STRUCTURAL PROPERTIES OF FULLERENES Klavdija Kutnar University of - - PowerPoint PPT Presentation

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STRUCTURAL PROPERTIES OF FULLERENES Klavdija Kutnar University of - - PowerPoint PPT Presentation

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STRUCTURAL PROPERTIES OF FULLERENES Klavdija Kutnar University of Primorska, Slovenia July, 2010 Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES Fullerenes In chemistry: carbon sphere-shaped

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STRUCTURAL PROPERTIES OF FULLERENES

Klavdija Kutnar University of Primorska, Slovenia

July, 2010

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Fullerenes

In chemistry: carbon ‘sphere’-shaped molecules In mathematics: cubic planar graphs, all of whose faces are pentagons and hexagons.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Euler’s formula for planar graphs

# faces = # edges - # vertices + 2

  • ⇒ In a fullerene: 12 pentagons and all other faces hexagonal.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Buckminsterfullerene

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Our motivation for the study of fullerenes - structural properties of fullerenes

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Existence of Hamilton (or long) cycles or paths in graphs

The question of finding or proving existence of Hamilton (or long) cycles

  • r paths in graphs has long been an active area of research.

Hamilton cycle = simple cycle traversing every vertex Hamilton path = simple path traversing every vertex Two particular instances of this general problem are: Hamilton cycles/paths in vertex-transitive graphs (Lovasz,’69) Hamilton cycles in fullerenes - a special case of one of Barnette’s conjectures.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Lovasz, 1969

Does every connected vertex-transitive graph have a Hamilton path? A graph X = (V , E) is vertex-transitive if for any pair of vertices u,v there exists an automorphism α such that α(u) = v.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Examples

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Examples

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Examples

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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VTG without Hamilton cycle Only four connected VTG (n > 2) without Hamilton cycle are known: Petersen graph truncated Petersen graph Coxeter graph truncated Coxeter graph

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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VTG without Hamilton cycle

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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The truncation of the Petersen graph

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Hamiltonicity of Cayley graphs

Given a group G and a subset S of G \ {1} such that S = S−1, the Cayley graph Cay(G, S) has vertex set G and edges of the form {g, gs} for all g ∈ G and s ∈ S. Every Cayley graph is vertex-transitive. There exist vertex-transitive graphs that are not Cayley. Conjecture Every connected Cayley graph has a Hamilton cycle.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Example

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Hamiltonicity of cubic Cayley graphs

Given a group G and a generating set S of G, the Cayley graph Cay(G, S) is cubic iff |S| = 3 and S = {a, b, c | a2 = b2 = c2 = 1} or S = {a, b, b−1 | a2 = bs = 1}.

  • Klavdija Kutnar University of Primorska, Slovenia

STRUCTURAL PROPERTIES OF FULLERENES

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Hamiltonicity of cubic Cayley graphs

Theorem (Glover, Maruˇ siˇ c, 2007) Let s ≥ 3 be an integer, let G be a group with a presentation G = a, b | a2 = bs = (ab)3 = 1, ect., and let S = {a, b, b−1}. Then if |G| ≡ 2(mod 4) the Cayley graph Cay(G, S) has a Hamilton cycle, and if |G| ≡ 0(mod 4) the Cayley graph Cay(G, S) has a cycle missing

  • ut only two adjacent vertices and therefore a Hamilton path.

Theorem (Glover, KK, Maruˇ siˇ c, 2009) If s ≡ 0(mod 4) or s is odd then the Cayley graph Cay(G, S) has a Hamilton cycle.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Method of proof I

Each Cayley graph we study has a canonical Cayley map given by an embedding of the Cayley graph X = Cay(G, {a, b, b−1}) of the (2, s, 3)-presentation of a group G = a, b|a2 = 1, bs = 1, (ab)3 = 1, etc. in the closed orientable surface of genus 1 + (s − 6) |G| 12s with faces |G|

s

disjoint s-gons and |G|

3 hexagons.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Method of proof II

How is this done? By finding a tree of faces in this canonical Cayley map whose boundary encompasses all vertices of the graph.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Hamilton cycle in Buckminsterfullerene

The Buckminsterfullerene is one of only two vertex-transitive fullerenes (the other is the Dodecahedron) and it is in fact a Cayley graph of A5.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Hamiltonian tree of faces method Essential ingredient in this Hamiltonian tree of faces method is the concept of cyclic edge-connectivity and to use a similar method in the context of fullerenes cyclic edge-connectivity of fullerenes need to be studied.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Cyclic edge connectivity of fullerenes

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Cyclically k-edge-connected graphs

Cycle-separating subset A subset F ⊆ E(X) of edges of X is said to be cycle-separating (or cyclic-edge cutset) if X − F is disconnected and at least two of its components contain cycles. A cycle-separating subset F of size k is trivial if at least one of the resulting components induces a single k-cycle.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Cyclically k-edge-connected graphs

Cycle-separating subset A subset F ⊆ E(X) of edges of X is said to be cycle-separating (or cyclic-edge cutset) if X − F is disconnected and at least two of its components contain cycles. A cycle-separating subset F of size k is trivial if at least one of the resulting components induces a single k-cycle. Cyclically k-edge-connected graphs A graph X is cyclically k-edge-connected, if no set of fewer than k edges is cycle-separating in X.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Is it c.4.c?

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Is it c.4.c?

No.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Cyclic edge connectivity of fullerenes

clearly the cyclic edge-connectivity ≤ 5, (since by deleting 5 edges connecting a 5-gonal face, two components each containing a cycle are obtained) It was proven that it is in fact precisely 5 (Doˇ sli´ c, 2003). The girth of a fullerene is 5.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Cyclic edge connectivity of fullerenes

Let F be a fullerene admitting a nontrivial cycle-separating subset of size

  • 5. Then F contains a ring R of five faces.
  • ⇒ All faces in R are hexagonal.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Types of rings of five hexagonal faces

  • Klavdija Kutnar University of Primorska, Slovenia

STRUCTURAL PROPERTIES OF FULLERENES

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The pentacap A planar graph on 15 vertices with 7 faces of which one is a 10-gon and six are pentagons.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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2 pentacaps = the dodecahedron

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Fullerenes with a nontrivial cyclic-5-cutset

Theorem (Maruˇ siˇ c, KK, 2008 & Kardoˇ s, ˇ Skrekovski, 2008) Let F be a fullerene admitting a nontrivial cyclic-5-cutset. Then F contains a pentacap, more precisely, it contains two disjoint antipodal pentacaps. Recently Shiu, Li and Chan (Australasian J. Combin., 2010) characterized the spectrum of fullerenes admitting a nontrivial cyclic-5-cutset.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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(Non)-degenerate cyclic cutsets in fullerenes

A cyclic cutset F of a fullerene graph X is non-degenerate, if both components of X − F contain precisely six pentagons. Otherwise, F is degenerate.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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(Non)-degenerate cyclic cutsets in fullerenes

A cyclic cutset F of a fullerene graph X is non-degenerate, if both components of X − F contain precisely six pentagons. Otherwise, F is degenerate. Trivial cyclic cutsets are degenerate. Non-trivial cyclic-5-edge cutsets are non-degenerate.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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(Non)-degenerate cyclic cutsets in fullerenes

A cyclic cutset F of a fullerene graph X is non-degenerate, if both components of X − F contain precisely six pentagons. Otherwise, F is degenerate. Trivial cyclic cutsets are degenerate. Non-trivial cyclic-5-edge cutsets are non-degenerate. In 2008 Kardoˇ s and ˇ Skrekovski characterized fullerenes admitting a nontrivial cyclic-6-cutset.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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(Non)-degenerate cyclic cutsets in fullerenes

A cyclic cutset F of a fullerene graph X is non-degenerate, if both components of X − F contain precisely six pentagons. Otherwise, F is degenerate. Trivial cyclic cutsets are degenerate. Non-trivial cyclic-5-edge cutsets are non-degenerate. In 2008 Kardoˇ s and ˇ Skrekovski characterized fullerenes admitting a nontrivial cyclic-6-cutset. Not all the non-trivial cyclic-6-cutsets of fullerenes are non-degenerate.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Non-degenerate cyclic-7-cutsets in fullerenes

Kardoˇ s, Krnc, Luˇ zar, ˇ Skrekovski, 2010 If there exists a non-degenerate cyclic-7-cutset in a fullerene then the graph is a nanotube unless it is one of the two exceptions given in their paper. A fullerene is a nanotube, if it can be divided into a cylindrical part containing only hexagons, and two caps, each containing six pentagons and maybe some hexagons. Moreover, at least one of the pentagons should have an edge incident to the outer face of a cap.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Hamilton cycles in fullerenes

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Barnette’s conjecture Barnette’s conjecture Every 3-connected planar graph with largest face size 6 contains a Hamilton cycle. Weaker conjecture Every fullerene contains a Hamilton cycle.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Hamilton cycles in fullerenes

Theorem (Maruˇ siˇ c, KK, 2008) Let X be a fullerene admitting a nontrivial cyclic 5-cutset. Then X has a Hamilton cycle.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Leapfrog fullerenes

Leapfrog Leap is a composite operation which can be written as Leap(F) = Tr(Du(F))

  • Klavdija Kutnar University of Primorska, Slovenia

STRUCTURAL PROPERTIES OF FULLERENES

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Leapfrog fullerenes

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Hamilton cycles in leapfrog-fullerenes

Theorem (Maruˇ siˇ c, 2007) Let X be a fullerene with n vertices. Then the leapfrog-fullerene Le(X) has a Hamilton cycle if n ≡ 2 (mod 4) and contains a long cycle missing

  • ut only two adjacent vertices if n ≡ 0 (mod 4).

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Hamilton cycles in leapfrog-fullerenes Leap(C24)

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Hamilton cycles in leapfrog-fullerenes Leap(C26)

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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The method in the proof depend on purely graph-theoretic result Theorem (Payan, Sakarovitch, 1975) Let X be a cyclically 4-connected cubic graph of order n, and let S be a maximum cyclically stable subset of V (X). Then |S| = ⌊(3n − 2)/2⌋ and more precisely, the following hold. If n ≡ 2 (mod 4) then |S| = (3n − 2)/4, and X[S] is a tree and V (X) \ S is an independent set of vertices; If n ≡ 0 (mod 4) then |S| = (3n − 4)/4, and either X[S] is a tree and V (X) \ S induces a graph with a single edge, or X[S] has two components and V (X) \ S is an independent set of vertices.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Cyclically stable subsets

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Semiregular automorphisms in fullerenes

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Semiregular element of a permutation group

A semiregular element of a permutation group is a non-identity element having all cycles of equal length in its cycle decomposition. The Petersen graph has a semiregular automorphism with two orbits of size 5.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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The dodecahedron given in Frucht’s notation relative to a semiregular automorphism with 4 orbits of size 5.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Semiregular automorphisms in vertex-transitive (di)graphs

Open problem (Maruˇ siˇ c, 1981) Is it true that a vertex-transitive digraph contains a semiregular automorphism? In the context of vertex-transitive graphs the existence of semiregular automorphisms helps proving the existence of Hamilton paths/cycles for some classes of such graphs. It seems reasonable to expect that methods similar to those used for finding Hamilton paths/cycles in vertex-transitive graphs could be applied, at least in some cases, to fullerenes as well. Motivated by this problem we recently characterized fullerenes with regards to the existence of semiregular automorphisms in their automorphism groups.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Semiregular automorphisms in fullerenes

Theorem (Janeˇ ziˇ c, Maruˇ siˇ c, KK, 2010) Let F be a fullerene with non-trivial automorphism group. Then either F admits a semiregular automorphism or Aut(F) ∼ = Z2, Z3 or S3. The automorphism group Aut(F) of a fullerene F is a subgroup of a {2, 3, 5}-group (Fowler, Manolopoulos, Redmond and Ryan, 1993). Let F be a fullerene admitting an automorphism α ∈ Aut(F) of

  • rder 5. Then α is a semiregular automorphism of F.

Leapfrog transformation enables us to construct an infinite family of fullerenes with a prescribed non-trivial automorphism group and having a semiregular automorphism. On the other hand, there are also infinitely many fullerenes having non-trivial automorphism groups without semiregular automorphisms.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Example of a fullerene having a semiregular automorphism

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Example of a fullerene without semiregular automorphisms

A fullerene of order 40 without a semiregular automorphism with the full automorphism group isomorphic to the cyclic group Z3.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Infinite family of fullerenes without semiregular automorphisms

The first fullerene (k = 0) in an infinite family of fullerenes without a semiregular automorphism with the full automorphism group isomorphic to the symmetric group S3.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Infinite family of fullerenes without semiregular automorphisms

The second fullerene (k = 1) in an infinite family of fullerenes without a semiregular automorphism with the full automorphism group isomorphic to the symmetric group S3.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Infinite family of fullerenes without semiregular automorphisms

The third fullerene (k = 2) in an infinite family of fullerenes without a semiregular automorphism with the full automorphism group isomorphic to the symmetric group S3.

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES

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Thank you !

Klavdija Kutnar University of Primorska, Slovenia STRUCTURAL PROPERTIES OF FULLERENES