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  • Graph Theoretic Approaches to Atomic Vibrations in

    Fullerenes

    ERNESTO ESTRADA Department of Mathematics & Statistics,

    Department of Physics University 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 2 θ

    =

    (1)

    (2)

  • β

    m=1

    θ

    Edge weight: βθ

    1

    2

    3

    4

      

      

    −−− −− −−− −−

    =−= 3111 1210 1131 1012

    ADL

  • ( )∫=≡∆ xdxPxxx iii 22

    ∫= xdxPxxxx jiji )(

    . 2

    exp

    2 exp

    2

    1  

      

    −=

     

      

    −=

    ∏∫

    =

    ∞+

    ∞− αα α

    α µ βθ

    βθ

    ydy

    yyydZ

    n

    T  M

    nµµµ ≤≤

  • .2

    2 exp~

    2

    2

    2

    ∏∫

    =

    =

    ∞+

    ∞−

    =

     

      

    −=

    n

    n

    ydyZ

    α α

    αα α

    α

    βθµ π

    µβθ

    01 =µ

    ( )

    ( )

    .~

    28 2

    2 exp

    2 exp~

    2

    2

    22 3

    2

    2

    2

    22

    2

    ∏∑

    ∏∫∑∫

    =

    ≠ ==

    ≠ =

    ∞+

    ∞− =

    ∞+

    ∞−

    ×=

    ×=

     

      

    −× 

      

    −≡

    n i

    nn i

    n

    i

    n

    i

    U Z

    U

    ydyyyUdyI

    ν ν

    ν

    να α νν ν

    ν

    αα

    να α

    ανννν ν

    ν

    βθµ

    βθµ π

    βθµ π

    µβθµβθ

    (6)

    (7)

  • ,~ ~

    2

    2 2 ∑

    =

    ==≡∆ n

    ii ii

    U Z I

    xx ν ν

    ν

    βθµ

    ( ) ( )iiix +=∆ Lβθ 12

    +L

    (8)

    (9)

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

  • ( )ij

    n ji

    ji

    UU xx

    +

    =

    =

    =∑

    L βθ

    βθλν ν νν

    1 2

    (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

    i i j

    ji RGKf ,2 1

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

  • ( ) ( )   

       −==∆ +

    n KfR

    n x iiii

    12 L

    ( ) ( )  

      

     ++−Ω−== + 2 21

    2 1

    n KfRR

    n xx jiijijji L

    ( ) ( )∑∑ ∈=

    Ω−+−= Eji

    ijji

    n

    i ii nRRn

    Rk n

    xV ,1 2

    1 2 1

    (11)

    (12)

    (13)

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

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

    ( ) ( )222 xnxnKf n

    ii i ∆=∆= ∑

    =

    ( ) ( ) ( ) ( )[ ]∑ =

    ∆+∆=∆+∆= n

    i iiii 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

    S S

    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 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 jj=3

    n

    ∑ 

     

     

    i=1

    n

    ∑ = 1n 1 ∆ +

    1 λ1 − λ jj=3

    n

    ∑ 

     

      .

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

  • ( ) ( ) 2 ,

    2 1 222

    ˆˆ s

    s s

    sr rssr

    s

    s xkAxx m

    p H ∑∑∑ −−−+= ωω

    dx d

    i ps

     ≡ˆ sk

     

      

     −=+ sss pm

    imxa ˆ 2 1ˆ

    ϖ ϖ

       

      

     +=− sss pm

    imxa ˆ 2 1ˆ

    ϖ ϖ

    m ωϖ ≡

    (18)

  • ( ) ( )−+−+−+ ++− 

      

     +=

    − 

      

     +=

    ∑∑

    ∑∑

    ssrs sr

    rrss s

    sr srsr

    s s

    s

    aaAaaaa

    xAxx m

    pH

    ˆˆˆˆ 22

    1ˆˆ

    22 ˆˆ

    ,

    ,

    2

    ϖϖ

    ωω

     

    ∑= j

    jHH ˆˆ

    (19)

    (20)

  • ( ) ( )1ˆˆ2ˆˆ

    22 1ˆˆ

    ˆˆ 22

    1ˆˆˆ

    22

    2

    +++− 

      

     +=

    +− 

      

     +≡

    −+−+−+

    −+−+

    jjjjjjj

    jjjjjj

    bbbbbb

    bbbbH

    λϖϖ

    λϖϖ

     

     

    ( ) ++ ∑= j s

    jj asb ˆˆ ϕ ( ) −− ∑= j s

    jj asb ˆˆ ϕ

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

    ( ) ( ) ,

    ˆ,ˆˆ,ˆ

    ,

    ,

    jll s

    j

    rsl sr

    j

    sr slrjjj

    ss

    sr

    asarbb

    δϕϕ

    δϕϕ

    ϕϕ

    ==

    =

    =

    ∑ +−+− Eigenvectors of H

    (21)

  • 001 ˆ s H

    rsrrs xexZ xxG β−≡

    .00 ĤeZ β−≡

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

    ( )

    ( ) ( ) ( ) ( )

    ( ) 

    

     +−×

     

     +−=

    ×=

    − 

      

     +=

    ∏∑

    ∏∑

    ∏∑

    −+−−

    −+−−+

    i ji

    jj j

    j

    ji

    H j

    H jjll

    lj j

    ll i

    H jjl

    lj jrs

    sr Z

    ebebsr Z

    bbebbsr Z

    G

    ij

    i

    λϖβλϖβϕϕ

    δϕϕ

    ϕϕ

    ββ

    β

    1 2

    exp1 2

    3exp1

    000ˆˆ01

    0ˆˆˆˆ01

    ˆˆ

    ,

    ˆ

    ,

    

    (22)

    (23)

    (24)

  • ( ) ( ) ( ) .jesrCeCG j j

    jrsrs βλβ ϕϕ∑== A

    ( ) ( ) ∑==≡ j

    jeetrGEEZ βλβA

    ( )[ ]∑=∆ j

    jr jerx 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 molecules can 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 graph representing the molecular system.

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

  • NAOMICHI HATANO Institute of Industrial Sciences University of Tokyo Japan

    ADELIO R. MATAMALA Department of Chemistry University of Concepcion Chile

    PATRICK W. FOWLER Department of Chemistry University of Sheffield U.K.

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