Invariant Kekulé structures in fullerene graphs

Transcription

1 Available online at Electronic Notes in Discrete Mathematics 40 (2013) Invariant Kekulé structures in fullerene graphs Mathieu Bogaerts 1 Faculté des Sciences Appliquées Université Libre de Bruxelles Bruxelles, Belgium Giuseppe Mazzuoccolo 2 G-Scop Laboratory Grenoble, France Gloria Rinaldi 3 Dipartimento di Scienze e Metodi dell Ingegneria Università di Modena e Reggio Emilia Reggio Emilia, Italy Abstract Fullerene graphs are trivalent plane graphs with only hexagonal and pentagonal faces. They are often used to model large carbon molecules: each vertex represents a carbon atom and the edges represent chemical bonds. A totally symmetric Kekulé structure in a fullerene graph is a set of independent edges which is fixed by all symmetries of the fullerene and molecules with totally symmetric Kekulé structures could have special physical and chemical properties, as suggested in [1] and [8]. All fullerenes with at least ten symmetries were studied in [4] and a complete catalog was given in [5]. Starting from this catalog in [2] we established exactly which of them have at least one totally symmetric Kekulé structure. Keywords: Fullerene graphs, 1 factors, automorphism groups /$ see front matter 2013 Elsevier B.V. All rights reserved.

2 324 M. Bogaerts et al. / Electronic Notes in Discrete Mathematics 40 (2013) Introduction A fullerene graph is a trivalent plane graph Γ = (V,E,F) with only hexagonal and pentagonal faces. From Euler s formula easily follows that each fullerene has exactly 12 pentagonal faces. The simplest one is the graph of the dodecahedron with 12 pentagonal faces and no hexagonal ones and their existence is guaranteed for all even v with v 24 and for v = 20. Graph theoretic fullerenes can be used to model large carbon molecules: each vertex represents a carbon atom and the edges represent chemical bonds. Since a carbon atom has chemical valence 4, one edge at each vertex of the graph must represent a double chemical bond. The study of these particular graphs have achieved considerable interest since Kroto, Smalley and co-workers discovered the icoshaedral molecule C 60, [7], the so called Buckminsterfullerene, and through the analysis of the products obtained from the laser vaporizations of graphite, numerous fullerenes with icosahedral symmetry, such as C 60, C 80, C 140, C 180, C 240, C 260, C 420, C 540, etc., were discovered, [6], [3]. Figure 1. the Buckminsterfullerene A perfect matching, or1 factor of Γ is a set of independent edges (i.e., edges not shearing vertices) covering all vertices of Γ. In the chemical literature a perfect matching is often called a Kekulé structure andinthemodelof carbon molecules the edges of a perfect matching correspond to double bonds. AKekulé structure is said to be totally symmetric if it is fixed by the full automorphism group of Γ. 1 mbogaert@ulb.ac.be 2 mazzuoccolo@unimore.it 3 gloria.rinaldi@unimore.it

3 M. Bogaerts et al. / Electronic Notes in Discrete Mathematics 40 (2013) It was suggested in [8] and[1] that molecules (fullerenes) with totally symmetric Kekulé structures could have special physical and chemical properties. For instance, this is the case of C 60 : it has Kekulé structures, however its special physical and chemical properties are compatible with the dominance of just one Kekulé structure: the only one of the to be totally symmetric (see [1] for more details). The main idea in [8] is that any totally symmetric Kekulé structure could correspond to a minimum on the potential surface: for that reason, in the cited paper, a complete catalog of all fullerenes with at most 40 vertices which admit a totally symmetric Kekulé structure is obtained. It is straightforward that in a fullerene with trivial automorphism group each Kekulé structure is totally symmetric. On the other hand, it seems reasonable to have a few or no totally symmetric Kekulé structure in a fullerene with a significant automorphism group. Fullerenes with at least ten symmetries were studied, classified and listed in a complete catalog by J.E. Graver in [4], [5]. The aim of our work was to discover which of these fullerenes admit at least one totally symmetric Kekulé structure. For brevity, we will denote a totally symmetric Kekulé structure by TSKS. There is one class of fullerenes for which a TSKS can always be found: the leapfrog fullerenes. A leapfrog fullerene Γ l is obtained by the classical construction of truncating the dual of a fullerene Γ. It is easy to check that the number of vertices of Γ l is three times the number of vertices of Γ and the full automorphism group of Γ l coincides with that of Γ. The catalog of [5] includes both leapfrog and not leapfrog fullerenes. 2 Fullerenes with at least ten symmetries All fullerenes with at least ten symmetries are listed in the catalog of [5]. This catalog contains 112 different infinite families, in details: 24 families of fullerenes with Icosahedral or Tetrahedral automorphism groups; 27 families with automorphism group which contains a rotation of order 6; 25 families with automorphism group which contains a rotation of order 5 and 36 families with automorphism group which contains a rotation of order 3, but not a rotation of order 6. The catalog presents a classification scheme for all fullerenes. In particular, to perform his contruction, Graver assigns to each fullerene a 12 vertex planar graph with edge and angle labels which is called the signature of the fullerene. It turns out that the signature graph and the fullerene have the same full automorphism group and each fullerene, together with its planar embedding

4 326 M. Bogaerts et al. / Electronic Notes in Discrete Mathematics 40 (2013) ,5 (2) 1,5 (2,2) (2,1) 4 2 (4,1) (3,3) 0 Figure 2. Coxeter coordinates on the sphere, can be reconstructed from its signature graph in a unique manner. To work with fullerenes and their signature, Coxeter coordinates of segments and angles between segments must be defined. Theese coordinates will be used to label edges and angles of a signature graph in order to reconstruct the associated fullerene. Namely, let Λ be a plane together with a regular hexagonal tesselation. Fix the center of each hexagon. By a segment in Λ we mean a straight line segment that joins two centers. We assign Coxeter coordinates to a segment as follows: if it lies on a line perpendicular to hexagon edges, the single Coxeter coordinate (p) is assigned, where p + 1 is the number of centers of hexagons on the segment. If not, take the first line to the right of the segment which is perpendicular to an hexagon edge to identify the first coordinate direction, then turn left with an angle of 60 o to find the second coordinate direction. The number p+1 (resp. q + 1) of centers of hexagons we pass through in the first (resp. second) direction when connecting the two endpoints of the segment, give the Coxeter coordinates (p, q) of the segment itself. The type of an angle between two segments is the number of centers of the edges of the central hexagons between the segments. Segments which runs to successive centers contribute 1/2 to each of the angle types on either side. See for example Figure 2. Essentially Graver s construction works as follows: denote by Λ the plane with the hexagonal tessellation, take the signature of Γ, draw each face of the signature on Λ putting each vertex of the face in the center of an hexagon of Λ in such a way that the polygonal region of Λ corresponding to that face is completely determined by the Coxeter coordinates which label edges of the face and by the types of angles between them. By gluing together the regions of Λ corresponding to the faces of the signature, we reconstitute the geodesic dome and the graph model of Γ together with its planar embedding on the sphere. In particular, during this process, the twelve vertices of the signature give rise to the twelve pentagonal faces of the corresponding fullerene.

5 Through a deep analysis of the catalog of [5], we were able to establish the following hexaustive result, [2]: Theorem 2.1 A fullerene with icoshedral or tetrahedral symmetry group has a TSKS if and only if it is leapfrog. Leapfrog condition coincides with the property that no order 3 symmetry of the group fixes a vertex of the fullerene. A fullerene graph with automorphism group which contains either a rotation of order 5 or a rotation of order 6 has a TSKS and no automorphism of order 3 fixes a vertex of the fullerene. Except for some sporadic cases, a fullerene graph with automorphism group which contains a rotation of order 3 but not a rotation of order 6 has a TSKS if and only if each automorphism of order 3 does not fix any vertex of the fullerene. Referring to the catalog notations, the sporadic cases mentioned above correspond to families P 4 and P 5 with r =3,s =1andp even, for which a TSKS does not exist. References M. Bogaerts et al. / Electronic Notes in Discrete Mathematics 40 (2013) [1] Austin, S.J, and J. Baker, P. W. Fowler, D. E. Manolopoulos, Bond stretch Isomerism and the Fullerenes, J. Chem. Soc. Perkin Trans. 2 (1994), [2] Bogaerts, M., and G. Mazzuoccolo, G.Rinaldi Totally symmetric Kekulé structures in fullerene graphs with ten or more symmetries, MATCH Communications in Mathematical and in Computer Chemistry 69 (2013), [3] Curl, R. F. and R. E. Smalley, Fullerenes, Sci.Am.265 (1991), [4] Graver, J.E. The Structure of Fullerene Signature, DIMACS Series of Discrete Mathematics and Theoretical Computer Science 64 AMS (2005), [5] Graver, J. E. Catalog of All Fullerene with Ten or More Symmetries DIMACS Series of Discrete Mathematics and Theoretical Computer Science 64 AMS (2005), [6] Kratschmer, W. and L. D. Lamb, K. Fostiropoulos, D. R. Huffman, Solid C60: a new form of carbon, Nature347 (1991), [7] Kroto, H. W. and J. R. Health, S. C. O Brien, R. F. Curl, R. E. Smalley, C 60 : Buckminsterfullerene, Nature (London) 318 (1985), 162. [8] Rogers, K.M., and P. W. Fowler Leapfrog fullerenes, Huckel bond order and Kekulé structures, J.Chem.Soc.PerkinTrans.2 (2001),