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Graph Theoretic Approaches to Atom ic Vibrations in Fullerenes

ERNESTO ESTRADA Department of Mathematics & Statistics, Department of Physics University of Strathclyde, Glasgow

w w w .estradalab.org

“A fullerene, by definition, is a closed convex cage molecule containing only hexagonal and pentagonal faces.”

• R. E. Smalley

β

m= 1 θ

= + Lx x M  

( )

x x x V

T

   L 2 θ =

(1) (2)

β

m= 1 θ

Edge weight: βθ

1 2 3 4

          − − − − − − − − − − = − = 3 1 1 1 1 2 1 1 1 3 1 1 1 2 A D L

( )

∫

= ≡ ∆ x d x P x x x

i i i

 

2 2

∫

= x d x P x x x x

j i j i

 ) (

. 2 exp 2 exp

2 1

     − =      − =

∏∫ ∫

= ∞ + ∞ − α α α α

µ βθ βθ y dy y y y d Z

n T

   M

n

µ µ µ ≤ ≤ < = 

2 1

(3) (4) (5)

. 2 2 exp ~

2 2 2

∏ ∏∫

= = ∞ + ∞ −

=      − =

n n

y dy Z

α α α α α α

βθµ π µ βθ

1 =

µ

( ) ( )

. ~ 2 8 2 2 exp 2 exp ~

2 2 2 2 3 2 2 2 2 2 2

∑ ∏ ∑ ∏∫ ∑∫

= ≠ = = ≠ = ∞ + ∞ − = ∞ + ∞ −

× = × =      − ×      − ≡

n i n n i n i n i

U Z U y dy y y U dy I

ν ν ν ν α α ν ν ν ν α α ν α α α ν ν ν ν ν ν

βθµ βθµ π βθµ π µ βθ µ βθ

(6) (7)

, ~ ~

2 2 2

∑

=

= = ≡ ∆

n i i i i

U Z I x x

ν ν ν

βθµ

( )

( )ii

i

x

+

= ∆ L βθ 1

2 +

L

(8) (9)

Estrada, E., Hatano, N. Chem. Phys. Lett. 486 2 0 1 0 , 166.

( )

ij n j i j i

U U x x

+ =

= =∑ L βθ βθλ

ν ν ν ν

1

2

(10)

Estrada, E., Hatano, N. Chem. Phys. Lett. 486 2 0 1 0 , 166.

Estrada, E., Hatano, N. Chem. Phys. Lett. 486 2 0 1 0 , 166.

Estrada, E., Hatano, N. Chem. Phys. Lett. 486 2 0 1 0 , 166.

1 2 3 4

2 , 1

Ω

( ) ( ) ( ) 2

, 1 2 , 2 1 , 1 2 , 1

2

+ + +

− + = Ω L L L

( )

∑

∈

Ω =

G V j j i i

R

,

( )

∑ ∑∑

= Ω =

i i i j j i

R G Kf

,

2 1

Klein, D.J. , Randić, M. J. Math. Chem. 1 2 (1993) 81.

( )

( )

      − = = ∆

+

n Kf R n x

i ii i

1

2

L

( ) ( )

      + + − Ω − = =

+ 2

2 1 2 1 n Kf R R n x x

j i ij ij j i

L

( )

( )

∑ ∑

∈ =

Ω − + − =

E j i ij j i n i i i

n R R n R k n x V

, 1

2 1 2 1 

(11) (12) (13)

Estrada, E., Hatano, N. Physica A. 389 2 0 1 0 , 3648.

( )

( )

[ ]

( )

2 2 2

2

j i j i j i ij

x x x x x x − = − ∆ + ∆ = Ω

( ) ( )

2 2 2

x n x n Kf

n i i i

∆ = ∆ = ∑

=

( ) ( ) ( ) ( )

[ ] ∑

=

∆ + ∆ = ∆ + ∆ =

n i i i i i

x x n x x n R

1 2 2 2 2

(14) (15) (16)

Estrada, E., Hatano, N. Physica A. 389 2 0 1 0 , 3648.

A graph G is said to have edge expansion (K, φ) if

( )

{ }

S v S u U v u E U S ∈ ∈ ⇒ ∈ ⊂ = ∂ , ,

Set of edges connecting S to its complement

( )

K S V S S S ≤ ⊆ ∀ ≥ ∂ with , φ

S S

n S S

∂ ≡

≤ 2 / :min

φ

S

S

j j

λ λ µ − =

1

∆xi

( )

2 = φ2 i

( )

   

2

∆ + φ j(i)2 λ1 − λ j

j=3 n

∑

.

n

λ λ λ ≥ ≥ ≥ 

2 1

1 ≡ βθ

2 1

λ λ − = ∆

(17)

Estrada, E.; Hatano, N.; Matamala, A. R. I n the book: Mathematics and Topology of Fullerenes, A. Graovac; O. Ori; F . Cataldo, Eds.; Springer, 2010 to appear.

Among all graphs with n nodes, those having good expansion display the smallest topological displacements for their nodes.

∆xi

( )

2 = 1

n φ2 i

( )

   

2

∆ + φ j(i)2 λ1 − λ j

j=3 n

∑

     

i=1 n

∑

= 1 n 1 ∆ + 1 λ1 − λ j

j=3 n

∑

      .

Estrada, E.; Hatano, N.; Matamala, A. R. I n the book: Mathematics and Topology of Fullerenes, A. Graovac; O. Ori; F . Cataldo, Eds.; Springer, 2010 to appear.

( ) ( ) 2

, 2

1 2 2 2 ˆ ˆ

s s s s r rs s r s s

x k A x x m p H

∑ ∑ ∑

− − − + = ω ω

dx d i ps  ≡ ˆ

s

k

        − =

+ s s s

p m i m x a ˆ 2 1 ˆ ϖ ϖ 

        + =

− s s s

p m i m x a ˆ 2 1 ˆ ϖ ϖ 

m ω ϖ ≡

(18)

( ) ( )

− + − + − +

+ + −       + = −         + =

∑ ∑ ∑ ∑

s s rs s r r r s s s s r s rs r s s s

a a A a a a a x A x x m p H ˆ ˆ ˆ ˆ 2 2 1 ˆ ˆ 2 2 ˆ ˆ

, , 2

ϖ ϖ ω ω  

∑

=

j j

H H ˆ ˆ

(19) (20)

( )

( )

1 ˆ ˆ 2 ˆ ˆ 2 2 1 ˆ ˆ ˆ ˆ 2 2 1 ˆ ˆ ˆ

2 2 2

+ + + −       + = + −       + ≡

− + − + − + − + − + j j j j j j j j j j j j j

b b b b b b b b b b H λ ϖ ϖ λ ϖ ϖ     ( ) +

+

∑

=

j s j j

a s b ˆ ˆ ϕ

( ) −

− ∑

=

j s j j

a s b ˆ ˆ ϕ

[ ]

( ) ( )

[ ]

( ) ( ) ( ) ( )

, ˆ , ˆ ˆ , ˆ

, , jl l s j rs l s r j s r s l r j j j

s s s r a s a r b b δ ϕ ϕ δ ϕ ϕ ϕ ϕ = = = =

∑ ∑ ∑

+ − + −

Eigenvectors of H

(21)

1

ˆ s H r s r rs

x e x Z x x G

β −

≡

.

ˆ H

e Z

β −

≡

( ) ( ) (

) ( )

( ) ( )

( )

( ) ( )

( )

( )

( )

     + − ×       + − = × = −         + =

∏ ∑ ∏ ∑ ∏ ∑

≠ ≠ − + − − − + − − + i j i j j j j j i H j H j jl l l j j l l i H j j l l j j rs

s r Z e b e b s r Z b b e b b s r Z G

i j i

λ ϖ β λ ϖ β ϕ ϕ δ ϕ ϕ ϕ ϕ

β β β

1 2 exp 1 2 3 exp 1 ˆ ˆ 1 ˆ ˆ ˆ ˆ 1

ˆ ˆ , ˆ ,

 

(22) (23) (24)

( )

( ) ( )

.

j

e s r C e C G

j j j rs rs βλ β

ϕ ϕ

∑

= =

A

( )

( ) ∑

= = ≡

j

j

e e tr G EE Z

βλ βA

( )

[ ]

∑

= ∆

j j r

j

e r x

βλ

ϕ

2

(25) (26) (27)

Estrada, E., Rodriguez-Velazquez, A. Phys. Rev. E 71 2 0 0 5 , 056103. Estrada, E., Hatano, N. Phys. Rev. E 77 2 0 0 8 , 036111. Estrada, E., Hatano, N. Chem. Phys. Lett 439 2 0 0 7 , 247.

1 ≡ ϖ  A H − =

A A B C B C

P Q Q P

A classical m echanics approach to vibrations in m olecules can be form ulated on the basis of the generalised Moore-Penrose inverse of the graph Laplacian. A quantum m echanics approach to vibrations in m olecules based on coupled harm onic oscillators can be form ulated

• n the basis of the exponential adjacency m atrix of the graph

representing the m olecular system . Both approaches give im portant inform ation about the energetic and stability of fullerene isom ers, as w ell as provide a theoretical fram ew ork for em pirical observations such as the ‘isolated pentagon rule’.

NAOMI CHI HATANO Institute of Industrial Sciences University of Tokyo Japan ADELI O R. MATAMALA Department of Chemistry University of Concepcion Chile PATRI CK W . FOW LER Department of Chemistry University of Sheffield U.K.