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Research Article

Fullerenes as Tilings of Surfaces

M. Deza,*^† P. W. Fowler,*^‡ A. Rassat,*^§ and K. M. Rogers*^‡

Laboratoire d'Informatique, Ecole Normale Suprieure, 45 Rue d'Ulm, 75230 Paris, France, School of Chemistry, University of Exeter, Stocker Road, Exeter EX4 4QD, U.K., and Department of Chemistry, Ecole Normale Suprieure, Rue Lhomond, 75005 Paris, France

J. Chem. Inf. Comput. Sci., 2000, 40 (3), pp 550–558

DOI: 10.1021/ci990066h

Publication Date (Web): March 18, 2000

To whom correspondence should be addressed. E-mail: deza@dmi.ens.fr; p.w.fowler@ex.ac.uk; rassat@chimene.ens.fr; k.m.rogers@ex.ac.uk.

†

Laboratoire d'Informatique, Ecole Normale Supérieure.

‡

University of Exeter.

Department of Chemistry, Ecole Normale Supérieure.

Abstract

If a fullerene is defined as a finite trivalent graph made up solely of pentagons and hexagons, embedding in only four surfaces is possible: the sphere, torus, Klein bottle, and projective (elliptic) plane. The usual spherical fullerenes have 12 pentagons; elliptic fullerenes, 6; and toroidal and Klein-bottle fullerenes, none. Klein-bottle and elliptic fullerenes are the antipodal quotients of centrosymmetric toroidal and spherical fullerenes, respectively. Extensions to infinite systems (plane fullerenes, cylindrical fullerenes, and space fullerenes) are indicated. Eigenvalue spectra of all four classes of finite fullerenes, are reviewed. Leapfrog fullerenes have equal numbers of positive and negative eigenvalues, with 0, 0, 2, or 4 eigenvalues zero for spherical, elliptic, Klein-bottle, and toroidal cases, respectively.

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