The topology of fullerenes

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

[1]  H. Whitney.  Non-Separable and Planar Graphs. , 1931, Proceedings of the National Academy of Sciences of the United States of America.

[2]  Linus Pauling,et al.  The Dependence of Interatomic Distance on Single Bond-Double Bond Resonance1 , 1935 .

[3]  Michael Goldberg,et al.  A Class of Multi-Symmetric Polyhedra , 1937 .

[4]  F. Strassmann,et al.  Hat ein Caesium-Isotop langer Halbwertszeit existiert? , 1943 .

[5]  H. Wiener.  Structural determination of paraffin boiling points. , 1947, Journal of the American Chemical Society.

[6]  Branko Grünbaum,et al.  The Number of Hexagons and the Simplicity of Geodesics on Certain Polyhedra , 1963, Canadian Journal of Mathematics.

[7]  M. Dewar,et al.  RESONANCE ENERGIES OF AROMATIC HYDROCARBONS , 1964 .

[8]  Harry P. Schultz,et al.  Topological Organic Chemistry. Polyhedranes and Prismanes , 1965 .

[9]  S. A. Robertson,et al.  On the Platonic and Archimedean Solids , 1970 .

[10]  P. Mani.  Automorphismen von polyedrischen Graphen , 1971 .

[11]  H. G. Forder,et al.  A Spectrum of mathematics : essays presented to H. G. Forder , 1971 .

[12]  D. Cvetkovic,et al.  Graph theory and molecular orbitals , 1974 .

[13]  D. Hearn,et al.  The Minimum Covering Sphere Problem , 1972 .

[14]  W. C. Herndon.  Resonance energies of aromatic hydrocarbons. Quantitative test of resonance theory , 1973 .

[15]  M. L. Ellzey,et al.  Resonance theory. V. Resonance energies of benzenoid and nonbenzenoid .pi. systems , 1974 .

[16]  Nenad Trinajstić,et al.  Molecular graphs having identical spectra , 1976 .

[17]  J. Aihara.  A new definition of Dewar-type resonance energies , 1976 .

[18]  I. Gutman,et al.  Graph theory and molecular orbitals. 19. Nonparametric resonance energies of arbitrary conjugated systems , 1977 .

[19]  Sumio Iijima,et al.  Direct observation of the tetrahedral bonding in graphitized carbon black by high resolution electron microscopy , 1980 .

[20]  Alexandru T. Balaban,et al.  Topological indices based on topological distances in molecular graphs , 1983 .

[21]  Ivan V. Stankevich,et al.  The Structural Chemistry of Crystalline Carbon: Geometry, Stability, and Electronic Spectrum , 1984 .

[22]  S. C. O'brien,et al.  C60: Buckminsterfullerene , 1985, Nature.

[23]  Cohen,et al.  Calculation of bulk moduli of diamond and zinc-blende solids. , 1985, Physical review. B, Condensed matter.

[24]  Douglas J. Klein,et al.  Resonance in C60, buckminsterfullerene , 1986 .

[25]  Louis E. Brus,et al.  Rehybridization and π-orbital alignment: the key to the existence of spheroidal carbon clusters , 1986 .

[26]  Robert C. Haddon,et al.  Electronic structure and bonding in icosahedral C60 , 1986 .

[27]  Robert F. Curl,et al.  Reactivity of large carbon clusters: spheroidal carbon shells and their possible relevance to the formation and morphology of soot , 1986 .

[28]  Frank K. Tittel,et al.  Negative carbon cluster ion beams: New evidence for the special nature of C60 , 1986 .

[29]  D. Wales,et al.  Theoretical studies of icosahedral C60 and some related species , 1986 .

[30]  Thomas F. George,et al.  Vibrational motions of buckminsterfullerene , 1987 .

[31]  Ph. Gerhardt,et al.  Polyhedral carbon ions in hydrocarbon flames , 1987 .

[32]  H. W. Kroto,et al.  The stability of the fullerenes Cn, with n = 24, 28, 32, 36, 50, 60 and 70 , 1987, Nature.

[33]  Sven J. Cyvin,et al.  Molecular vibrations of footballene , 1988 .

[34]  Haruo Hosoya,et al.  On some counting polynomials in chemistry , 1988, Discret. Appl. Math..

[35]  Douglas J. Klein,et al.  Elemental carbon cages , 1988 .

[36]  David E. Weeks,et al.  Rotation–vibration spectra of icosahedral molecules. II. Icosahedral symmetry, vibrational eigenfrequencies, and normal modes of buckminsterfullerene , 1989 .

[37]  T. Hopp,et al.  An Algorithm for Computing the Minimum Covering Sphere in Any Dimension , 1989 .

[38]  Shinsaku Fujita,et al.  Application of coset representations to the construction of symmetry adapted functions , 1990 .

[39]  W. Krätschmer,et al.  Solid C60: a new form of carbon , 1990, Nature.

[40]  Vudhichai Parasuk,et al.  C20: the smallest fullerene? , 1991 .

[41]  Helmut Schwarz,et al.  Endohedral Cluster Compounds: Inclusion of Helium within C 60•⊕ and C 70•⊕ through Collision Experiments , 1991 .

[42]  Douglas J. Klein,et al.  Generation of carbon‐cage polyhedra , 1991 .

[43]  D. Manolopoulos,et al.  Theoretical studies of the fullerenes: C34 to C70 , 1991 .

[44]  Smith,et al.  Orientational ordering transition in solid C60. , 1991, Physical review letters.

[45]  Helmut Schwarz,et al.  Formation of endohedral carbon-cluster noble-gas compounds with high-energy bimolecular reactions : C60Hen+ (n=1, 2) , 1991 .

[46]  M. Kappes,et al.  The Structure of the C60 Molecule: X-Ray Crystal Structure Determination of a Twin at 110 K , 1991, Science.

[47]  Kosmas Prassides,et al.  Crystal structure and bonding of ordered C60 , 1991, Nature.

[48]  Rodney S. Ruoff,et al.  The bulk modulus of C60 molecules and crystals: A molecular mechanics approach , 1991 .

[49]  A. Mackay,et al.  Diamond from graphite , 1991, Nature.

[50]  A. McGhie,et al.  Orientational Disorder in Solvent-Free Solid C70 , 1991, Science.

[51]  Walter Thiel,et al.  MNDO Study of Large Carbon Clusters , 1991 .

[52]  Edward M. Reingold,et al.  Graph drawing by force‐directed placement , 1991, Softw. Pract. Exp..

[53]  Paul A. Heiney,et al.  Solid‐State Chemistry of Fullerene‐Based Materials , 1992 .

[54]  Che Ting Chan,et al.  The geometry of small fullerene cages: C20 to C70 , 1992 .

[55]  Helmut Schwarz,et al.  Endohedral fullerene-noble gas clusters formed with high-energy bimolecular reactions of Cxn+ (x=60, 70; n=1,2,3) , 1992 .

[56]  Walter Thiel,et al.  Quantum-chemical study of C84 fullerene isomers , 1992 .

[57]  Walter Thiel,et al.  Quantum-chemical study of C78 fullerene isomers , 1992 .

[58]  J. Tersoff,et al.  Negative-curvature fullerene analog of C60. , 1992, Physical review letters.

[59]  Bertil Sundqvist,et al.  Compressibility, specific heat capacity, and Grüneisen parameter for C60/C70 , 1992 .

[60]  Valerie J. Kuck,et al.  Fullerenes : synthesis, properties, and chemistry of large carbon clusters , 1992 .

[61]  Alan L. Mackay,et al.  The geometry of hypothetical curved graphite structures , 1992 .

[62]  Ottorino Ori,et al.  A topological study of the structure of the C76 fullerene , 1992 .

[63]  Xavier Gonze,et al.  Energetics of negatively curved graphitic carbon , 1992, Nature.

[64]  Patrick W. Fowler,et al.  Localised models and leapfrog structures of fullerenes , 1992 .

[65]  D. Ugarte.  Curling and closure of graphitic networks under electron-beam irradiation , 1992, Nature.

[66]  Dresselhaus,et al.  Force-constant model for the vibrational modes in C60. , 1992, Physical review. B, Condensed matter.

[67]  Tomonari Wakabayashi,et al.  A model for the C60 and C70 growth mechanism , 1992 .

[68]  Broughton,et al.  Intramolecular-force-constant model for C60. , 1992, Physical review. B, Condensed matter.

[69]  Morinobu Endo,et al.  Formation of Carbon Nanofibers , 1992 .

[70]  Harold W. Kroto.  C60: BUCKMINSTERFULLERENE, THE CELESTIAL SPHERE THAT FELL TO EARTH , 1992 .

[71]  D. R. Woodall,et al.  Electronic stability of fullerenes: eigenvalue theorems for leapfrog carbon clusters , 1992 .

[72]  R. Smalley,et al.  Self-assembly of the fullerenes , 1992 .

[73]  Helmut Schwarz,et al.  The Neutralization of HeC in the Gas Phase: Compelling Evidence for the Existence of an Endohedral Structure for He@C60 , 1992 .

[74]  Harold W. Kroto,et al.  A postbuckminsterfullerene view of carbon in the galaxy , 1992 .

[75]  Yoshinori Ando,et al.  Pentagons, heptagons and negative curvature in graphite microtubule growth , 1992, Nature.

[76]  Robert Hettich,et al.  Fullerenes from the Geological Environment , 1992, Science.

[77]  Huang,et al.  Electronic and optical properties of the Vanderbilt-Tersoff model of negative-curvature fullerene. , 1992, Physical review. B, Condensed matter.

[78]  Helmut Schwarz,et al.  Sequential Insertion of 3He and 4He in C 60 , 1992 .

[79]  P. Fowler,et al.  Molecular graphs, point groups, and fullerenes , 1992 .

[80]  J. Bohr,et al.  C60 a new form of carbon , 1992 .

[81]  T. Kikegawa,et al.  Solid C70 high pressure and high temperature , 1993 .

[82]  E. Wasserman,et al.  The Fullerenes: The evolution of the football structure for the C60 molecule: a retrospective , 1993 .

[83]  P. Fowler,et al.  A fullerene without a spiral , 1993 .

[84]  E. Ōsawa,et al.  The evolution of the football structure for the C60 molecule: a retrospective , 1993, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[85]  Krishnan Balasubramanian,et al.  Spectral moments of fullerene cages , 1993 .

[86]  Michael T. Bowers,et al.  Experimental evidence for the formation of fullerenes by collisional heating of carbon rings in the gas phase , 1993, Nature.

[87]  D. Ugarte,et al.  Canonical structure of large carbon clusters : Cn, n > 100 , 1993 .

[88]  D. Tománek,et al.  Stability of multishell fullerenes. , 1993, Physical review. B, Condensed matter.

[89]  V. L. Turova-Botkina,et al.  An algorithm for finding the Chebyshev center of a convex polyhedron , 1994, System Modelling and Optimization.

[90]  Tomonari Wakabayashi,et al.  Ring-stacking consideration on higher fullerene growth , 1993 .

[91]  Gustavo E. Scuseria,et al.  Role of sp 3 carbon and 7-membered rings in fullerene annealing and fragmentation , 1993, Nature.

[92]  Patrick W. Fowler,et al.  Cylindrical fullerenes: The smallest nanotubes? , 1993 .

[93]  Christoph Rüchardt,et al.  C70 ist stabiler als C60: experimentelle Bestimmung der Bildungswärme von C70 , 1994 .

[94]  Patrick W. Fowler,et al.  Predictions of special signatures of fullerenes. Second-order Jahn–Teller effects on the structures of C44, C56, C68 and C92 , 1994 .

[95]  Arnout Ceulemans,et al.  C60 vibrates as a hollow sphere , 1994 .

[96]  Alan L. Mackay,et al.  Negatively curved graphite and triply periodic minimal surfaces , 1994 .

[97]  Krishnan Balasubramanian.  Exhaustive Generation and Analytical Expressions of Matching Polynomials of Fullerenes C20-C50 , 1994, J. Chem. Inf. Comput. Sci..

[98]  Helmut Schwarz,et al.  High-energy collisions of Kr@C60+. with helium. Evidence for the formation of HeKr@C60+. , 1994 .

[99]  Christoph Rüchardt,et al.  C70 Is More Stable than C60: Experimental Determination of the Heat of Formation of C70 , 1994 .

[100]  Pierre Hansen,et al.  Fullerene isomers of C60. Kekulé counts versus stability , 1994 .

[101]  Patrick W. Fowler,et al.  Closed-shell carbon frameworks: Leapfrog fullerenes and decorated spheriphane hydrocarbons , 1994, J. Chem. Inf. Comput. Sci..

[102]  M. Wohlers,et al.  Elastic Properties of Fullerenes , 1994 .

[103]  Patrick W. Fowler,et al.  Symmetry properties of the leapfrog transformation for fullerenes and benzenoids , 1994 .

[104]  B. Borštnik,et al.  Molecular mechanics of toroidal carbon molecules , 1994 .

[105]  Ioannis G. Tollis,et al.  Graph Drawing , 1994, Lecture Notes in Computer Science.

[106]  Martin Saunders,et al.  Incorporation of helium, neon, argon, krypton, and xenon into fullerenes using high pressure , 1994 .

[107]  Daniel Ugarte.  Onion-like graphitic particles , 1995 .

[108]  J. Grossman,et al.  Structure and stability of molecular carbon: Importance of electron correlation. , 1995, Physical review letters.

[109]  Francesco Zerbetto,et al.  THE STONE-WALES MAP FOR C60 , 1995 .

[110]  Patrick W. Fowler,et al.  Structural Motifs and the Stability of Fullerenes , 1995 .

[111]  Bryant W. York,et al.  Generalized Stone-Wales Transformations , 1995 .

[112]  Walter Thiel,et al.  Ab initio helium NMR chemical shifts of endohedral fullerene compounds He@Cn (n = 32−180) , 1995 .

[113]  Jerzy Cioslowski,et al.  Electronic Structure Calculations on Fullerenes and Their Derivatives , 1995 .

[114]  Walter Thiel,et al.  Can Large Fullerenes Be Spherical , 1995 .

[115]  Franz-Josef Brandenburg,et al.  An Experimental Comparison of Force-Directed and Randomized Graph Drawing Algorithms , 1995, GD.

[116]  D. Manolopoulos,et al.  An Atlas of Fullerenes , 1995 .

[117]  Walter Thiel,et al.  A density functional study on the shape of C180 and C240 fullerenes , 1995 .

[118]  Avron,et al.  Tiling rules for toroidal molecules. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[119]  Douglas J. Klein,et al.  Graph Invariants for Fullerenes , 1995, J. Chem. Inf. Comput. Sci..

[120]  Douglas J. Klein,et al.  Nomenclature and Coding of Fullerenes , 1995, J. Chem. Inf. Comput. Sci..

[121]  Alexandru T. Balaban,et al.  Chemical Graphs: Looking Back and Glimpsing Ahead , 1995, J. Chem. Inf. Comput. Sci..

[122]  Umpei Nagashima,et al.  Stability of small fullerenes Cn (n=36,40 and 60): A topological and molecular orbital approach , 1995 .

[123]  Jorge Berger,et al.  Classification scheme for toroidal molecules , 1995 .

[124]  Arnout Ceulemans,et al.  Symmetry extensions of Euler's theorem for polyhedral, toroidal and benzenoid molecules , 1995 .

[125]  Peter Senn.  Computation of the cartesian coordinates of Buckminsterfullerene , 1995 .

[126]  Yong Kiel Sung,et al.  The atom-atom potential. Exohedral and endohedral complexation energies of complexes of X@C60 between fullerene and rare-gas atoms (X = He, Ne, Ar, Kr, and Xe) , 1995 .

[127]  L. B. Ebert.  Science of fullerenes and carbon nanotubes , 1996 .

[128]  Douglas J. Klein,et al.  Graph-Geometric Invariants for Molecular Structures , 1996, J. Chem. Inf. Comput. Sci..

[129]  Douglas J. Klein,et al.  GENERALIZATIONS OF THE STONE-WALES REARRANGEMENT FOR CAGE COMPOUNDS, INCLUDING FULLERENES , 1996 .

[130]  Charlier,et al.  Structural and electronic properties of pentagon-heptagon pair defects in carbon nanotubes. , 1996, Physical review. B, Condensed matter.

[131]  Burke,et al.  Generalized Gradient Approximation Made Simple. , 1996, Physical review letters.

[132]  Bor Plestenjak.  An Algorithm for Drawing Schlegel Diagrams , 1996 .

[133]  Patrick W. Fowler,et al.  Fullerene stability and structure , 1996 .

[134]  Patrick W. Fowler,et al.  Systematic relationships between fullerenes without spirals , 1997 .

[135]  Mauricio Terrones,et al.  Electronic properties of fullerenes with nonpositive Gaussian curvature: Finite zeolites , 1997 .

[136]  P. Fowler,et al.  Dihedral fullerenes of threefold symmetry with and without face spirals , 1997 .

[137]  Tomaz Pisanski,et al.  Isoperimetric Quotient for Fullerenes and Other Polyhedral Cages , 1997, J. Chem. Inf. Comput. Sci..

[138]  Eiji Ōsawa,et al.  Endohedral Metallofullerenes. Are the Isolated Pentagon Rule and Fullerene Structures Always Satisfied , 1997 .

[139]  Harold W. Kroto.  Symmetry, Space, Stars, and C60 (Nobel Lecture) , 1997 .

[140]  Walter Thiel,et al.  Interaction energies and NMR chemical shifts of noble gases in C60 , 1997 .

[141]  Patrick W. Fowler,et al.  Non-bonding orbitals in graphite, carbon tubules, toroids andfullerenes , 1997 .

[142]  Andreas W. M. Dress,et al.  A Constructive Enumeration of Fullerenes , 1997, J. Algorithms.

[143]  P. W. Fowler.  Fullerene graphs with more negative than positiveeigenvalues: The exceptions that prove the rule of electrondeficiency? , 1997 .

[144]  Patrick W. Fowler,et al.  C62: Theoretical Evidence for a Nonclassical Fullerene with a Heptagonal Ring , 1996 .

[145]  Robert F. Curl,et al.  Dawn of the fullerenes: experiment and conjecture , 1997 .

[146]  Andreas W. M. Dress,et al.  Topological Resonance Energy of Fullerenes , 1997, J. Chem. Inf. Comput. Sci..

[147]  Mauricio Terrones,et al.  Quasiperiodic icosahedral graphite sheets and high-genus fullereneswith nonpositive Gaussian curvature , 1997 .

[148]  Mauricio Terrones,et al.  The transformation of polyhedral particles into graphitic onions , 1997 .

[149]  Gunnar Brinkmann,et al.  Problems and scope of spiral algorithms and spiral codes for polyhedral cages , 1997 .

[150]  P. Fowler,et al.  Fullerenes containing fused triples of pentagonalrings , 1997 .

[151]  Christopher D. Latham,et al.  Quantitative density-functional study of nested fullerenes , 1998 .

[152]  Tomaž Pisanski,et al.  Aspects of topology, genus and isomerism in closed 3-valent networks , 1998 .

[153]  Xiangyu Zhang,et al.  Macroscopic synthesis of onion-like graphitic particles , 1998 .

[154]  W. Thurston.  Shapes of polyhedra and triangulations of the sphere , 1998, math/9801088.

[155]  Patrick W. Fowler,et al.  A generalized ring spiral algorithm for coding fullerenes and other cubic polyhedra , 1998, Discrete Mathematical Chemistry.

[156]  G. A. Vinogradov,et al.  New isomerization operations for fullerene graphs , 1998 .

[157]  Adrian Hernandez,et al.  Fast Partial-Differential Synthesis of the Matching Polynomial of C72-100 , 1998, J. Chem. Inf. Comput. Sci..

[158]  Tiffany R. Walsh,et al.  Relaxation dynamics of C60 , 1998 .

[159]  Gustavo E. Scuseria,et al.  Why are buckyonions round? , 1998 .

[160]  Giorgio Benedek,et al.  GROWTH OF NANOSTRUCTURED CARBON FILMS BY CLUSTER ASSEMBLY , 1999 .

[161]  Keiichi Yamamoto,et al.  Electron energy-loss spectroscopy of carbon onions , 1999 .

[162]  Andreas Hirsch,et al.  Spherical Aromaticity in Ih Symmetrical Fullerenes: The 2(N+1)2 Rule. , 2000, Angewandte Chemie.

[163]  Patrick W. Fowler,et al.  Fullerenes as Tilings of Surfaces , 2000, J. Chem. Inf. Comput. Sci..

[164]  John Shawe-Taylor,et al.  Characterizing Graph Drawing with Eigenvectors , 2000, J. Chem. Inf. Comput. Sci..

[165]  Susumu Narita,et al.  Linear relationship between the bond lengths and the Pauling bond orders in fullerene molecules , 2000 .

[166]  Rodney S. Ruoff,et al.  Fullerenes : chemistry, physics, and technology , 2000 .

[167]  Patrick W. Fowler,et al.  Competition between Even and Odd Fullerenes: C118, C119, and C120 , 2000 .

[168]  David Moncrieff,et al.  Standard Enthalpies of Formation of Fullerenes and Their Dependence on Structural Motifs , 2000 .

[169]  Yuji Kobayashi,et al.  Materials science: C66 fullerene encaging a scandium dimer , 2000, Nature.

[170]  Lawrence T. Scott,et al.  Gas-phase production and photoelectron spectroscopy of the smallest fullerene, C20 , 2000, Nature.

[171]  Pierre Hansen,et al.  Distance Matrices, Wiener Indices, and Related Invariants of Fullerenes , 2001 .

[172]  Pierre Hansen,et al.  Facts and Conjectures about Fullerene Graphs: Leapfrog, Cylinder and Ramanujan Fullerenes , 2001 .

[173]  Walter Thiel,et al.  Theoretical investigation into structures and magnetic properties of smaller fullerenes and their heteroanalogues , 2001 .

[174]  Edward C. Kirby,et al.  THE ENERGETIC STABILITY OF TORI AND SINGLE-WALL TUBES , 2001 .

[175]  A. Hirsch,et al.  Spherical aromaticity of fullerenes. , 2001, Chemical reviews.

[176]  Patrick W. Fowler,et al.  Complete bond force fields for trivalent and deltahedral cages: Group theory and applications to cubane, closo-dodecaborane, and buckminsterfullerene , 2001 .

[177]  Rong-Bin Huang,et al.  A new carbon solid made of the world's smallest caged fullerene C20 , 2001 .

[178]  Masako Yudasaka,et al.  Raman scattering study of double-wall carbon nanotubes derived from the chains of fullerenes in single-wall carbon nanotubes , 2001 .

[179]  Patrick W. Fowler,et al.  Leapfrog fullerenes, Hückel bond order and Kekulé structures , 2001 .

[180]  Yuji Kobayashi,et al.  A catalytic synthesis and structural characterization of a new [84]fullerene isomer , 2001 .

[181]  Pierre Hansen,et al.  Boundary uniqueness of fusenes , 2002, Discret. Appl. Math..

[182]  J. van Leeuwen,et al.  Discrete and Computational Geometry , 2002, Lecture Notes in Computer Science.

[183]  Patrick W. Fowler,et al.  Constructing fullerene graphs from their eigenvalues and angles , 2002 .

[184]  M. Randic,et al.  On Characterization of 3D Molecular Structure , 2002 .

[185]  K. Sasaki,et al.  Distributions of C2 and C3 radical densities in laser-ablation carbon plumes measured by laser-induced fluorescence imaging spectroscopy , 2002 .

[186]  John Robertson,et al.  Negatively curved spongy carbon , 2002 .

[187]  Patrick W. Fowler,et al.  A Note on the Smallest Eigenvalue of Fullerenes , 2002 .

[188]  James Mack,et al.  A Rational Chemical Synthesis of C60 , 2002, Science.

[189]  M. F. Budyka,et al.  Is C2 cluster ingested by fullerene C60 , 2002 .

[190]  John Robertson,et al.  The structure of negatively curved spongy carbon , 2003 .

[191]  Patrick W. Fowler,et al.  A Catalogue of Growth Transformations of Fullerene Polyhedra , 2003, J. Chem. Inf. Comput. Sci..

[192]  Eva Zurek,et al.  Relativistic Density-Functional Computations of the Chemical Shift of 129Xe in Xe@C60 , 2003 .

[193]  Mauricio Terrones,et al.  Curved nanostructured materials , 2003 .

[194]  Patrick W. Fowler,et al.  A Catalogue of Isomerization Transformations of Fullerene Polyhedra , 2003, J. Chem. Inf. Comput. Sci..

[195]  Jack E. Graver.  The (m,k)-patch boundary code problem , 2003 .

[196]  Chun-Sing Lee,et al.  A thermodynamic and kinetic study of the formation of C20 compounds encapsulating H, He and Ne atoms , 2003 .

[197]  Walter Thiel,et al.  Performance of semiempirical methods in fullerene chemistry: relative energies and nucleus-independent chemical shifts , 2003 .

[198]  Thomas C. Hales.  A computer verification of the Kepler conjecture , 2003 .

[199]  Gunnar Brinkmann,et al.  Pentagon-hexagon-patches with short boundaries , 2003, Eur. J. Comb..

[200]  Beate Paulus,et al.  Electronic and structural properties of the cage-like molecules C20 to C36 , 2003 .

[201]  Henry E. Bass,et al.  Handbook of Elastic Properties of Solids, Liquids, and Gases , 2004 .

[202]  J. L. Dunn,et al.  A complete nearest-neighbor force field model for C60. , 2004, The Journal of chemical physics.

[203]  Mikael P Johansson,et al.  Au32: a 24-carat golden fullerene. , 2004, Angewandte Chemie.

[204]  Antonio Laganà,et al.  Computational Science and Its Applications – ICCSA 2004 , 2004, Lecture Notes in Computer Science.

[205]  Mathieu Dutour Sikiric,et al.  Goldberg-Coxeter Construction for 3- and 4-valent Plane Graphs , 2004, Electron. J. Comb..

[206]  Anatoly L. Buchachenko,et al.  Endofullerenes: size effects on structure and energy , 2004 .

[207]  Paul von Ragué Schleyer,et al.  Properties of fullerene[50] and D5h decachlorofullerene[50]: a computational study. , 2004, Journal of the American Chemical Society.

[208]  Andrew P. E. York.  Inorganic Fullerenes, Onions, and Tubes. , 2004 .

[209]  Susana Iglesias-Groth,et al.  Fullerenes and Buckyonions in the Interstellar Medium , 2004 .

[210]  Paul von Ragué Schleyer,et al.  Theoretical studies on the smallest fullerene: from monomer to oligomers and solid States. , 2004, Chemistry.

[211]  Peng Sun,et al.  Computation of Minimum Volume Covering Ellipsoids , 2002, Oper. Res..

[212]  Frank Nielsen,et al.  Approximating Smallest Enclosing Balls , 2004, ICCSA.

[213]  Aleksandar Donev,et al.  Unusually dense crystal packings of ellipsoids. , 2004, Physical Review Letters.

[214]  Masaru Tsukada,et al.  Positive magnetic susceptibility in polygonal nanotube tori , 2005 .

[215]  T. Hales.  The Kepler conjecture , 1998, math/9811078.

[216]  Xiao Cheng Zeng,et al.  Au42: an alternative icosahedral golden fullerene cage. , 2005, Journal of the American Chemical Society.

[217]  L. A. Openov,et al.  Stone-Wales transformation paths in fullerene C60 , 2005 .

[218]  A. Hirsch,et al.  Fullerenes: Chemistry and Reactions , 2005 .

[219]  Damir Vukičević,et al.  Atlas of kekulé valence structures of buckminsterfullerene , 2005 .

[220]  W. H. Powell,et al.  Numbering of Fullerenes (IUPAC Recommendations 2004) , 2005 .

[221]  G. Scheithauer,et al.  Packing of convex polytopes into a parallelepiped , 2005 .

[222]  Stephan Irle,et al.  Performance of the DFTB method in comparison to DFT and semiempirical methods for geometries and energies of C20–C86 fullerene isomers , 2005 .

[223]  Manuel Alcamí,et al.  Structure and electronic properties of fullerenes C(52)q+: is C(52)2+ an exception to the pentagon adjacency penalty rule? , 2005, Chemphyschem : a European journal of chemical physics and physical chemistry.

[224]  Zhongfang Chen,et al.  Curved pi-conjugation, aromaticity, and the related chemistry of small fullerenes (< C60) and single-walled carbon nanotubes. , 2005, Chemical reviews.

[225]  L.A.Openov,et al.  Stone-Wales Transformation Paths in Fullerene C60 , 2005 .

[226]  F. Weigend,et al.  Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy. , 2005, Physical chemistry chemical physics : PCCP.

[227]  Piyush Kumar,et al.  Minimum-Volume Enclosing Ellipsoids and Core Sets , 2005 .

[228]  Manuel Alcamí,et al.  Fullerene C50: Sphericity takes over, not strain , 2005 .

[229]  David G. Wagner.  Matroid Inequalities from Electrical Network Theory , 2005, Electron. J. Comb..

[230]  Nan Shao,et al.  Search for lowest-energy fullerenes: C98 to C110. , 2006, The journal of physical chemistry. A.

[231]  Mircea V. Diudea,et al.  The Chirality of Icosahedral Fullerenes: a Comparison of the Tripling (leapfrog), Quadrupling (chamfering), and Septupling (capra) Transformations , 2006 .

[232]  Patrick W. Fowler,et al.  Growing fullerenes from seed: Growth transformations of fullerene polyhedra , 2006 .

[233]  Dieter Kotschick,et al.  The Topology and Combinatorics of Soccer Balls , 2006 .

[234]  Xiao Cheng Zeng,et al.  Evidence of hollow golden cages. , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[235]  Stephan Irle,et al.  The C60 formation puzzle "solved": QM/MD simulations reveal the shrinking hot giant road of the dynamic fullerene self-assembly mechanism. , 2006, The journal of physical chemistry. B.

[236]  Tianming Zuo,et al.  Tb3N@C84: an improbable, egg-shaped endohedral fullerene that violates the isolated pentagon rule. , 2006, Journal of the American Chemical Society.

[237]  E. Alper Yildirim,et al.  On the Minimum Volume Covering Ellipsoid of Ellipsoids , 2006, SIAM J. Optim..

[238]  Robert P. Krawczyk,et al.  Extension of the Lennard-Jones potential: Theoretical investigations into rare-gas clusters and crystal lattices of He, Ne, Ar, and Kr using many-body interaction expansions , 2006 .

[239]  Xueguang Shao,et al.  Prediction of low-energy isomers of large fullerenes from C132 to C160. , 2006, The journal of physical chemistry. A.

[240]  Filip Uhlík,et al.  Relative Stabilities of C74 Isomers , 2007 .

[241]  J. A. Peña,et al.  Estimating the Estrada index , 2007 .

[242]  Milan Randic,et al.  Numerical Kekulé Structures of Fullerenes and Partitioning of π-Electrons to Pentagonal and Hexagonal Rings , 2007, J. Chem. Inf. Model..

[243]  Nan Shao,et al.  Search for Lowest-Energy Fullerenes 2: C38 to C80 and C112 to C120† , 2007 .

[244]  Pekka Pyykkö,et al.  A London-type formula for the dispersion interactions of endohedral A@B systems. , 2007, Physical chemistry chemical physics : PCCP.

[245]  Wendy Myrvold,et al.  Vertex spirals in fullerenes and their implications for nomenclature of fullerene derivatives. , 2007, Chemistry.

[246]  Stefan Grimme,et al.  Noncovalent Interactions between Graphene Sheets and in Multishell (Hyper)Fullerenes , 2007 .

[247]  Jack E. Graver.  Kekulé structures and the face independence number of a fullerene , 2007, Eur. J. Comb..

[248]  B. Yakobson,et al.  B80 fullerene: an Ab initio prediction of geometry, stability, and electronic structure. , 2007, Physical review letters.

[249]  Heping Zhang,et al.  An Upper Bound for the Clar Number of Fullerene Graphs , 2007 .

[250]  Yang Wang,et al.  Structural patterns in fullerenes showing adjacent pentagons: C20 to C72. , 2007, Journal of nanoscience and nanotechnology.

[251]  Ado Jorio,et al.  Carbon Nanotubes: Advanced Topics in the Synthesis, Structure, Properties and Applications , 2007 .

[252]  Michael J. Todd,et al.  On Khachiyan's algorithm for the computation of minimum-volume enclosing ellipsoids , 2007, Discret. Appl. Math..

[253]  E. Alper Yildirim,et al.  Two Algorithms for the Minimum Enclosing Ball Problem , 2008, SIAM J. Optim..

[254]  T. I. Borodina,et al.  Phase Transitions of C70 Fullerite with Hexagonal Closed‐packed Structure under Shock‐wave Loading , 2008 .

[255]  P. Kumar,et al.  Computing Minimum-Volume Enclosing Axis-Aligned Ellipsoids , 2008 .

[256]  Tapani A. Pakkanen,et al.  Icosahedral Au72: A Predicted Chiral and Spherically Aromatic Golden Fullerene , 2008 .

[257]  Pekka Pyykkö,et al.  Icosahedral Au(72): a predicted chiral and spherically aromatic golden fullerene. , 2008, Chemical communications.

[258]  Jean-Christophe Charlier,et al.  Electron and Phonon Properties of Graphene: Their Relationship with Carbon Nanotubes , 2008 .

[259]  G. Scuseria,et al.  C60 buckminsterfullerene high yields unraveled. , 2008, The journal of physical chemistry. A.

[260]  Michal Straka,et al.  Toward calculations of the 129Xe chemical shift in Xe@C60 at experimental conditions: relativity, correlation, and dynamics. , 2008, The journal of physical chemistry. A.

[261]  Tomislav Došlić.  Leapfrog fullerenes have many perfect matchings , 2008 .

[262]  Brendan D. McKay,et al.  A universal set of growth operations for fullerenes , 2008 .

[263]  Pablo A Denis,et al.  Mechanical properties of graphene nanoribbons , 2009, Journal of physics. Condensed matter : an Institute of Physics journal.

[264]  Peter Schwerdtfeger,et al.  A systematic search for minimum structures of small gold clusters Au(n) (n=2-20) and their electronic properties. , 2009, The Journal of chemical physics.

[265]  Jack E. Graver,et al.  Numbers of faces in disordered patches , 2009 .

[266]  Michel Deza,et al.  The symmetries of cubic polyhedral graphs with face size no larger than 6 , 2009 .

[267]  Jean-Sébastien Sereni,et al.  Fullerene graphs have exponentially many perfect matchings , 2009 .

[268]  Luis Echegoyen,et al.  Chemical, electrochemical, and structural properties of endohedral metallofullerenes. , 2009, Angewandte Chemie.

[269]  Ante Graovac,et al.  Topological Ranking of C28 Fullerenes Reactivity , 2009 .

[270]  Rosendo Valero,et al.  Consistent van der Waals radii for the whole main group. , 2009, The journal of physical chemistry. A.

[271]  Nan Shao,et al.  Search for lowest-energy nonclassical fullerenes III: C22. , 2009, The journal of physical chemistry. A.

[272]  Heping Zhang,et al.  Extremal fullerene graphs with the maximum Clar number , 2009, Discret. Appl. Math..

[273]  Ming-Hsuan Kang.  Toroidal fullerenes with the Cayley graph structures , 2009 .

[274]  Ali Iranmanesh,et al.  Computing Wiener Polynomial, Wiener Index and Hyper Wiener Index of C80 Fullerene by GAP Program , 2009 .

[275]  Ante Graovac,et al.  WIENER WAY TO DIMENSIONALITY , 2010 .

[276]  L. Gan,et al.  A global search for the lowest energy isomer of C(26). , 2010, The Journal of chemical physics.

[277]  Christina M. Graves,et al.  Fullerene patches I , 2010, Ars Math. Contemp..

[278]  Stefan Hildebrandt,et al.  Global Analysis of Minimal Surfaces , 2010 .

[279]  A. Rodríguez‐Fortea,et al.  The maximum pentagon separation rule provides a guideline for the structures of endohedral metallofullerenes. , 2010, Nature chemistry.

[280]  Marilyn M Olmstead,et al.  New tetraazaannulene hosts for fullerenes. , 2010, Inorganic chemistry.

[281]  Jan Cami,et al.  Detection of C60 and C70 in a Young Planetary Nebula , 2010, Science.

[282]  Dong Ye,et al.  A combination of Clar number and Kekulé count as an indicator of relative stability of fullerene isomers of C60 , 2010 .

[283]  Fengshan Bai,et al.  A Note on Fowler-Manolopoulos Predictor of Fullerene Stability , 2010 .

[284]  Takeshi Akasaka,et al.  Endohedral metal atoms in pristine and functionalized fullerene cages. , 2010, Accounts of chemical research.

[285]  Klaus Sattler,et al.  Handbook of Nanophysics : Clusters and Fullerenes , 2010 .

[286]  Yuan-Chia Fan,et al.  Systematics of Toroidal, Helically-Coiled Carbon Nanotubes, High-genus Fullernens, and Other Exotic Graphitic Materials , 2011 .

[287]  Valeri I. Kovalenko,et al.  Electronic Structure and Stability of C80 Fullerene IPR Isomers , 2011 .

[288]  Ante Graovac,et al.  The Mathematics and Topology of Fullerenes , 2011 .

[289]  A. Enyashin,et al.  Graphene allotropes , 2011 .

[290]  Alexander G. G. M. Tielens,et al.  The formation of buckminsterfullerene (C$_{60}$) in interstellar space , 2011 .

[291]  Pei Zhuang,et al.  Effect of Guanabenz on Rat AMD Models and Rabbit Choroidal Blood Flow , 2011, The open ophthalmology journal.

[292]  Y. Murata,et al.  A Single Molecule of Water Encapsulated in Fullerene C60 , 2011, Science.

[293]  Douglas J. Klein,et al.  Clarology for Conjugated Carbon NanoStructures : Molecules , Polymers , Graphene , Defected Graphene , Fractal Benzenoids , Fullerenes , NanoTubes , Nano-Cones , Nano-Tori , etc , 2011 .

[294]  Louis Esperet,et al.  Exponentially many perfect matchings in cubic graphs , 2011 .

[295]  Olivier Berné,et al.  Formation of buckminsterfullerene (C60) in interstellar space , 2011, Proceedings of the National Academy of Sciences.

[296]  Stephan Irle,et al.  Fullerenes: formation, stability, and reactivity , 2011 .

[297]  Ante Graovac,et al.  Graph Drawing with Eigenvectors , 2011 .

[298]  Jun Li,et al.  Carbon arc production of heptagon-containing fullerene[68] , 2011, Nature communications.

[299]  Jane Zundel.  MATCHING THEORY , 2011 .

[300]  Ottorino Ori,et al.  Topological Anisotropy of Stone-Wales Waves in Graphenic Fragments , 2011, International journal of molecular sciences.

[301]  Tatiana Korona,et al.  Small Molecules in C60 and C70: Which Complexes Could Be Stabilized? , 2011, Journal of chemical theory and computation.

[302]  Stephan Irle,et al.  Hot Giant Fullerenes Eject and Capture C2 Molecules: QM/MD Simulations with Constant Density , 2011 .

[303]  Eugenio Cinquanta,et al.  The Topological Background of Schwarzite Physics , 2011 .

[304]  Henryk A Witek,et al.  Relative isomer abundance of fullerenes and carbon nanotubes correlates with kinetic stability. , 2011, Physical review letters.

[305]  Ante Graovac,et al.  Topological efficiency of C66 fullerene , 2011 .

[306]  Mircea V. Diudea,et al.  C60 Structural Relatives – An Omega-Aided Topological Study , 2011 .

[307]  E. Sheka,et al.  Fullerenes: Nanochemistry, Nanomagnetism, Nanomedicine, Nanophotonics , 2011 .

[308]  Ante Graovac,et al.  Distance Property of Fullerenes , 2011 .

[309]  Ralf Tonner,et al.  Packed to the rafters: filling up C60 with rare gas atoms. , 2011, Chemphyschem : a European journal of chemical physics and physical chemistry.

[310]  Leo Gross,et al.  Bond-Order Discrimination by Atomic Force Microscopy , 2012, Science.

[311]  B. McKay,et al.  The smallest fullerene without a spiral , 2012 .

[312]  Yusuke Nakanishi,et al.  Closed network growth of fullerenes , 2012, Nature Communications.

[313]  J. Geelen.  ON HOW TO DRAW A GRAPH , 2012 .

[314]  Carlo Thilgen,et al.  A single water molecule trapped inside hydrophobic C60. , 2012, Angewandte Chemie.

[315]  H. Terrones,et al.  Beyond carbon nanopeapods. , 2012, Chemphyschem : a European journal of chemical physics and physical chemistry.

[316]  Clémence Corminboeuf,et al.  Is C60 buckminsterfullerene aromatic? , 2012, Physical chemistry chemical physics : PCCP.

[317]  Eiji Ōsawa,et al.  Formation Mechanism of C60 under Nonequilibrium and Irreversible Conditions — An Annotation , 2012 .

[318]  Brendan D. McKay,et al.  The Generation of Fullerenes , 2012, J. Chem. Inf. Model..

[319]  Hadrien Mélot,et al.  House of Graphs: A database of interesting graphs , 2012, Discret. Appl. Math..

[320]  Alessandro Bagno,et al.  Spectroscopic signatures of the carbon buckyonions C60@C180 and C60@C240: a dispersion-corrected DFT study. , 2013, Physical chemistry chemical physics : PCCP.

[321]  James Avery,et al.  Program Fullerene: A software package for constructing and analyzing structures of regular fullerenes , 2013, Journal of computational chemistry.

[322]  Takeshi Akasaka,et al.  Carbene additions to fullerenes. , 2013, Chemical reviews.

[323]  Ernesto Estrada,et al.  Chemical Graph Theory , 2013 .

[324]  Lothar Dunsch,et al.  Endohedral fullerenes. , 2013, Chemical reviews.

[325]  Daniel W Bearden,et al.  A missing link in the transformation from asymmetric to symmetric metallofullerene cages implies a top-down fullerene formation mechanism. , 2013, Nature chemistry.

[326]  Martin Jansen,et al.  Bottom-up C60 fullerene construction from a fluorinated C60H21F9 precursor by laser-induced tandem cyclization. , 2013, Chemistry.

[327]  Klaus Schulten,et al.  Mature HIV-1 capsid structure by cryo-electron microscopy and all-atom molecular dynamics , 2013, Nature.

[328]  Elizabeth J. Hartung.  Fullerenes with complete Clar structure , 2013, Discret. Appl. Math..

[329]  Bun Chan,et al.  Evaluation of the heats of formation of corannulene and C60 by means of high-level theoretical procedures. , 2013, The journal of physical chemistry. A.

[330]  Kim K Baldridge,et al.  Of graphs and graphenes: molecular design and chemical studies of aromatic compounds. , 2013, Angewandte Chemie.

[331]  Karl Weierstrass,et al.  Untersuchungen über die Flächen, deren mittlere Krümmung überall gleich Null ist , 2013 .

[332]  Nicolas Van Cleemput,et al.  The history of the generation of cubic graphs , 2013 .

[333]  Tomislav Došlić.  The Smallest Eigenvalue of Fullerene Graphs - Closing the Gap , 2013 .

[334]  Patrick Norman,et al.  Non-additivity of polarizabilities and van der Waals C6 coefficients of fullerenes. , 2013, The Journal of chemical physics.

[335]  Jack E. Graver,et al.  Clar and Fries numbers for benzenoids , 2013, Journal of Mathematical Chemistry.

[336]  Dage Sundholm,et al.  C72: gaudiene, a hollow and aromatic all-carbon molecule. , 2013, Physical chemistry chemical physics : PCCP.

[337]  Roberto Dovesi,et al.  Structural, electronic and energetic properties of giant icosahedral fullerenes up to C6000: insights from an ab initio hybrid DFT study. , 2014, Physical chemistry chemical physics : PCCP.

[338]  James Avery,et al.  Structure and Properties of the Nonface-Spiral Fullerenes T-C380, D3-C384, D3-C440, and D3-C672 and Their Halma and Leapfrog Transforms , 2014, J. Chem. Inf. Model..

[339]  M V Diudea,et al.  Carbon multi-shell cages. , 2014, Physical chemistry chemical physics : PCCP.

[340]  Milan Randić,et al.  Novel insight into Clar’s aromatic π-sextets , 2014 .

[341]  Jun Li,et al.  Observation of an all-boron fullerene. , 2014, Nature chemistry.

[342]  Jack E. Graver,et al.  Fullerene patches II , 2014, Ars Math. Contemp..

[343]  Stan Schein,et al.  Fourth class of convex equilateral polyhedron with polyhedral symmetry related to fullerenes and viruses , 2014, Proceedings of the National Academy of Sciences.

[344]  Dusanka Janezic,et al.  Graph-Theoretical Matrices in Chemistry , 2015 .

[345]  Nenad Raos.  Ante Graovac – Life and Works , 2016 .

[346]  Melanie Grunwald.  Platonic And Archimedean Solids , 2016 .

[347]  K.,et al.  Applications of Combinatorics and Graph Theory to Spectroscopy and Quantum Chemistry , 2022 .