2. Expanders and Ramanujan Graphs Mike Krebs, Cal State LA For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

3. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs A graph is a collection of vertices (dots) and edges (connections between vertices).

4. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Distances don’t matter. All we care about is how the vertices are connected. These two graphs are the same, for example

5. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Distances don’t matter. All we care about is how the vertices are connected. These two graphs are the same, for example

6. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Consider the graph below.

7. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs The blue vertex below has degree 4.

8. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs The red vertex has degree 2.

9. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs So this graph is not “regular.”

10. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Is this graph regular?

11. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Yes, every vertex has degree 3.

12. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs We say it is 3-regular.

13. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs In this talk, we will be concerned primarily with regular graphs.

14. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Think of a graph as a communications network. Two vertices can communcate directly with one another if they are connected by an edge.

15. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Communication is instantaneous across edges, but there may be delays at vertices. Edges are expensive.

16. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Goals: - Keep the degree fixed. - Let the number of vertices go to infinity. - Make sure the communications networks are “good.”

17. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Main questions: How do we measure how good a graph is as a communications network? How good can we make them?

18. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs CAI H G F E D B J R U XZS TV W Y Q Here are two graphs. Each has 10 vertices. Each has degree 4.

19. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Here are two graphs. Each has 10 vertices. Each has degree 4. Which one is a better communications network, and why? CAI H G F E D B J R U XZS TV W Y Q

20. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs I like the one on the right better. CAI H G F E D B J R U XZS TV W Y Q

21. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs You can get from any vertex to any other vertex in two steps. CAI H G F E D B J R U XZS TV W Y Q I like the one on the right better.

22. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs CAI H G F E D B J R U XZS TV W Y Q In the graph on the left, it takes at least three steps to get from A to F.

23. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs CAI H G F E D B J Let’s look at the set of vertices we can get to in n steps.

24. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs CAI H G F E D B J Let’s look at the set of vertices we can get to in n steps.

25. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs CAI H G F E D B J Here’s where we can get to in one step.

26. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs CAI H G F E D B J We would like to have many edges going outward from there.

27. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs CAI H G F E D B J Here’s where we can get to in two steps.

28. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs For any set S of vertices, we would like to have many edges from S to its complement.

29. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs CAI H G F E D B J The set of edges from S to its complement is called the boundary of S, denoted dS. Example: |S| = 5 |dS| = 6

30. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Let G be a graph. Define h(G) to be the minimum value of over all sets S containing no more than half the vertices.

31. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs h(G) is called the expansion constant of G.

32. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Example:

33. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Example: = 2/1= 2

34. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Example: = 2/2= 1

35. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Example: = 4/2= 2

36. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Example: 2/2=1 was the min. So h(X)=1.

37. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs The expansion constant tells us that even in a worst case, for any small set S, we have at least h(X)|S| edges going from S to its complement.

38. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs We want h(X) to be BIG! If a graph has small degree but many vertices, this is not easy.

39. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Consider cycle graphs. They are 2-regular. Number of vertices goes to infinity.

40. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Let’s see what happens to the expansion constants.

41. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Let C be a cycle graph with n vertices.

42. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Choose S to be the “bottom half.”

43. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs So S has n/2 or (n-1)/2 vertices.

44. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs But |dS|=2.

45. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs So h(C)<2/[(n-1)/2]=n-10

46. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs We say that a sequence of regular graphs is an expander family if:

47. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs We say that a sequence of regular graphs is an expander family if: (A) They all have the same degree.

48. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs We say that a sequence of regular graphs is an expander family if: (A) They all have the same degree. (2) The number of vertices goes to infinity.

49. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs We say that a sequence of regular graphs is an expander family if: (A) They all have the same degree. (2) The number of vertices goes to infinity. (iii) There exists a positive lower bound r such that the expansion constant is always at least r.

50. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Expander families of degree 2 do not exist, as we just saw.

51. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Expander families of degree 2 do not exist, as we just saw. Amazing fact: if d is any integer greater then 2, then an expander family of degree d exists.

52. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Expander families of degree 2 do not exist, as we just saw. Amazing fact: if d is any integer greater then 2, then an expander family of degree d exists. (Constructing them is highly nontrivial!)

53. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs So far, we’ve looked at expansion from a combinatorial point of view. Now let’s look at it from an algebraic point of view.

54. We form the adjacency matrix of a graph as follows:

55. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs The expansion constant of a graph is closely related to the eigenvalues of its adjacency matrix.

56. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Facts about graph eigenvalues:

57. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Facts about graph eigenvalues: - the eigenvalues of a graph are all real

58. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Facts about graph eigenvalues: - the eigenvalues of a graph are all real - the largest eigenvalue of a d-regular graph is d

59. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Facts about graph eigenvalues: - the eigenvalues of a graph are all real - the largest eigenvalue of a d-regular graph is d - ifis the second largest eigenvalue, then

60. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs We call d-the spectral gap of the graph. h(G) is big iff the spectral gap is big.

61. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs A sequence of regular graphs is an expander family if: (A) They all have the same degree. (2) The number of vertices goes to infinity. (iii) There exists a positive lower bound r such that the spectral gap is always at least r.

62. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs We call d-the spectral gap of the graph. So we want the spectral gap to be big. So we wantto be small. But there is a limit to how small it can be.

63. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs must lie above the red curve.

64. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs at. The red curve had a horizontal asymptote

65. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs In other words, can be. at. is essentially the smallest The red curve had a horizontal asymptote

66. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs In other words, can be. at. is essentially the smallest We say a graph is Ramanujan if: < The red curve had a horizontal asymptote

67. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Known: There exists a family of d-regular Ramanujan graphs if d-1 is a prime power.

68. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Known: There exists a family of d-regular Ramanujan graphs if d-1 is a prime power. Open problem: Does a family of d-regular Ramanujan graphs for all d >3?