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Fullerenes as tilings of surfaces
1Laboratoire d'Informatique, Ecole Normale Superieure, Paris, France. deza@dmi.ens.fr
Journal of Chemical Information and Computer Sciences
|June 13, 2000
Summary
Fullerenes, defined as trivalent graphs of pentagons and hexagons, can embed on four surfaces: sphere, torus, Klein bottle, and projective plane. Their eigenvalue spectra reveal distinct properties across these fullerene types.
Area of Science:
- Graph theory
- Computational chemistry
- Materials science
Background:
- Fullerenes are finite trivalent graphs composed of pentagons and hexagons.
- Their topological embedding is restricted to four specific surfaces: sphere, torus, Klein bottle, and projective plane.
Purpose of the Study:
- To review the eigenvalue spectra of fullerenes embedded on these four surfaces.
- To explore the relationship between fullerene structure and topological properties.
Main Methods:
- Topological analysis of fullerene structures.
- Review of eigenvalue spectra for different fullerene classes.
- Examination of antipodal quotients for Klein bottle and elliptic fullerenes.
Main Results:
- Spherical fullerenes contain 12 pentagons; elliptic, 6; toroidal and Klein-bottle, none.
- Klein-bottle and elliptic fullerenes are antipodal quotients of specific toroidal and spherical fullerenes.
- Leapfrog fullerenes exhibit specific numbers of zero eigenvalues (0, 0, 2, 4) for spherical, elliptic, Klein-bottle, and toroidal types, respectively.
Conclusions:
- The four distinct surfaces dictate the possible configurations and properties of finite fullerenes.
- Eigenvalue spectra provide a characteristic signature for each class of fullerene.
- The study indicates potential for extensions to infinite fullerene systems.
