λ n 1 0 = λ 0 λ 1 λ n 1 2. λ n 1 = 2 if and only if G is bipartite. Laplacian and Random Walks on Graphs Linyuan Lu 10 / 59

Facts General properties: The multiplicity of 0 is the number of connected components. Laplacian eigenvalues: λ 0,...,λ n 1 0 = λ 0 λ 1 λ n 1 2. λ n 1 = 2 if and only if G is bipartite. λ 1 > 1 if and only if G is the complete graph. Laplacian and Random Walks on Graphs Linyuan Lu 10 / 59

Rayleigh quotients The Laplacian eigenvalues can also be computed by Rayleigh quotients: for 0 i n 1, λ i = sup dim(m)=n i inf f M x y (f(x) f(y))2 x f(x)2 d x. Laplacian and Random Walks on Graphs Linyuan Lu 11 / 59

Rayleigh quotients The Laplacian eigenvalues can also be computed by Rayleigh quotients: for 0 i n 1, λ i = sup dim(m)=n i inf f M x y (f(x) f(y))2 x f(x)2 d x. In particular, λ 1 can be evaluated by λ 1 = inf f D1 x y (f(x) f(y))2 x f(x)2 d x. Laplacian and Random Walks on Graphs Linyuan Lu 11 / 59

An important parameter λ 1 is related to the mixing rate of random walks Laplacian and Random Walks on Graphs Linyuan Lu 12 / 59

An important parameter λ 1 is related to the mixing rate of random walks diameter Laplacian and Random Walks on Graphs Linyuan Lu 12 / 59

An important parameter λ 1 is related to the mixing rate of random walks diameter neighborhood/edge expansion Laplacian and Random Walks on Graphs Linyuan Lu 12 / 59

An important parameter λ 1 is related to the mixing rate of random walks diameter neighborhood/edge expansion Cheeger constant Laplacian and Random Walks on Graphs Linyuan Lu 12 / 59

An important parameter λ 1 is related to the mixing rate of random walks diameter neighborhood/edge expansion Cheeger constant quasi-randomness Laplacian and Random Walks on Graphs Linyuan Lu 12 / 59

An important parameter λ 1 is related to the mixing rate of random walks diameter neighborhood/edge expansion Cheeger constant quasi-randomness many other applications. Laplacian and Random Walks on Graphs Linyuan Lu 12 / 59

Random walks A walk on a graph is a sequence of vertices together a sequence of edges: v 0,v 1,v 2,v 3,...,v k,v k+1,... v 0 v 1,v 1 v 2,v 2 v 3,...,v k v k+1,... Laplacian and Random Walks on Graphs Linyuan Lu 13 / 59

Random walks A walk on a graph is a sequence of vertices together a sequence of edges: v 0,v 1,v 2,v 3,...,v k,v k+1,... v 0 v 1,v 1 v 2,v 2 v 3,...,v k v k+1,... Random walks on a graph G: f k+1 = f k D 1 A. D 1 A D 1/2 AD 1/2 = I L. λ determines the mixing rate of random walks. 1 d v 1 d v v 1 d v Laplacian and Random Walks on Graphs Linyuan Lu 13 / 59

Convergence row vector f k : the vertex probability distribution at time k. f k = f 0 (D 1 A) k. Laplacian and Random Walks on Graphs Linyuan Lu 14 / 59

Convergence row vector f k : the vertex probability distribution at time k. f k = f 0 (D 1 A) k. Stationary distribution π = 1 vol(g) (d 1,d 2,...,d n ). π(d 1 A) = π. Laplacian and Random Walks on Graphs Linyuan Lu 14 / 59

Convergence row vector f k : the vertex probability distribution at time k. f k = f 0 (D 1 A) k. Stationary distribution π = 1 vol(g) (d 1,d 2,...,d n ). Mixing: π(d 1 A) = π. (f k π)d 1/2 λ k (f 0 π)d 1/2. If G is bipartite, then the random walk does not mix. In this case, we will use the α-lazy random walk with the transition matrix αi +(1 α)d 1 A. Laplacian and Random Walks on Graphs Linyuan Lu 14 / 59

Diameter Suppose that G is not a complete graph. Then the diameter of G satisfies log(vol(g)/δ) diam(g) log λ n 1+λ 1, λ n 1 λ 1 where δ is the minimum degree of G. Laplacian and Random Walks on Graphs Linyuan Lu 15 / 59

Edge discrepancy Let vol(x) = x X d x and λ = max{1 λ 1,λ n 1 1}. Then E(X,Y) vol(x)vol(y) vol(g) λ vol(x)vol(y)vol( X)vol(Ȳ) vol(g). X Y E(X,Y) Laplacian and Random Walks on Graphs Linyuan Lu 16 / 59

Proof Let 1 X and 1 Y be the indicated vector of X and Y respectively. Let α 0,α 1,...,α n 1 be orthogonal unit eigenvectors of L. Write D 1/2 1 X = n 1 i=0 x iα i and D 1/2 1 Y = n 1 i=0 y iα i. Then E(X,Y) = 1 XA1 Y = (D 1/2 1 X ) (I L)D 1/2 1 Y n 1 = (1 λ i )x i y i. i=0 Laplacian and Random Walks on Graphs Linyuan Lu 17 / 59

continue Note x 0 = vol(x) and y 0 = vol(y). Hence, vol(g) vol(g) vol(x)vol(y) E(X,Y) vol(g) n = (1 λ i )x i y i i=1 λ n 1 i=1 x 2 n 1 i i=1 y 2 i = λ vol(x)vol(y)vol( X)vol( Ȳ). vol(g) Laplacian and Random Walks on Graphs Linyuan Lu 18 / 59

Cheeger Constant For a subset S V, we define h G (S) = E(S, S) min(vol(s),vol( S)). The Cheeger constant h G of a graph G is defined to be h G = min S h G (S). Laplacian and Random Walks on Graphs Linyuan Lu 19 / 59

Cheeger Constant For a subset S V, we define h G (S) = E(S, S) min(vol(s),vol( S)). The Cheeger constant h G of a graph G is defined to be h G = min S h G (S). Cheeger s inequality: 2h G λ 1 h2 G 2. Laplacian and Random Walks on Graphs Linyuan Lu 19 / 59

d-regular graph If G is d-regular graph, then adjacency matrix, combinatorial Laplacian, and normalize Laplacian are all equivalent. Laplacian and Random Walks on Graphs Linyuan Lu 20 / 59

d-regular graph If G is d-regular graph, then adjacency matrix, combinatorial Laplacian, and normalize Laplacian are all equivalent. Suppose A has eigenvalues µ 1,...,µ n. Then D A has eigenvalues d µ 1,...,d µ n. I D 1/2 AD 1/2 has eigenvalues 1 µ 1 /d,...,1 µ n /d. Laplacian and Random Walks on Graphs Linyuan Lu 20 / 59

d-regular graph If G is d-regular graph, then adjacency matrix, combinatorial Laplacian, and normalize Laplacian are all equivalent. Suppose A has eigenvalues µ 1,...,µ n. Then D A has eigenvalues d µ 1,...,d µ n. I D 1/2 AD 1/2 has eigenvalues 1 µ 1 /d,...,1 µ n /d. The theories of three matrices apply to the d-regular graphs. Laplacian and Random Walks on Graphs Linyuan Lu 20 / 59

An application Constructing Small Folkman Graphs. Laplacian and Random Walks on Graphs Linyuan Lu 21 / 59

Ramsey s theorem For integers k,l 2, there exists a least positive integer R(k,l) such that no matter how the complete graph K R(k,l) is two-colored, it will contain a blue subgraph K k or a red subgraph K l. R(3,3) = 6 R(4,4) = 18 43 R(5,5) 49 102 R(6,6) 165. ( ) 2 (1+o(1)) e n2n/2 R(n,n) (n 1) C log(n 1) log log(n 1) 2(n 1). n 1 Spencer[1975] Colon [2009] Laplacian and Random Walks on Graphs Linyuan Lu 22 / 59

Ramsey number R(3,3) = 6 If edges of K 6 are 2-colored then there exists a monochromatic triangle. Laplacian and Random Walks on Graphs Linyuan Lu 23 / 59

Ramsey number R(3,3) = 6 If edges of K 6 are 2-colored then there exists a monochromatic triangle. There exists a 2-coloring of edges of K 5 with no monochromatic triangle. Laplacian and Random Walks on Graphs Linyuan Lu 23 / 59

Rado s arrow notation G (H): if the edges of G are 2-colored then there exists a monochromatic subgraph of G isomorphic to H. K 6 (K 3 ) K 5 (K 3 ) Laplacian and Random Walks on Graphs Linyuan Lu 24 / 59

Rado s arrow notation G (H): if the edges of G are 2-colored then there exists a monochromatic subgraph of G isomorphic to H. K 6 (K 3 ) K 5 (K 3 ) Fact: If K 6 G, then G (K 3 ). Laplacian and Random Walks on Graphs Linyuan Lu 24 / 59

An Erdős-Hajnal Question Is there a K 6 -free graph G with G (K 3 )? Laplacian and Random Walks on Graphs Linyuan Lu 25 / 59

An Erdős-Hajnal Question Is there a K 6 -free graph G with G (K 3 )? Graham (1968): Yes! K 8 \C 5 Laplacian and Random Walks on Graphs Linyuan Lu 25 / 59

Graham s graph K 8 \C 5 Suppose G has no monochromatic triangle. Laplacian and Random Walks on Graphs Linyuan Lu 26 / 59

Graham s graph K 8 \C 5 Laplacian and Random Walks on Graphs Linyuan Lu 27 / 59

Graham s graph K 8 \C 5 Laplacian and Random Walks on Graphs Linyuan Lu 28 / 59

Graham s graph K 8 \C 5 Laplacian and Random Walks on Graphs Linyuan Lu 29 / 59

Graham s graph K 8 \C 5 Laplacian and Random Walks on Graphs Linyuan Lu 30 / 59

Graham s graph K 8 \C 5 (b, r) (r, b) Label the vertices of C 5 by either (r,b) or (b,r). Laplacian and Random Walks on Graphs Linyuan Lu 31 / 59

Graham s graph K 8 \C 5 (b, r) (b, r) Label the vertices of C 5 by either (r,b) or (b,r). A red triangle is unavoidable since χ(c 5 ) = 3. Laplacian and Random Walks on Graphs Linyuan Lu 32 / 59

K 5 -free G with G (K 3 ) Year Authors G 1969 Sch auble 42 1971 Graham, Spencer 23 1973 Irving 18 1979 Hadziivanov, Nenov 16 1981 Nenov 15 Laplacian and Random Walks on Graphs Linyuan Lu 33 / 59

K 5 -free G with G (K 3 ) Year Authors G 1969 Sch auble 42 1971 Graham, Spencer 23 1973 Irving 18 1979 Hadziivanov, Nenov 16 1981 Nenov 15 In 1998, Piwakowski, Radziszowski and Urbański used a computer-aided exhaustive search to rule out all possible graphs on less than 15 vertices. Laplacian and Random Walks on Graphs Linyuan Lu 33 / 59

General results Folkman s theorem (1970): For any k 2 > k 1 3, there exists a K k2 -free graph G with G (K k1 ). Laplacian and Random Walks on Graphs Linyuan Lu 34 / 59

General results Folkman s theorem (1970): For any k 2 > k 1 3, there exists a K k2 -free graph G with G (K k1 ). These graphs are called Folkman Graphs. Laplacian and Random Walks on Graphs Linyuan Lu 34 / 59

General results Folkman s theorem (1970): For any k 2 > k 1 3, there exists a K k2 -free graph G with G (K k1 ). These graphs are called Folkman Graphs. Nešetřil-Rödl s theorem (1976): For p 2 and any k 2 > k 1 3, there exists a K k2 -free graph G with G (K k1 ) p. Here G (H) p : if the edges of G are p-colored then there exists a monochromatic subgraph of G isomorphic to H. Laplacian and Random Walks on Graphs Linyuan Lu 34 / 59

f(p,k 1,k 2 ) Let f(p,k 1,k 2 ) denote the smallest integer n such that there exists a K k2 -free graph G on n vertices with G (K k1 ) p. Graham f(2,3,6) = 8. Laplacian and Random Walks on Graphs Linyuan Lu 35 / 59

f(p,k 1,k 2 ) Let f(p,k 1,k 2 ) denote the smallest integer n such that there exists a K k2 -free graph G on n vertices with G (K k1 ) p. Graham f(2,3,6) = 8. Nenov, Piwakowski, Radziszowski and Urbański f(2,3,5) = 15. Laplacian and Random Walks on Graphs Linyuan Lu 35 / 59

f(p,k 1,k 2 ) Let f(p,k 1,k 2 ) denote the smallest integer n such that there exists a K k2 -free graph G on n vertices with G (K k1 ) p. Graham f(2,3,6) = 8. Nenov, Piwakowski, Radziszowski and Urbański What about f(2,3,4)? f(2,3,5) = 15. Laplacian and Random Walks on Graphs Linyuan Lu 35 / 59

Upper bound of f(2,3,4) Folkman, Nešetřil-Rödl s upper bound is huge. Frankl and Rödl (1986) f(2,3,4) 7 10 11. Laplacian and Random Walks on Graphs Linyuan Lu 36 / 59

Upper bound of f(2,3,4) Folkman, Nešetřil-Rödl s upper bound is huge. Frankl and Rödl (1986) f(2,3,4) 7 10 11. Erdős set a prize of $100 for the challenge f(2,3,4) 10 10. Laplacian and Random Walks on Graphs Linyuan Lu 36 / 59

Upper bound of f(2,3,4) Folkman, Nešetřil-Rödl s upper bound is huge. Frankl and Rödl (1986) f(2,3,4) 7 10 11. Erdős set a prize of $100 for the challenge f(2,3,4) 10 10. Spencer (1988) claimed the prize. f(2,3,4) 3 10 9. Laplacian and Random Walks on Graphs Linyuan Lu 36 / 59

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 There is no efficient algorithm to test whether G (K 3 ). Laplacian and Random Walks on Graphs Linyuan Lu 39 / 59

Difficulty There is no efficient algorithm to test whether G (K 3 ). For moderate n, Folkman graphs are very rare among all K 4 -free graphs on n vertices. Laplacian and Random Walks on Graphs Linyuan Lu 39 / 59

Difficulty There is no efficient algorithm to test whether G (K 3 ). For moderate n, Folkman graphs are very rare among all K 4 -free graphs on n vertices. Probabilistic methods are generally good choices for asymptotic results. However, it is not good for moderate size n. Laplacian and Random Walks on Graphs Linyuan Lu 39 / 59

Our approach Find a simple and sufficient condition for G (K 3 ), and an efficient algorithm to verify this condition. Laplacian and Random Walks on Graphs Linyuan Lu 40 / 59

Our approach Find a simple and sufficient condition for G (K 3 ), and an efficient algorithm to verify this condition. Search a special class of graphs so that we have a better chance of finding a Folkman graph. Use spectral analysis instead of probabilistic methods. Laplacian and Random Walks on Graphs Linyuan Lu 40 / 59

Our approach Find a simple and sufficient condition for G (K 3 ), and an efficient algorithm to verify this condition. Search a special class of graphs so that we have a better chance of finding a Folkman graph. Use spectral analysis instead of probabilistic methods. Localization and δ-fairness. Laplacian and Random Walks on Graphs Linyuan Lu 40 / 59

Our approach Find a simple and sufficient condition for G (K 3 ), and an efficient algorithm to verify this condition. Search a special class of graphs so that we have a better chance of finding a Folkman graph. Use spectral analysis instead of probabilistic methods. Localization and δ-fairness. Circulant graphs and L(m, s). Laplacian and Random Walks on Graphs Linyuan Lu 40 / 59

My result I received $100-award by proving Theorem [Lu, 2007]: f(2,3,4) 9697. Laplacian and Random Walks on Graphs Linyuan Lu 41 / 59

My result I received $100-award by proving Theorem [Lu, 2007]: f(2,3,4) 9697. Shortly Dudek and Rödl improved it to f(2,3,4) 941. They also received a $50-award. Laplacian and Random Walks on Graphs Linyuan Lu 41 / 59

My result I received $100-award by proving Theorem [Lu, 2007]: f(2,3,4) 9697. Shortly Dudek and Rödl improved it to f(2,3,4) 941. They also received a $50-award. Lange-Radziszowski-Xu [2012+]: f(2, 3, 4) 786. Laplacian and Random Walks on Graphs Linyuan Lu 41 / 59

Spencer s Lemma Notations: - G v : the induced graph on the the neighborhood of v. - b(h): the maximum size of edge-cuts for H. Laplacian and Random Walks on Graphs Linyuan Lu 42 / 59

Spencer s Lemma Notations: - G v : the induced graph on the the neighborhood of v. - b(h): the maximum size of edge-cuts for H. Lemma (Spencer) If v b(g v) < 2 3 v E(G v), then G (K 3 ). Laplacian and Random Walks on Graphs Linyuan Lu 42 / 59

Localization For 0 < δ < 1 2, a graph H is δ-fair if b(h) < ( 1 2 +δ) E(H). Laplacian and Random Walks on Graphs Linyuan Lu 43 / 59

Localization For 0 < δ < 1 2, a graph H is δ-fair if b(h) < ( 1 2 +δ) E(H). G is a Folkman graph if for each v - G v is 1 6 -fair. - G v is K 3 -free. Laplacian and Random Walks on Graphs Linyuan Lu 43 / 59

Localization For 0 < δ < 1 2, a graph H is δ-fair if b(h) < ( 1 2 +δ) E(H). G is a Folkman graph if for each v - G v is 1 6 -fair. - G v is K 3 -free. For vertex transitive graph G, all G v s are isomorphic. Laplacian and Random Walks on Graphs Linyuan Lu 43 / 59

Spectral lemma - H: a graph on n vertices - A: the adjacency matrix of H - d = (d 1,d 2,...,d n ): degrees of H - Vol(S) = v S d v: the volume of S - d = Vol(H) n : the average degree Laplacian and Random Walks on Graphs Linyuan Lu 44 / 59

Spectral lemma - H: a graph on n vertices - A: the adjacency matrix of H - d = (d 1,d 2,...,d n ): degrees of H - Vol(S) = v S d v: the volume of S - d = Vol(H) n : the average degree Lemma (Lu) If the smallest eigenvalue of M = A 1 Vol(H) d d is greater than 2δ d, then H is δ-fair. Laplacian and Random Walks on Graphs Linyuan Lu 44 / 59

Spectral lemma - H: a graph on n vertices - A: the adjacency matrix of H - d = (d 1,d 2,...,d n ): degrees of H - Vol(S) = v S d v: the volume of S - d = Vol(H) n : the average degree Lemma (Lu) If the smallest eigenvalue of M = A 1 Vol(H) d d is greater than 2δ d, then H is δ-fair. Similar results hold for A and L. However, they are weaker than using M in experiments. Laplacian and Random Walks on Graphs Linyuan Lu 44 / 59

Corollary 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. Laplacian and Random Walks on Graphs Linyuan Lu 45 / 59

Corollary 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. Laplacian and Random Walks on Graphs Linyuan Lu 45 / 59

The proof of the Lemma V(H) = X Y: a partition of the vertex-set. Laplacian and Random Walks on Graphs Linyuan Lu 46 / 59

The proof of the Lemma V(H) = X Y: a partition of the vertex-set. 1 X, 1 Y : indicated functions of X and Y. 1 X +1 Y = 1. Laplacian and Random Walks on Graphs Linyuan Lu 46 / 59

The proof of the Lemma V(H) = X Y: a partition of the vertex-set. 1 X, 1 Y : indicated functions of X and Y. 1 X +1 Y = 1. We observe M1 = 0. Laplacian and Random Walks on Graphs Linyuan Lu 46 / 59

The proof of the Lemma V(H) = X Y: a partition of the vertex-set. 1 X, 1 Y : indicated functions of X and Y. 1 X +1 Y = 1. We observe M1 = 0. For each t (0,1), let α(t) = (1 t)1 X t1 Y. We have α(t) M α(t) = e(x,y)+ 1 Vol(H) Vol(X)Vol(Y). Laplacian and Random Walks on Graphs Linyuan Lu 46 / 59

The proof of the Lemma 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 α t 2. Laplacian and Random Walks on Graphs Linyuan Lu 47 / 59

The proof of the Lemma 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 α t 2. Choose t = X n so that α(t) 2 reaches its minimum X Y n. Laplacian and Random Walks on Graphs Linyuan Lu 47 / 59

The proof of the Lemma 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 α t 2. Choose t = X n so that α(t) 2 reaches its minimum X Y We have e(x,y) Vol(X)Vol(Y) Vol(H) Vol(H) ρ n 4 4 +ρ X Y. n < ( 1 2 +δ) E(H), since ρ > 2δ d. n. Laplacian and Random Walks on Graphs Linyuan Lu 47 / 59

Circulant graphs - Z n = Z/nZ - S: a subset of Z n satisfying S = S and 0 S. We define a circulant graph H by - V(H) = Z n - E(H) = {xy x y S}. Example: A circulant graph withn = 8andS = {±1,±3}. Laplacian and Random Walks on Graphs Linyuan Lu 48 / 59

Spectrum of circulant graphs Lemma: The eigenvalues of the adjacency matrix for the circulant graph generated by S Z n are for i = 0,...,n 1. s S cos 2πis n Laplacian and Random Walks on Graphs Linyuan Lu 49 / 59

Spectrum of circulant graphs Lemma: The eigenvalues of the adjacency matrix for the circulant graph generated by S Z n are s S cos 2πis n for i = 0,...,n 1. Proof: Note A = g(j), where g(x) = s S x s. J = 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0.......... 0 0 0 0 1 1 0 0 0 0 Laplacian and Random Walks on Graphs Linyuan Lu 49 / 59

Proof continues... Let φ = e 2π 1 n denote the primitive n-th unit root. J has eigenvalues 1,φ,φ 2,...,φ n 1. Laplacian and Random Walks on Graphs Linyuan Lu 50 / 59

Proof continues... Let φ = e 2π 1 n denote the primitive n-th unit root. J has eigenvalues 1,φ,φ 2,...,φ n 1. Thus, the eigenvalues of A = g(j) are g(1),g(φ),...,g(φ n 1 ). For i = 0,1,2,...,n 1, we have g(φ i ) = R(g(φ i )) = s S cos 2πis n. Laplacian and Random Walks on Graphs Linyuan Lu 50 / 59

Graph L(m,s) Suppose s and m are relatively prime to each other. Let n be the least positive integer satisfying s n 1 mod m. Laplacian and Random Walks on Graphs Linyuan Lu 51 / 59

Graph L(m,s) Suppose s and m are relatively prime to each other. Let n be the least positive integer satisfying s n 1 mod m. We define the graph L(m,s) to be a circulant graph on m vertices with S = {s i mod m i = 0,1,2,...,n 1}. Laplacian and Random Walks on Graphs Linyuan Lu 51 / 59

Graph L(m,s) Suppose s and m are relatively prime to each other. Let n be the least positive integer satisfying s n 1 mod m. We define the graph L(m,s) to be a circulant graph on m vertices with S = {s i mod m i = 0,1,2,...,n 1}. Proposition: The local graph G v of L(m,s) is also a circulant graph. Laplacian and Random Walks on Graphs Linyuan Lu 51 / 59

Algorithm For each L(m,s), compute the local graph G v. If G v 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 G v is less than 1 3, reject it and try a new graph L(m, s). Output a Folkman graph L(m, s). Laplacian and Random Walks on Graphs Linyuan Lu 52 / 59

Computational results L(m, s) σ L(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 ratio of the smallest eigenvalue to the largest eigenvalue in the local graph. All graphs on the left are K 4 -free. Graphs in red are Folkman graphs. Graphs in black are good candidates. Laplacian and Random Walks on Graphs Linyuan Lu 53 / 59

Improvements Our method has inspired two improvements. Dudek-Rodl [2008]: f(2,3,4) 941. Laplacian and Random Walks on Graphs Linyuan Lu 54 / 59

Improvements Our method has inspired two improvements. Dudek-Rodl [2008]: f(2,3,4) 941. Lange-Radziszowski-Xu [2012+]: f(2, 3, 4) 786. Laplacian and Random Walks on Graphs Linyuan Lu 54 / 59

Dudek and Rodl Given a graph G, a triangle graph H G is defined as V(H G ) = E(G) e 1 e 2 in H G if e 1 and e 2 belong to the same triangle of G. Laplacian and Random Walks on Graphs Linyuan Lu 55 / 59

Dudek and Rodl Given a graph G, a triangle graph H G is defined as V(H G ) = E(G) e 1 e 2 in H G if e 1 and e 2 belong to the same triangle of G. Dudek, Rodl, 2008 If b(h G ) < 2 3 E(H G), then G (K 3 ). If H G is 1 6 -fair, then G (K 3). Laplacian and Random Walks on Graphs Linyuan Lu 55 / 59

Dudek and Rodl Theorem [Dudek, Rodl, 2008] f(2,3,4) 941. Laplacian and Random Walks on Graphs Linyuan Lu 56 / 59

Dudek and Rodl Theorem [Dudek, Rodl, 2008] f(2,3,4) 941. Proof: Let G = L(941,12). Then G is 188 regular. The triangle graph H has 941 188/2 = 88454 vertices and 2122896 edges. Laplacian and Random Walks on Graphs Linyuan Lu 56 / 59

Dudek and Rodl Theorem [Dudek, Rodl, 2008] f(2,3,4) 941. Proof: Let G = L(941,12). Then G is 188 regular. The triangle graph H has 941 188/2 = 88454 vertices and 2122896 edges. Using Matlab, they calculate the least eigenvalue So H is 1 6-fair. Done. µ n 15.196 > ( 1 2 + 1 6 )24. Laplacian and Random Walks on Graphs Linyuan Lu 56 / 59

Lange-Radziszowski-Xu Instead of spectral methods, they use semi-definite program (SDP) to approximate the MAX-CUT problem. First they try the graph G 1 obtained from L(941,12) by deleting 81 vertices. They showed 3b(H G1 ) < 1084985 < 1085028 = 2 E(H G1 ). This implies f(2,3,4) 860. Second they try the graph G 2 obtained from L(785,53) by one vertex and some 60 edges. They showed 3b(H G2 ) < 857750 < 857762 = 2 E(H G2 ). This implies f(2,3,4) 786. Laplacian and Random Walks on Graphs Linyuan Lu 57 / 59

Open questions Exoo conjectured L(127, 5) is a Folkman graph. In 2012 SIAMDM, Ronald Graham announced a $100 award for determining if f(2,3,4) < 100. A open problem on 3-colors: prove or disprove f(3,3,4) 3 34. Laplacian and Random Walks on Graphs Linyuan Lu 58 / 59

References 1. Linyuan Lu, Explicit Construction of Small Folkman Graphs. SIAM Journal on Discrete Mathematics, 21(4):1053-1060, January 2008. 2. Andrzej Dudek and Vojtech Rödl. On the Folkman Number f(2, 3, 4), Experimental Mathematics, 17(1):63-67, 2008. 3. A. Lange, S. Radziszowski, and X. Xu, Use of MAX-CUT for Ramsey Arrowing of Triangles, http://arxiv.org/pdf/1207.3750.pdf. Homepage: http://www.math.sc.edu/ lu/ Thank You Laplacian and Random Walks on Graphs Linyuan Lu 59 / 59