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

2. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Expanders and Ramanujan Graphs 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/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Think of a graph For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

4. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Think of a graph For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

5. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Think of a graph as a communications network. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

6. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Two vertices can communicate directly with one another

7. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Two vertices can communicate directly with one another iff they are connected by an edge. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

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

9. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Edges are expensive. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

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

11. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs That is, same degree (number of edges) at each vertex. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

12. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Goals:

13. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Goals: ● Keep the degree fixed

14. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Goals: ● Let the number of vertices go to infinity. ● Keep the degree fixed

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

16. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Main questions: For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

17. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor 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?

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

19. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Remark: For regular graphs, this “communications network” business is more of an analogy than an application. But expanders have many real-world applications, including: Cryptographic hash functions Structural engineering Error-correcting codes Derandomization and many more...

20. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Which one is a better communications network, and why? For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Consider the two graphs below. Each has 46 vertices and is 3-regular.

21. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs In some sense, the graph on the left is better; you can get from any vertex to any other vertex in 8 steps or fewer. (The diameter is 8.) For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

22. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Whereas in the graph on the right, you need at least 11 steps to get from point 1 to point 12.

23. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs So the graph on the left is a faster network. (It has a smaller diameter.) For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

24. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs The graph on the left has a fatal flaw, though. Deleting just one edge separates 15 of the vertices from the other 31 vertices. (It has connectivity 1.) For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

25. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs You can delete any two edges from the graph on the right, though, and it will still be connected. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

26. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs So the graph on the right is a more reliable network. (It has larger connectivity.) For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

27. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Remarkably, we can capture both of these concepts with a single measurement. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

28. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Let’s look at the set of vertices we can get to in n steps. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

29. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Here’s where we can get to in one step.

30. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Here’s where we can get to in one step.

31. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Two steps.

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

33. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs At each stage, we would like to have many edges going outward from the points we’ve been to so far. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

34. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs But we run into “overlap”---many red vertices but relatively few green edges. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

35. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Similarly for the other graph. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

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

37. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs CAI H G F E D B J For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

38. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs CAI H G F E D B J For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

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

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

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

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

43. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Take-home Message #1: The expansion constant is one measure of how good a graph is as a communications network. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

44. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs The expansion constant captures both the speed and the reliability of the communications network. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

45. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs We want h(X) to be BIG! For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs

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

47. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Consider cycle graphs.

48. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Consider cycle graphs. They are 2-regular.

49. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor 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.

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

51. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Let S be the “bottom half”...

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

53. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor 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:

54. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor 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.

55. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor 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.

56. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs (iii) There exists a positive lower bound r such that the expansion constant is always at least r. 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.

57. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor 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.

58. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor 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.

59. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor 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 explicitly is highly nontrivial!) Existence: Pinsker 1973 First explicit construction: Margulis 1973

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

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

62. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs COMING THIS FALL!!!

63. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Shameless self-promotion!!! COMING THIS FALL!!!

64. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor 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 another point of view.

65. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs We form the adjacency matrix of a graph as follows:

66. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor 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.

67. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Facts about eigenvalues of a d-regular graph G:

68. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Facts about eigenvalues of a d-regular graph G: ● They are all real.

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

70. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs ● If Facts about eigenvalues of a d-regular graph G: For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs is the second largest eigenvalue, then (Alon-Dodziuk-Milman-Tanner) ● They are all real. ● The largest eigenvalue is d.

71. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs (Alon-Dodziuk-Milman-Tanner)

72. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs (Alon-Dodziuk-Milman-Tanner)

73. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs (Alon-Dodziuk-Milman-Tanner)

74. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Take-home Message #1: The expansion constant is one measure of how good a graph is as a communications network. Take-home Message #2:

75. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Student Research Projects

76. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Student Research Projects One way to learn about eigenvalues: count closed walks in the graph.

77. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Student Research Projects One way to learn about eigenvalues: count closed walks in the graph. Number of closed walks of length n equals the sum of the nth powers of the eigenvalues.

78. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Student Research Projects Matthew AivazianKeith Dsouza Natalie Martinez Certain Cayley graphs Cycle graphs Small graphs

79. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Shameless self-promotion!!! COMING THIS FALL!!!

80. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Take-home Message #2:

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

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

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

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

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

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

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

88. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs. (Actually, this definition is slightly inaccurate, but never mind...)

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

90. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Take-home Message #1: The expansion constant is one measure of how good a graph is as a communications network.

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

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

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