Graph Theoretic Approaches to Atomic Vibrations in

Fullerenes

ERNESTO ESTRADADepartment of Mathematics & Statistics,

Department of PhysicsUniversity of Strathclyde, Glasgow

www.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

0=+ LxxM

( ) xxxV T L

2θ

=

(1)

(2)

β

m=1

θ

Edge weight: βθ

1

2

3

4

−−−−−−−−−−

=−=3111121011311012

ADL

( )∫=≡∆ xdxPxxx iii22

∫= xdxPxxxx jiji)(

.2

exp

2exp

2

1

−=

−=

∏∫

∫

=

∞+

∞− ααα

α µβθ

βθ

ydy

yyydZ

n

T M

nµµµ ≤≤<= 210

(3)

(4)

(5)

.2

2exp~

2

2

2

∏

∏∫

=

=

∞+

∞−

=

−=

n

n

ydyZ

α α

ααα

α

βθµπ

µβθ

01 =µ

( )

( )

.~

282

2exp

2exp~

2

2

223

2

2

2

22

2

∑

∏∑

∏∫∑∫

=

≠==

≠=

∞+

∞−=

∞+

∞−

×=

×=

−×

−≡

ni

nni

n

i

n

i

UZ

U

ydyyyUdyI

ν ν

ν

ναα νν ν

ν

αα

ναα

αννννν

ν

βθµ

βθµπ

βθµπ

µβθµβθ

(6)

(7)

,~~

2

22 ∑

=

==≡∆n

iiii

UZI

xxν ν

ν

βθµ

( ) ( )iiix +=∆ Lβθ12

+L

(8)

(9)

Estrada, E., Hatano, N. Chem. Phys. Lett. 486 2010, 166.

( )ij

nji

ji

UUxx

+

=

=

=∑

Lβθ

βθλν ν

νν

12

(10)

Estrada, E., Hatano, N. Chem. Phys. Lett. 486 2010, 166.

Estrada, E., Hatano, N. Chem. Phys. Lett. 486 2010, 166.

Estrada, E., Hatano, N. Chem. Phys. Lett. 486 2010, 166.

1

2

3

4

2,1Ω( ) ( ) ( ) 2,12,21,12,1 2 +++ −+=Ω LLL

( )∑∈

Ω=GVj

jiiR ,

( ) ∑∑∑ =Ω=i

ii j

ji RGKf ,21

Klein, D.J. , Randić, M. J. Math. Chem. 12 (1993) 81.

( ) ( )

−==∆ +

nKfR

nx iiii

12 L

( ) ( )

++−Ω−== +2

2121

nKfRR

nxx jiijijji L

( ) ( )∑∑∈=

Ω−+−=Eji

ijji

n

iii nRR

nRk

nxV

,1 21

21

(11)

(12)

(13)

Estrada, E., Hatano, N. Physica A. 389 2010, 3648.

( ) ( )[ ] ( )222 2 jijijiij xxxxxx −=−∆+∆=Ω

( ) ( )222 xnxnKfn

iii ∆=∆= ∑

=

( ) ( ) ( ) ( )[ ]∑=

∆+∆=∆+∆=n

iiiii xxnxxnR

1

2222

(14)

(15)

(16)

Estrada, E., Hatano, N. Physica A. 389 2010, 3648.

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

( ) SvSuUvuEUS ∈∈⇒∈⊂=∂ ,,

Set of edges connecting S to its complement

( ) KSVSSS ≤⊆∀≥∂ with ,φ

SS

nSS

∂≡

≤ 2/:minφ

S

S

jj λλµ −= 1

∆xi( )2 = φ2 i( ) 2

∆+

φ j (i)2

λ1 − λ jj=3

n

∑ .

nλλλ ≥≥≥ 21

1≡βθ 21 λλ −=∆

(17)

Estrada, E.; Hatano, N.; Matamala, A. R. In the book: Mathematics andTopology of Fullerenes, A. Graovac; O. Ori; F. Cataldo, Eds.; Springer, 2010 toappear.

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

∆xi( )2 = 1n

φ2 i( ) 2

∆+

φ j (i)2

λ1 − λ jj=3

n

∑

i=1

n

∑ =1n

1∆+

1λ1 − λ jj=3

n

∑

.

Estrada, E.; Hatano, N.; Matamala, A. R. In the book: Mathematics andTopology of Fullerenes, A. Graovac; O. Ori; F. Cataldo, Eds.; Springer, 2010 toappear.

( ) ( ) 2

,

2 1222

ˆˆs

ss

srrssr

s

s xkAxxm

pH ∑∑∑ −−−+=

ωω

dxd

ips

≡ˆ sk

−=+

sss pmimxa ˆ

21ˆ

ϖϖ

+=−

sss pmimxa ˆ

21ˆ

ϖϖ

mωϖ ≡

(18)

( ) ( )−+−+−+ ++−

+=

−

+=

∑∑

∑∑

ssrssr

rrsss

srsrsr

ss

s

aaAaaaa

xAxxm

pH

ˆˆˆˆ22

1ˆˆ

22ˆˆ

,

,

2

ϖϖ

ωω

∑=j

jHH ˆˆ

(19)

(20)

( )( )1ˆˆ2ˆˆ

221ˆˆ

ˆˆ22

1ˆˆˆ

22

2

+++−

+=

+−

+≡

−+−+−+

−+−+

jjjjjjj

jjjjjj

bbbbbb

bbbbH

λϖϖ

λϖϖ

( ) ++ ∑= js

jj asb ˆˆ ϕ ( ) −− ∑= js

jj asb ˆˆ ϕ

[ ] ( ) ( )[ ]( ) ( )

( ) ( ) ,

ˆ,ˆˆ,ˆ

,

,

jlls

j

rslsr

j

srslrjjj

ss

sr

asarbb

δϕϕ

δϕϕ

ϕϕ

==

=

=

∑

∑

∑ +−+−

Eigenvectors of H

(21)

001 ˆs

Hrsrrs xex

ZxxG β−≡

.00 HeZ β−≡

( ) ( ) ( ) ( )( ) ( )

( )

( ) ( ) ( )( )

( )

+−×

+−=

×=

−

+=

∏∑

∏∑

∏∑

≠

≠

−+−−

−+−−+

iji

jjj

j

ji

Hj

Hjjll

ljj

lli

Hjjl

ljjrs

srZ

ebebsrZ

bbebbsrZ

G

ij

i

λϖβλϖβϕϕ

δϕϕ

ϕϕ

ββ

β

12

exp12

3exp1

000ˆˆ01

0ˆˆˆˆ01

ˆˆ

,

ˆ

,

(22)

(23)

(24)

( ) ( ) ( ) .jesrCeCG jj

jrsrsβλβ ϕϕ∑== A

( ) ( ) ∑==≡j

jeetrGEEZ βλβA

( )[ ]∑=∆j

jrjerx βλϕ 2

(25)

(26)

(27)

Estrada, E., Rodriguez-Velazquez, A. Phys. Rev. E 71 2005, 056103.Estrada, E., Hatano, N. Phys. Rev. E 77 2008, 036111.Estrada, E., Hatano, N. Chem. Phys. Lett 439 2007, 247.

1≡ϖ AH −=

A

A

B C

B

C

P Q

Q

P

A classical mechanics approach to vibrations in moleculescan be formulated on the basis of the generalised Moore-Penrose inverse of the graph Laplacian.

A quantum mechanics approach to vibrations in molecules based on coupled harmonic oscillators can be formulated on the basis of the exponential adjacency matrix of the graphrepresenting the molecular system.

Both approaches give important information about the energetic and stability of fullerene isomers, as well asprovide a theoretical framework for empirical observationssuch as the ‘isolated pentagon rule’.

NAOMICHI HATANOInstitute of Industrial SciencesUniversity of TokyoJapan

ADELIO R. MATAMALADepartment of ChemistryUniversity of ConcepcionChile

PATRICK W. FOWLERDepartment of ChemistryUniversity of SheffieldU.K.