1. Ramanujan Graphs of Every Degree Adam Marcus (Crisply, Yale) Daniel Spielman (Yale) Nikhil Srivastava (MSR India)

2. Expander Graphs Sparse, regular well-connected graphs with many properties of random graphs. Random walks mix quickly. Every small set of vertices has many neighbors. Pseudo-random generators. Error-correcting codes. Sparse approximations of complete graphs. Major theorems in Theoretical Computer Science.

3. Spectral Expanders Let G be a graph and A be its adjacency matrix a c d e b 0 1 0 0 1 1 0 1 0 1 0 1 0 1 0 0 0 1 0 1 1 1 0 1 0 Eigenvalues If d -regular (every vertex has d edges), “trivial”

4. Spectral Expanders If bipartite (all edges between two parts/colors) eigenvalues are symmetric about 0 “trivial” a cd f b e If d -regular and bipartite, 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 0 0 0 1 0 1 0 0 0

5. Spectral Expanders 0 [] -d d G is a good spectral expander if all non-trivial eigenvalues are small

6. Bipartite Complete Graph Adjacency matrix has rank 2, so all non-trivial eigenvalues are 0 a cd f b e 0 0 0 1 1 1 1 1 1 0 0 0

7. Spectral Expanders 0 [] -d d Challenge: construct infinite families of fixed degree G is a good spectral expander if all non-trivial eigenvalues are small

8. Spectral Expanders G is a good spectral expander if all non-trivial eigenvalues are small 0 [] -d d Challenge: construct infinite families of fixed degree Alon-Boppana ‘86: Cannot beat ) (

9. G is a Ramanujan Graph if absolute value of non-trivial eigs 0 [] -d d ) ( Ramanujan Graphs:

10. G is a Ramanujan Graph if absolute value of non-trivial eigs 0 [] -d d ) ( Ramanujan Graphs: Margulis, Lubotzky-Phillips-Sarnak’88: Infinite sequences of Ramanujan graphs exist for

11. G is a Ramanujan Graph if absolute value of non-trivial eigs 0 [] -d d ) ( Ramanujan Graphs: Margulis, Lubotzky-Phillips-Sarnak’88: Infinite sequences of Ramanujan graphs exist for Can be quickly constructed: can compute neighbors of a vertex from its name

12. G is a Ramanujan Graph if absolute value of non-trivial eigs 0 [] -d d ) ( Ramanujan Graphs: Friedman’08: A random d -regular graph is almost Ramanujan :

13. Ramanujan Graphs of Every Degree Theorem: there are infinite families of bipartite Ramanujan graphs of every degree.

14. Theorem: there are infinite families of bipartite Ramanujan graphs of every degree. And, are infinite families of (c,d) -biregular Ramanujan graphs, having non-trivial eigs bounded by Ramanujan Graphs of Every Degree

15. Find an operation that doubles the size of a graph without creating large eigenvalues. 0 [] -d d Bilu-Linial ‘06 Approach ( )

16. 0 [] -d d Bilu-Linial ‘06 Approach Find an operation that doubles the size of a graph without creating large eigenvalues. ( )

17. 2-lifts of graphs a c d e b

18. a c d e b a c d e b duplicate every vertex

19. 2-lifts of graphs duplicate every vertex a0a0 c0c0 d0d0 e0e0 b0b0 a1a1 c1c1 d1d1 e1e1 b1b1

20. 2-lifts of graphs a0a0 c0c0 d0d0 e0e0 b0b0 a1a1 d1d1 e1e1 b1b1 c1c1 for every pair of edges: leave on either side (parallel), or make both cross

21. 2-lifts of graphs for every pair of edges: leave on either side (parallel), or make both cross a0a0 c0c0 d0d0 e0e0 b0b0 a1a1 d1d1 e1e1 b1b1 c1c1

22. 2-lifts of graphs 0 1 0 0 1 1 0 1 0 1 0 1 0 1 0 0 0 1 0 1 1 1 0 1 0

23. 2-lifts of graphs 0 1 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 1 0

24. 2-lifts of graphs 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 0

25. Eigenvalues of 2-lifts (Bilu-Linial) Given a 2-lift of G, create a signed adjacency matrix A s with a -1 for crossing edges and a 1 for parallel edges 0 -1 0 0 1 -1 0 1 0 1 0 1 0 -1 0 0 0 -1 0 1 1 1 0 1 0 a0a0 c0c0 d0d0 e0e0 b0b0 a1a1 d1d1 e1e1 b1b1 c1c1

26. Eigenvalues of 2-lifts (Bilu-Linial) Theorem: The eigenvalues of the 2-lift are the union of the eigenvalues of A (old) and the eigenvalues of A s (new) 0 -1 0 0 1 -1 0 1 0 1 0 1 0 -1 0 0 0 -1 0 1 1 1 0 1 0 a0a0 c0c0 d0d0 e0e0 b0b0 a1a1 d1d1 e1e1 b1b1 c1c1

27. Eigenvalues of 2-lifts (Bilu-Linial) Conjecture: Every d -regular graph has a 2-lift in which all the new eigenvalues have absolute value at most Theorem: The eigenvalues of the 2-lift are the union of the eigenvalues of A (old) and the eigenvalues of A s (new)

28. Eigenvalues of 2-lifts (Bilu-Linial) Conjecture: Every d -regular graph has a 2-lift in which all the new eigenvalues have absolute value at most Would give infinite families of Ramanujan Graphs: start with the complete graph, and keep lifting.

29. Eigenvalues of 2-lifts (Bilu-Linial) Conjecture: Every d -regular graph has a 2-lift in which all the new eigenvalues have absolute value at most We prove this in the bipartite case. a 2-lift of a bipartite graph is bipartite

30. Eigenvalues of 2-lifts (Bilu-Linial) Theorem: Every d -regular graph has a 2-lift in which all the new eigenvalues have absolute value at most Trick: eigenvalues of bipartite graphs are symmetric about 0, so only need to bound largest

31. Eigenvalues of 2-lifts (Bilu-Linial) Theorem: Every d -regular bipartite graph has a 2-lift in which all the new eigenvalues have absolute value at most

32. First idea: a random 2-lift Specify a lift by Pick s uniformly at random

33. First idea: a random 2-lift Specify a lift by Pick s uniformly at random Are graphs for which this usually fails

34. First idea: a random 2-lift Specify a lift by Pick s uniformly at random Are graphs for which this usually fails Bilu and Linial proved G almost Ramanujan, implies new eigenvalues usually small. Improved by Puder and Agarwal-Kolla-Madan

35. The expected polynomial Consider

36. The expected polynomial Consider Conclude there is an s so that Prove Prove is an interlacing family

37. The expected polynomial Theorem (Godsil-Gutman ‘81): the matching polynomial of G

38. The matching polynomial (Heilmann-Lieb ‘72) m i = the number of matchings with i edges

40. one matching with 0 edges

41. 7 matchings with 1 edge

44. Proof that x ±1 0 0 ±1 ±1 ±1 x ±1 0 0 0 0 ±1 x ±1 0 0 0 0 ±1 x ±1 0 ±1 0 0 ±1 x ±1 ±1 0 0 0 ±1 x Expand using permutations

45. Proof that x ±1 0 0 ±1 ±1 ±1 x ±1 0 0 0 0 ±1 x ±1 0 0 0 0 ±1 x ±1 0 ±1 0 0 ±1 x ±1 ±1 0 0 0 ±1 x Expand using permutations same edge: same value

46. Proof that x ±1 0 0 ±1 ±1 ±1 x ±1 0 0 0 0 ±1 x ±1 0 0 0 0 ±1 x ±1 0 ±1 0 0 ±1 x ±1 ±1 0 0 0 ±1 x Expand using permutations same edge: same value

47. Proof that x ±1 0 0 ±1 ±1 ±1 x ±1 0 0 0 0 ±1 x ±1 0 0 0 0 ±1 x ±1 0 ±1 0 0 ±1 x ±1 ±1 0 0 0 ±1 x Expand using permutations Get 0 if hit any 0s

48. Proof that x ±1 0 0 ±1 ±1 ±1 x ±1 0 0 0 0 ±1 x ±1 0 0 0 0 ±1 x ±1 0 ±1 0 0 ±1 x ±1 ±1 0 0 0 ±1 x Expand using permutations Get 0 if take just one entry for any edge

49. Proof that x ±1 0 0 ±1 ±1 ±1 x ±1 0 0 0 0 ±1 x ±1 0 0 0 0 ±1 x ±1 0 ±1 0 0 ±1 x ±1 ±1 0 0 0 ±1 x Expand using permutations Only permutations that count are involutions

50. Proof that x ±1 0 0 ±1 ±1 ±1 x ±1 0 0 0 0 ±1 x ±1 0 0 0 0 ±1 x ±1 0 ±1 0 0 ±1 x ±1 ±1 0 0 0 ±1 x Expand using permutations Only permutations that count are involutions

51. Proof that x ±1 0 0 ±1 ±1 ±1 x ±1 0 0 0 0 ±1 x ±1 0 0 0 0 ±1 x ±1 0 ±1 0 0 ±1 x ±1 ±1 0 0 0 ±1 x Expand using permutations Only permutations that count are involutions Correspond to matchings

52. The matching polynomial (Heilmann-Lieb ‘72) Theorem (Heilmann-Lieb) all the roots are real

53. The matching polynomial (Heilmann-Lieb ‘72) Theorem (Heilmann-Lieb) all the roots are real and have absolute value at most

54. The matching polynomial (Heilmann-Lieb ‘72) Theorem (Heilmann-Lieb) all the roots are real and have absolute value at most Implies

55. Interlacing Polynomial interlaces if

56. Common Interlacing and have a common interlacing if can partition the line into intervals so that each interval contains one root from each poly

57. Common Interlacing and have a common interlacing if can partition the line into intervals so that each interval contains one root from each poly ) ) ) ) ( (( (

58. If p 1 and p 2 have a common interlacing, for some i. Largest root of average Common Interlacing

59. If p 1 and p 2 have a common interlacing, for some i. Largest root of average Common Interlacing

60. is an interlacing family If the polynomials can be placed on the leaves of a tree so that when put average of descendants at nodes siblings have common interlacings Interlacing Family of Polynomials

61. is an interlacing family Interlacing Family of Polynomials If the polynomials can be placed on the leaves of a tree so that when put average of descendants at nodes siblings have common interlacings

62. Interlacing Family of Polynomials There is an s so that Theorem:

63. An interlacing family Theorem: Let is an interlacing family

64. Interlacing and have a common interlacing iff is real rooted for all

65. To prove interlacing family Let

66. To prove interlacing family Let Need to prove that for all, is real rooted

67. To prove interlacing family Let are fixed is 1 with probability, -1 with are uniformly Need to prove that for all, is real rooted

68. Generalization of Heilmann-Lieb We prove is real rooted for every independent distribution on the entries of s

69. Generalization of Heilmann-Lieb We prove is real rooted for every independent distribution on the entries of s By using mixed characteristic polynomials

70. Mixed Characteristic Polynomials For independently chosen random vectors is their mixed characteristic polynomial. Theorem: Mixed characteristic polynomials are real rooted. Proof: Using theory of real stable polynomials.

71. Mixed Characteristic Polynomials For independently chosen random vectors is their mixed characteristic polynomial. Obstacle: our matrix is a sum of random rank-2 matrices 0 -1 -1 0 0 1 1 0 or

72. Mixed Characteristic Polynomials For independently chosen random vectors is their mixed characteristic polynomial. Solution: add to the diagonal 1 -1 -1 1 1 1 or

73. Generalization of Heilmann-Lieb We prove is real rooted for every independent distribution on the entries of s Implies is an interlacing family

74. Generalization of Heilmann-Lieb We prove is real rooted for every independent distribution on the entries of s Conclude there is an s so that Implies is an interlacing family

75. Universal Covers The universal cover of a graph G is a tree T of which G is a quotient. vertices map to vertices edges map to edges homomorphism on neighborhoods Is the tree of non-backtracking walks in G. The universal cover of a d -regular graph is the infinite d -regular tree.

76. Quotients of Trees d -regular Ramanujan as quotient of infinite d -ary tree Spectral radius and norm of inf d -ary tree are

77. Godsil’s Proof of Heilmann-Lieb T(G,v) : the path tree of G at v vertices are paths in G starting at v edges to paths differing in one step

78. Godsil’s Proof of Heilmann-Lieb a c d e b a a b e a e a b e a b c a b d e a

79. T(G,v) : the path tree of G at v vertices are paths in G starting at v edges to paths differing in one step Theorem: The matching polynomial divides the characteristic polynomial of T(G,v)

80. Godsil’s Proof of Heilmann-Lieb T(G,v) : the path tree of G at v vertices are paths in G starting at v edges to paths differing in one step Theorem: The matching polynomial divides the characteristic polynomial of T(G,v) Is a subgraph of infinite tree, so has smaller spectral radius

81. Quotients of Trees (c,d) -biregular bipartite Ramanujan as quotient of infinite (c,d) -ary tree Spectral radius For (c,d) -regular bipartite Ramanujan graphs

82. Irregular Ramanujan Graphs (Greenberg-Lubotzky) Def: G is Ramanujan if its non-trivial eigenvalues have abs value less than the spectral radius of its cover Theorem: If G is bipartite and Ramanujan, then there is an infinite family of Ramanujan graphs with the same cover.

83. Questions Non-bipartite Ramanujan Graphs of every degree? Efficient constructions? Explicit constructions?