Laplacian and Random Walks on Graphs

Transcription

1 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

2 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

3 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

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

5 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

6 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

7 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

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

9 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

10 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

11 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 Laplacian and Random Walks on Graphs Linyuan Lu 6 / 59

12 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

13 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 S 4 L(S 4 ) = Laplacian and Random Walks on Graphs Linyuan Lu 6 / 59

14 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 S 4 L(S 4 ) = Combinatorial Laplacian eigenvalues of S 4 : 0,1,1,4. Laplacian and Random Walks on Graphs Linyuan Lu 6 / 59

15 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

16 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

17 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

18 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

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

20 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

21 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 ) L(S 4 ) = (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

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

23 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

24 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

25 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

26 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

27 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

28 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

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

30 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

31 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

32 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

33 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

34 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

35 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

36 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

37 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

38 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

39 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. Laplacian and Random Walks on Graphs Linyuan Lu 14 / 59

40 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

41 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

42 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

43 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

44 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

45 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

46 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

47 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

48 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

49 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

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

51 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. Laplacian and Random Walks on Graphs Linyuan Lu 22 / 59

52 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) = R(5,5) 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

53 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

54 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

55 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

56 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

57 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

58 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

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

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

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

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

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

64 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

65 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

66 K 5 -free G with G (K 3 ) Year Authors G 1969 Sch auble Graham, Spencer Irving Hadziivanov, Nenov Nenov 15 Laplacian and Random Walks on Graphs Linyuan Lu 33 / 59

67 K 5 -free G with G (K 3 ) Year Authors G 1969 Sch auble Graham, Spencer Irving Hadziivanov, Nenov 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

68 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

69 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

70 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

71 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

72 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

73 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

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

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

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

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

78 Most wanted Folkman Graph Laplacian and Random Walks on Graphs Linyuan Lu 37 / 59

79 Most wanted Folkman Graph Laplacian and Random Walks on Graphs Linyuan Lu 38 / 59

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

81 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

82 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

83 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

84 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. Laplacian and Random Walks on Graphs Linyuan Lu 40 / 59

85 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

86 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

87 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

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

89 My result I received $100-award by proving Theorem [Lu, 2007]: f(2,3,4) 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

90 My result I received $100-award by proving Theorem [Lu, 2007]: f(2,3,4) 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

91 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

92 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

93 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

94 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

95 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

96 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

97 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

98 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

99 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

100 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

101 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

102 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

103 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

104 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

105 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

106 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

107 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

108 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

109 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

110 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

111 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 = Laplacian and Random Walks on Graphs Linyuan Lu 49 / 59

112 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

113 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

114 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

115 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

116 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

117 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

118 Computational results L(m, s) σ L(127,5) L(761,3) L(785,53) L(941,12) L(1777,53) L(1801,125) L(2641,2) L(9697,4) L(30193,53) L(33121,2) L(57401,7) 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

119 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

120 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

121 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

122 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

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

124 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 /2 = vertices and edges. Laplacian and Random Walks on Graphs Linyuan Lu 56 / 59

125 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 /2 = vertices and edges. Using Matlab, they calculate the least eigenvalue So H is 1 6-fair. Done. µ n > ( )24. Laplacian and Random Walks on Graphs Linyuan Lu 56 / 59

126 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 ) < < = 2 E(H G1 ). This implies f(2,3,4) 860. Laplacian and Random Walks on Graphs Linyuan Lu 57 / 59

127 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 ) < < = 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 ) < < = 2 E(H G2 ). This implies f(2,3,4) 786. Laplacian and Random Walks on Graphs Linyuan Lu 57 / 59

128 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) Laplacian and Random Walks on Graphs Linyuan Lu 58 / 59

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