Bipartizing fullerenes Zden ek Dvo rk, Bernard Lidick and Riste - - PowerPoint PPT Presentation

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Bipartizing fullerenes Zden ek Dvo rk, Bernard Lidick and Riste - - PowerPoint PPT Presentation

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Bipartizing fullerenes Zden ek Dvo rk, Bernard Lidick and Riste krekovski Charles University University of Illinois University of Ljubljana Second Postdoctoral Research Symposium Urbana-Champaign Jan 27 2012 Bipartizing

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SLIDE 1

Bipartizing fullerenes

Zdenˇ ek Dvoˇ rák, Bernard Lidický and Riste Škrekovski

Charles University University of Illinois University of Ljubljana

Second Postdoctoral Research Symposium Urbana-Champaign Jan 27 2012

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SLIDE 2

Bipartizing fullerenes

What is a fulleren?

A fullerene is a polyhedral molecule consisting only of carbon atoms and containing only pentagonal and hexagonal faces.

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SLIDE 3

Bipartizing fullerenes

What is a fulleren?

A fullerene is a polyhedral molecule consisting only of carbon atoms and containing only pentagonal and hexagonal faces. buckyball (C60)

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SLIDE 4

Bipartizing fullerenes

What is a fulleren?

A fullerene is a polyhedral molecule consisting only of carbon atoms and containing only pentagonal and hexagonal faces. buckyball (C60) soccer ball

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SLIDE 5

Bipartizing fullerenes

What is a fulleren?

A fullerene is a polyhedral molecule consisting only of carbon atoms and containing only pentagonal and hexagonal faces. nanotubes

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SLIDE 6

Bipartizing fullerenes

History

  • 1965 - C60 first mentioned H.P

. Schultz (1965)

  • 70’s - theoretical study (prediction) of C60
  • 1985 - C60 exists! H. Kroto, J. R. Heath, S. O’Brien, R. Curl

and R. Smalley

  • 1991 - possible to produce C60, nanotubes
  • 1996 - Nobel prize in chemistry for C60
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SLIDE 7

Bipartizing fullerenes

History

  • 1965 - C60 first mentioned H.P

. Schultz (1965)

  • 70’s - theoretical study (prediction) of C60
  • 1985 - C60 exists! H. Kroto, J. R. Heath, S. O’Brien, R. Curl

and R. Smalley

  • 1991 - possible to produce C60, nanotubes
  • 1996 - Nobel prize in chemistry for C60
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SLIDE 8

Bipartizing fullerenes

History

  • 1965 - C60 first mentioned H.P

. Schultz (1965)

  • 70’s - theoretical study (prediction) of C60
  • 1985 - C60 exists! H. Kroto, J. R. Heath, S. O’Brien, R. Curl

and R. Smalley

  • 1991 - possible to produce C60, nanotubes
  • 1996 - Nobel prize in chemistry for C60
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SLIDE 9

Bipartizing fullerenes

History

  • 1965 - C60 first mentioned H.P

. Schultz (1965)

  • 70’s - theoretical study (prediction) of C60
  • 1985 - C60 exists! H. Kroto, J. R. Heath, S. O’Brien, R. Curl

and R. Smalley

  • 1991 - possible to produce C60, nanotubes
  • 1996 - Nobel prize in chemistry for C60
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SLIDE 10

Bipartizing fullerenes

History

  • 1965 - C60 first mentioned H.P

. Schultz (1965)

  • 70’s - theoretical study (prediction) of C60
  • 1985 - C60 exists! H. Kroto, J. R. Heath, S. O’Brien, R. Curl

and R. Smalley

  • 1991 - possible to produce C60, nanotubes
  • 1996 - Nobel prize in chemistry for C60
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SLIDE 11

Bipartizing fullerenes

History

  • 50’s - Select manufactured the first "C60" soccer ball
  • 1965 - C60 first mentioned H.P

. Schultz (1965)

  • 70’s - theoretical study (prediction) of C60
  • 1985 - C60 exists! H. Kroto, J. R. Heath, S. O’Brien, R. Curl

and R. Smalley

  • 1991 - possible to produce C60, nanotubes
  • 1996 - Nobel prize in chemistry for C60
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SLIDE 12

Bipartizing fullerenes

Theoretical prediction of fullerenes

  • glue together pentagonal and

hexagonal faces

  • is the result stable?
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SLIDE 13

Bipartizing fullerenes

Fullerens as graphs (in graph theory)

  • atoms - vertices
  • adjacency - edges
  • molecule - planar graph
  • 12 pentagonal faces, unbounded number of hexagonal

faces C60

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SLIDE 14

Bipartizing fullerenes

Stability of fullerenes predicted by graphs

Not all graphs correspond to fullerenes (resulting molecules are not stable)

Conjecture

Stability of fullerenes corresponds to some graph property.

  • number of perfect matchings
  • independence number
  • ....
  • isolated pentagon rule - close pentagons are trouble
  • What is the distance between pentagons?

(No good correspondence is know yet)

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SLIDE 15

Bipartizing fullerenes

Stability of fullerenes predicted by graphs

Not all graphs correspond to fullerenes (resulting molecules are not stable)

Conjecture

Stability of fullerenes corresponds to some graph property.

  • number of perfect matchings
  • independence number
  • ....
  • isolated pentagon rule - close pentagons are trouble
  • What is the distance between pentagons?

(No good correspondence is know yet)

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SLIDE 16

Bipartizing fullerenes

How distant?

How far can the pentagons be from each other?

Conjecture (Došli´ c, Vukiˇ cevi´ c)

Distance is at most

  • 12n/5.

Theorem (Dvoˇ rák, L., Škrekovski)

Distance is at most c√n for some constant c. Fullerene graph can be made bipartite by removing c√n edges.

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SLIDE 17

Bipartizing fullerenes

How distant?

How far can the pentagons be from each other?

Conjecture (Došli´ c, Vukiˇ cevi´ c)

Distance is at most

  • 12n/5.

Theorem (Dvoˇ rák, L., Škrekovski)

Distance is at most c√n for some constant c. Fullerene graph can be made bipartite by removing c√n edges.

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SLIDE 18

Bipartizing fullerenes

Our result

Theorem (Dvoˇ rák, L., Škrekovski)

Let F be a pentagonal face. There are 5 other pentagonal faces in distance at most c√n from F.

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SLIDE 19

Bipartizing fullerenes

Our result

Theorem (Dvoˇ rák, L., Škrekovski)

Let F be a pentagonal face. There are 5 other pentagonal faces in distance at most c√n from F.

Corollary (Dvoˇ rák, L., Škrekovski)

"Fullerenes look like nanotubes."

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SLIDE 20

Bipartizing fullerenes

Thank you for your attention!