Eigenvalues of (3, 6)-Fullerenes. Luis Goddyn. Matt DeVos, Bojan Mohar, Robert Šámal. Coast-to-coast Seminar:

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1 Eigenvalues of (3, 6)-Fullerenes Luis Goddyn joint work with: Matt DeVos, Bojan Mohar, Robert Šámal Simon Fraser University, Burnaby, BC Coast-to-coast Seminar:

2 Richard Buckminster Fuller Buckyball - C 60 molecule 1985

3 Definition: Fullerene (5,6)-Fullerene: 3-regular, planar, (12) pentagons & hexagons

4 Definition: (3,6)-Fullerene (3, 6)-Fullerene: 3-regular, planar, (4) triangles & hexagons Chemistry/Physics Interest: They modify to tetrahedral (5,6)-Fullerenes Can model nanotubes Easier to characterize Critical Question: Find Energy Spectrum = eigenvalues of adjacency matrix

5 Tube Construction of all (3,6)-Fullerenes (4, 6, 0) tube (4, 6, 1) tube Specification is 3-to-1. (length, girth, twist)

6 Maple Tests GraphTheory package

7 Maple Tests Adjacency Matrix

8 Maple Tests Spectrum

9 Fowler conjecture Conjecture (Fowler, 1995) The spectrum of every (3, 6)-Fullerene takes the form {3, 1, 1, 1} Λ Λ. where Λ is a multiset of reals.

10 (3, 6)-Fullerenes are spectrally almost bipartite? Lemma G has spectrum Λ Λ iff G is bipartite. Eigenvector x : V C uv E x(u) =λ x(v) x x x x x x 5 6 x x x x x x x 1 + x 2 + x 3 = λx 5 =( λ)( x 5 )

11 Why {3, 1, 1, 1}? Lemma Every (3, 6)-Fullerene is a cover of K 4. { Fibre of v v

12 Why {3, 1, 1, 1}? Lemma Every (3, 6)-Fullerene is a cover of K 4. { Fibre of v v

13 Why {3, 1, 1, 1}? Eigenvector lift to covers Eigenvalue = 1 1 = ( 1)(1+1 1)

14 Why {3, 1, 1, 1}? Eigenvalue = 3 Spec( K ) = {3, 1, 1, 1} 4

15 Experimenting... Idea Where x, x are eigenvectors for λ, λ, plot vertex v at coordinates (x v, x v ).

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21 Suspicion Where x, x are (appropriately scaled) eigenvectors for λ, λ, the complex numbers x(v)+ix (v) are (phase shifted) roots of unity.

22 A Similar Phenomenon Fact Abelian Cayley graphs have complex-root-of-unity eigenvectors (not phase shifted).

23 Cayley Graph Γ = additive abelian group. S Γ, the symbol set. S = S Cay(Γ, S) : V =Γ, E = {uv u v S} Cay(Z2 3, {100, 010, 001}) Cay(Z 9, {±1, ±2})

24 Cayley Sum Graph Γ=additive abelian group S Γ, the symbol set. S ///////////// = S Cay(Γ, S) : V =Γ E = {uv u v S} CaySum(Γ, S) : V =Γ E = {uv u + v S} Cay(Z 3 2, {100, 010, 001}) Cay(Z 9, {±1, ±2}) CaySum(Z 9, {±1, 2}) CaySum(Z2 3, {100, 010, 001})

25 Geometric Cayley Sum Graphs Concurrence of edges Cay(Z 2, {(±1, 0), (0, ±2)}) CaySum(Z 2, {(1, 0), (0, 2)})

26 Cayley Sum Graphs - Facts CaySum(Γ, S) is S -regular (counting loops once) CaySum(Γ, S) CaySum(Γ, S + 2t) (via x x + t) CaySum(Γ, S) (2) = Cay(Γ, S S) (F. Chung, 1989) Good expansion (B. Green, 2003) Pseudorandom clique size (D. Grynkiewicz et al, 2007) Connectivity (N. Alon, 2007+) Independence Number (B. Cheyne et al 2003; V. Lev, 2008+) Hamiltonicity

27 Abelian Group Characters χ :Γ C is a homomorphism. 1 eg. Γ=Z 5 χ 1 : x e 2iπ x χ 2 : x e 2iπ 2x 5 χ 0 : x 1 3 4

28 Abelian Group Characters χ :Γ C is a homomorphism. eg. Γ=Z χ 1 : x e 2iπ x 5 0 χ 2 : x e 2iπ 2x 5 χ 0 : x 1 4 2

29 Abelian Group Characters χ :Γ C is a homomorphism. eg. Γ=Z 5 χ 1 : x e 2iπ x 5 χ 2 : x e 2iπ 2x 5 χ 0 : x

30 Abelian Group Characters χ :Γ C is a homomorphism. eg. Γ=Z 5 χ 1 : x e 2iπ x 5 χ 2 : x e 2iπ 2x 5 χ 0 : x 1 For a Z n, χ a (x) =e 2iπ ax n

31 Abelian Group Characters eg. Γ=Z 5 Z 3 χ (2,0) : (x 1, x 2 ) e 2iπ 2x 1 5 χ (0,1) :(x 1, x 2 ) e 2iπ x 2 3 χ (3,2) :(x 1, x 2 ) e 2iπ( 3x x 2 3 ) (1,0) (1,1) (1,2) (4,0) (4,1) (4,2) (3,0) (3,1) (3,2) (2,0) (2,1) (2,2) (0,0) (0,1) (0,2)

32 Abelian Group Characters eg. Γ=Z 5 Z 3 χ (2,0) :(x 1, x 2 ) e 2iπ 2x 1 5 χ (0,1) : (x 1, x 2 ) e 2iπ x 2 3 χ (3,2) :(x 1, x 2 ) e 2iπ( 3x x 2 3 ) (0,1) (1,1) (2,1) (3,1) (4,1) (0,2) (1,2) (2,2) (3,2) (4,2) (0,0) (1,0) (2,0) (3,0) (4,0)

33 Abelian Group Characters (0,2) (1,1) (2,0) (3,2) eg. Γ=Z 5 Z 3 χ (2,0) :(x 1, x 2 ) e 2iπ 2x 1 5 χ (0,1) :(x 1, x 2 ) e 2iπ x 2 3 (4,0) (3,1) (2,2) (4,1) (0,0) χ (3,2) : (x 1, x 2 ) e 2iπ( 3x x 2 3 ) (1,0) (0,1) (4,2) (3,0) (1,2) (2,1)

34 Abelian Group Characters eg. Γ=Z 5 Z 3 χ (2,0) :(x 1, x 2 ) e 2iπ 2x 1 5 χ (0,1) :(x 1, x 2 ) e 2iπ x 2 3 χ (3,2) :(x 1, x 2 ) e 2iπ( 3x x 2 3 ) For (a 1, a 2 ) Z 5 Z 3, χ (a1,a 2 )(x 1, x 2 )=e 2iπ( a 1 x a 2 x 2 3 )

35 Abelian Group Characters eg. Γ=Z 5 Z 3 χ (2,0) :(x 1, x 2 ) e 2iπ 2x 1 5 χ (0,1) :(x 1, x 2 ) e 2iπ x 2 3 χ (3,2) :(x 1, x 2 ) e 2iπ( 3x x 2 3 ) For a Z n1 Z n2 Z nk, χ a (x) =e 2iπ( a 1 x 1 n a 2 x 2 n k )

36 Spectra of Cayley Graphs Theorem (Classic) { } Spec(Cay(Γ, {s 1,...,s k })) = χ a (s i ) a Γ i Cay(Z 5 Z 3, {(1, 1), (2, 1), (3, 2)}) λ (3,2) = i (3,1) (2,2) (4,0) (1,0) (0,2) (0,1) (1,1) χ (3,2) (4,2) (2,0) (3,0) i (3,2) (4,1) (0,0) (1,2) (2,1)

37 Spectra of Cayley Graphs Theorem (Classic) { } Spec(Cay(Γ, {s 1,...,s k })) = χ a (s i ) a Γ i Cay(Z 5 Z 3, {(1, 1), (2, 1), (3, 2)}) λ (3,2) = i R(λ (3,2) )= I(λ (3,2) )= to-2 (unless χ a is real-valued) (1,1) (2,0) (0,2) (4,0) (3,1) (2,2) χ (3,2) (1,0) (0,1) (4,2) (3,0) i (3,2) (4,1) (0,0) (1,2) (2,1)

38 Spectra of Cayley Graphs Theorem (Classic) { } Spec(Cay(Γ, {s 1,...,s k })) = χ a (s i ) a Γ i Proof: (A χ a )(u) = χ a (v) v N(u) = χ a (u + s i ) i u u + s 1 = χ a (u) i χ a (s i ) u + s 2 u + s 3

39 Spectra of Cayley Graphs Theorem (Classic) χ S a(s) {}}{ { }}{ Spec(Cay(Γ, {s 1,...,s k })) = χ a (s i ) a Γ i Proof: (A χ a )(u) = χ a (v) v N(u) = χ a (u + s i ) i u u + s 1 = χ a (u) χ a (s i ) i }{{} u + s u + s χ a(s) 2 3

40 Spectra of Cayley Sum graphs χ a is real-valued (±1 - valued) iff a Inv(Γ). Theorem (Chung 89, DGMS) Spec(CaySum(Γ, S)) = { χ a (S) a Inv(Γ) } {± χ a (S) a, a / Inv(Γ) } Eigenvectors are phase-shifted characters χ a e arg(χa(s))/2.

41 (3,1) (2,2) (4,0) (1,0) (0,2) (0,1) (1,1) χ (3,2) (4,2) (2,0) (3,0) i (3,2) (4,1) (0,0) (1,2) (2,1)

42 (3,1) (2,2) (4,0) (1,0) (0,2) (0,1) (1,1) χ (3,2) (4,2) (2,0) (3,0) (3,2) (2,1) (4,1) (0,0) (1,2) i θ = arg χ (3,2)(S) 2

43 (4,0) (0,2) (1,1) (2,0) i (3,1) (3,2) (2,2) (4,1) (1,0) χ (3,2) (0,0) θ (0,1) (1,2) (4,2) (3,0) (2,1)

44 (4,0) (0,2) (1,1) (2,0) i (3,1) (3,2) (2,2) (4,1) (1,0) χ (3,2) (0,0) θ (0,1) (1,2) (4,2) (3,0) (2,1) Eigenvalues: ± i = ± to-2

45 Proof: (A χ a )(u) = χ a (v) = χ a (s u) v N(u) s S = χ a (s) χ a (u) 1 = χ a (S) χ a (u) Not quite! s S

46 Proof: Phase change: ρ := e arg χ a(s) 2.Soχ a (S)ρ 2 = χ a (S). (A χ a )(u) = χ a (s u) χ a (v) = v N(u) s S = χ a (s) χ a (u) 1 = χ a (S) χ a (u) Not quite! s S

47 Proof: Phase change: ρ := e arg χ a(s) 2.Soρ 2 χ a (S) = χ a (S). (A ρχ a )(u) = ρχ a (s u) ρχ a (v) = v N(u) s S = ρ 2 χ a (s) (ρχ a (u)) 1 = χ a (S) ρχ a (u) s S So R(ρχ a ) has eigenvalue + χ a (S) I(ρχ a ) has eigenvalue χ a (S) 2 to 2 (unless a is an involution).

48 Proof: Phase change: ρ := e arg χ a(s) 2.Soρ 2 χ a (S) = χ a (S). (A ρχ a )(u) = ρχ a (s u) ρχ a (v) = v N(u) s S = ρ 2 χ a (s) (ρχ a (u)) 1 = χ a (S) ρχ a (u) s S So R(ρχ a ) has eigenvalue + χ a (S) I(ρχ a ) has eigenvalue χ a (S) 2 to 2 (unless a is an involution).

49 Proof: Phase change: ρ := e arg χ a(s) 2.Soρ 2 χ a (S) = χ a (S). (A ρχ a )(u) = ρχ a (s u) ρχ a (v) = v N(u) s S = ρ 2 χ a (s) (ρχ a (u)) 1 = χ a (S) ρχ a (u) s S So R(ρχ a ) has eigenvalue + χ a (S) I(ρχ a ) has eigenvalue χ a (S) 2 to 2 (unless a is an involution).

50 (3,6)-Fullerenes are Cayley Sum Graphs Theorem (DGMS) Clincher! Every (3,6)-Fullerene is a Cayley Sum Graph on Z 2m Z 2n. Involutions a {(0, 0), (m, 0), (0, n), (m, n)} yield λ = χ a (S) {3, 1, 1, 1} NonInvolutions a = ±(x, y) yield eigenvalue pairs λ = ± χ a (S)

51 Grid Construction of (3, 6)-Fullerenes C B A A C B ABC any acute triangle on Eisenstein grid Midpoints A, B, C are grid points Fold tetrahedron Take the dual graph

52 Grid Construction of (3, 6)-Fullerenes C B A A C B ABC any acute triangle on Eisenstein grid Midpoints A, B, C are grid points Fold tetrahedron Take the dual graph

53 Proof Setup C D A B A C B

54 Proof Setup Vertices are Up-triangles and Down-triangles

55 Proof Setup Folding identifies vertices and connects edges

56 Proof Setup (B, B ) 1 2 (A, A ) 1 2 A =(A 1, A 2 ) and B =(B 1, B 2 ) generate lattice L

57 Proof Setup Translated lattice of pivot points

58 Proof Setup Reflect (rotate) down-triangles about any pivot point

59 Universal Cover is an Eisenstein Cayley Sum Graph

60 Universal Cover is an Eisenstein Cayley Sum Graph

61 Universal Cover is an Eisenstein Cayley Sum Graph

62 Universal Cover is an Eisenstein Cayley Sum Graph

63 Universal Cover is an Eisenstein Cayley Sum Graph

64 Universal Cover is an Eisenstein Cayley Sum Graph

65 Universal Cover is an Eisenstein Cayley Sum Graph

66 Universal Cover is an Eisenstein Cayley Sum Graph

67 Universal Cover is an Eisenstein Cayley Sum Graph

68 Universal Cover is an Eisenstein Cayley Sum Graph

69 Universal Cover is an Eisenstein Cayley Sum Graph Three Points of Concurrence!

70 Universal Cover is an Eisenstein Cayley Sum Graph The Cayley Sum Graph

71 General Procedure L L/ <A,B> L + L + /<A,B>

72 Finding the Homomorphism E L Z 2m Z 2n (0,0) (m,0) (0,0) (m,n) (0,n) (m,n) (0,0) (m,0) (0,0)

73 Finding the Homomorphism E L Z 2m Z 2n (1,1) (0,0) (0,1) (1,0) (0,9) (1,8) (1,1) (0,0) (1,9) (0,8) (1,7) (0,6) (1,5) (1,8) (0,7) (1,6) (0,5) (1,4) (0,3) (0,6) (1,5) (0,4) (1,3) (0,2) (1,1) (0,0) (1,2) (0,3) (0,1) (1,0) (0,9) (1,8) (1,1) (0,0) (1,9) (0,8) (1,7) (0,9) (1,8) (0,7)

74 Finding the Homomorphism E L Z 2m Z 2n (0,0) (0,0) (1,7) (1,1) (1,8) (0,7) (0,3) (1,4) (0,5) (1,6) (0,6) (1,5) (0,4) (1,3) (0,2) (0,0) (0,6) (1,7) (0,8) (1,9) (0,0) (0,3) (1,2) (0,1) (1,0) (0,9) (1,8) (1,8) (0,9) (1,0) (0,1) (1,1) (0,0) (1,9) (0,8) (1,7) (0,0) (1,1) (0,9) (1,8) Rotate

75 Finding the Homomorphism E L Z 2m Z 2n (1,1) (0,0) (0,0) (0,9) (1,8) (1,8) (0,9) (1,1) (0,0) (1,1) (0,9) (1,8) Generators are Γ-coords of Eisenstein [ 1, 0], [0, 1], [ 1, 1]

76 Computing It: Lattice Basis Change ( ) Z E =[a, b] AB = 4a + 2b AC = 2a + 4b Z ( ) 4 2 Put M = 2 4 SNF: UMV = D, (for some U, V unimodular, D diagonal) ( ) ( ) ( ) ( ) = Hence Γ=Z 2 Z 10. If U =(u 1 u 2 ), then S = { u 1, u 2, u 1 u 2 } = {(1, 8), (1, 1), (0, 9)}

77 Computing an eigenvalue Let a =(1, 3) / Inv(Z 2 Z 10 ),say. S = {(1, 8), (1, 1), (0, 9)} χ (1,3) (S) =e 2iπ1(1) + 2iπ3(8) e 2iπ1(1) + 2iπ3(1) e 2iπ1(0) + 2iπ3(9) 2 10 =(2cos(π/5) cos 2π/5) i sin 2π/5 = e π/5 Corresponding eigenvalues: ± χ (1,3) (S) = ± Phase shift for eigenvectors: 1 2 arg(χ (1,3)(S)) = π/10.

78 What if A, B, C are not gridpoints? B A C C B A Semiloops! (0, 3, 6)-Fullerene: 3-regular, planar, hexagons & triangles & (up to 4) semiloops

79 What if A, B, C are not gridpoints? B A C C A B (5, 2, 1) Semiloops! (0, 3, 6)-Fullerene: 3-regular, planar, hexagons & triangles & (up to 4) semiloops

80 Some (0, 3, 6)-Fullerenes Eigenvalues = {3, 1, 1, 1}.

81 Fowler s other conjecture Conjecture (Fowler et al., 2002) The spectrum of every (0, 3, 6)-Fullerene has the form A Λ Λ, where A contains one 3 and at most three ±1. This is true! They are all Cayley Sum Graphs.

82 Proof for (0, 3, 6)-polyhedra G a(0, 3, 6)-polyhedron G K 2 bipartite double cover embedding of G gives an embedding of G K 2 (on a different surface) this embedding has all faces of size 6, d1 a0 a1 d0 b1 c1 hence it is on torus next we proceed as for (3, 6)-polyhedra: U black vertices of the universal cover (infinite hex. grid) L black vertices that correspond to a given vertex of G etc. c0 b0

83 Other Possibilities L L/ <A,B> L + L + /<A,B> L = A 2 L + = A + 2 : triangular grid ( (3,6)-Fullerenes ) L = D 2 L + = Z 2 : checkerboard ( requires 4 semiloops ) L = D 3 L + = D + 3 : diamond packing (8 unpaired eigenvalues)

84 Summary We proved Fowler s conjecture and described precisely eigenvalues and eigenvectors of (0, 3, 6)-polyhedra. Other lattices yield families with A Λ Λ Other natural families of spectrally nearly bipartite graphs? Much Harder: (5, 6)-Fullerenes. One problem: When are there more negative than positive eigenvalues?