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WIREs Comput Mol Sci
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The topology of fullerenes

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Fullerenes are carbon molecules that form polyhedral cages. Their bond structures are exactly the planar cubic graphs that have only pentagon and hexagon faces. Strikingly, a number of chemical properties of a fullerene can be derived from its graph structure. A rich mathematics of cubic planar graphs and fullerene graphs has grown since they were studied by Goldberg, Coxeter, and others in the early 20th century, and many mathematical properties of fullerenes have found simple and beautiful solutions. Yet many interesting chemical and mathematical problems in the field remain open. In this paper, we present a general overview of recent topological and graph theoretical developments in fullerene research over the past two decades, describing both solved and open problems. WIREs Comput Mol Sci 2015, 5:96–145. doi: 10.1002/wcms.1207

Conflict of interest: The authors have declared no conflicts of interest for this article.

One of the Hamiltonian cycles out of the many for (a) C60Ih (NHC = 1090), and (b) the C60D5h carbon nanotube (NHC = 3040).
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One of the Fries structures of C60, and superimposed the structure with highest Clar number (Clar(C60) = 8). Double bonds are shown in red and isolated aromatic hexagons are shaded in.
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Atomic force microscopy (AFM) image for C60 by Gross and co‐workers at a tip height of z = 3.3 Å showing the different bond orders of individual carbon‐carbon bonds in a hexagon. (Copyright © 2012, American Association for the Advancement of Science)
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Minimum, median, and maximum perfect matching count for all isomers of CN up to C120. (a) When looking at the plot for all N, we notice three distinct series depending on the value of N mod 6. (b) For each of the three series, both the maximum and minimum counts follow a simple exponential function (shown as dashed lines). Here, the ‘peak series’ at N mod 6 = 2 is shown. (c) The bounds for the IPR isomers (solid curves) are shown together with the bound for all isomers (dashed curves) for the series N mod 6 = 0. The other two series behave similarly.
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The 15 basic shapes for the two‐ and three‐ring (face) adjacencies on the surface of a fullerene; (l) denotes linear, (b) bent, (o) open, and (c) closed ring patterns (see Ref ).
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fcc lattice constant afcc for a hard sphere model as a function of the number of carbon atoms N.
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Various deformation parameters D (in percent) for a series of fullerenes selected according to stability. For larger fullerenes, Goldberg‐Coxeter transformed structures of C20 were chosen. Geometries were obtained from DFT (up to C540) or force field optimizations. IPQ: isoperimetric quotient; MCS: Minimum covering sphere; MDS: Minimum distance sphere; DTP: Diaz‐Tendero parameter; FAP: Fowler asymmetry parameter; TEP: (1 − ρ) from the topological efficiency parameter ρ.
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The decision tree for determining the symmetry point group for any fullerene from the group order and orbit counts.
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Examples of surfaces with zero (a), positive (b), and negative (c) Gaussian curvature around a point.
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Subdivision of a triangle in the dual graph; (a) for the leapfrog transformation (k = l = 1), and (b) for the lowest few halmas (l = 0).
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The steps in the GC2,1 Goldberg‐Coxeter transformation from C32D3h(5) to C224D3h. The transform and the diagrams of the unfolded fullerenes were automatically generated using the program Fullerene.
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Illustration of GC‐transform. (a) Original figure of a hexagonal sheet from Goldberg's paper. (b) The GC‐transform acting on a face in the fullerene dual for various values of k and l.
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(a) Endo‐Kroto (EK) 2‐vertex insertion. (b) Stone‐Wales (SW) transformation. (c) Example of an extended Stone‐Wales transformation. (d) A patch replacement which introduces one heptagon.
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Planar embeddings (no crossing edges) for three different fullerenes obtained from a perspective projection. (a) icosahedron C20Ih; (b) truncated icosahedron C60Ih; (c) C540Ih.
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B3LYP optimized structures for the two smallest non‐spiralable fullerenes (a) C380T and (b) C384D3 (see Ref for details).
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(a) Canonical face spiral for C60Ih. (b) The first fullerene (C28D2) with a failing face spiral.
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(a) Schlegel projection of C60Ih. (b) Cone projection of disk‐shaped C72D6d.
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Planar embeddings of fullerene graph and dual (blue color and dotted lines for the dual representation), and 3D embeddings of the duals: (a) C20Ih, for which the dual is the icosahedron; (b) C60Ih, for which the dual is the pentakis‐dodecahedron.
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Torus consisting of hexagons only with 1/3 of the edges tangential (a) and perpendicular (b) to the toroidal direction.
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A selection of different 3D shapes for regular fullerenes (distribution of the pentagons DP are set in parentheses). ‘Spherically’ shaped (icosahedral), for example, (a) C20Ih, (b) C60Ih, and (c) C960Ih (DP = 12 × 1); barrel shaped, for example, (d) C140D3h (DP = 6 × 2); trigonal pyramidally shaped (tetrahedral structures), for example, (e) C1140Td (DP = 4 × 3); (f) trihedrally shaped C440D3 (DP = 3 × 4); (g) nano‐cone or menhir C524C1 (DP = 5 + 7 × 1); cylindrically shaped (nanotubes), for example, (h) C360D5h, (i) C1152D6d, (j) C840D5d (DP = 2 × 6). The fullerenes shown in this figure and throughout the paper have been generated automatically using the Fullerene program.
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Number of distinct (non‐isomorphic) fullerene isomers CN (with and without fulfilling the IPR) with increasing number of carbon atoms N up to N = 400 (double logarithmic scale). (Data taken from the House of Graphs Ref )
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All boron fullerene B40 (adapted from Ref ). (a) 3D structure; (b) 2D graph.
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Comparison between a small portion of a TEM image of a P‐type Schwarzite and the surface described by Eq projected onto a plane normal to the z‐direction (see Ref for details). (Copyright © 2013 AIP Publishing LLC)
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Periodic P‐type (a) and D‐type (b) Schwarzite surfaces.
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Peanut shaped fulleroids. (a) C120D5d[5, 6, 7] with 10 heptagons and 22 pentagons. (b) C168D3d[5, 6, 7] with 18 heptagons and 30 pentagons.
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Spiky fulleroids with negative curvature containing heptagons derived from fullerenes through patch replacement. (a) C260I[5, 7] fulleroid derived from C140I. (b) C300D6h[5, 6, 7] derived from C180D6h. (c) C310D5h[5, 6, 7] derived from C190D5h.
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C28[5,7]‐D7d, the smallest fulleroid with heptagon extension, compared to the most stable (by ΔE= −102 kcal/mol) C28Td[5,6] fullerene isomer.
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The buckyonion structure C60@C240@C540.
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The stability of fullerenes CN in comparison to C60 obtained from density functional calculations up to the graphene limit (N → ∞). Topological stability indices from resonance energies (TRE using β = − 216 kcal/mol) or ring patterns by Alcami and Cioslowski are also shown. The graphene limit is estimated from the heat of formation of C60.
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