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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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Under the assumptions that no two sp 3 carbon atoms are adjacent in the end product of bromination of a fullerene and that the residual π system is a closed shell, graph theory predicts maximum stoichiometries C 60 Br 24 , C 70 Br 26 , C 76 Br 28 , C 84 Br 32 and rules out all but 58 of the ~10 23 addition patterns conceivable for these molecules.

An exhaustive search of C 36 H 6 and symmetrical C 36 F 6 isomers based on the cylindrical C 36 fullerene predicts the same D 3h structure to have the lowest energy in each series. Some competitive structures of C 36 H 6 have only trivial symmetry but share the feature of 1,4 pairing of addends with the best isomer. Isostructural C 36 X 6 (X = H, F) molecules are predicted to have constant energy difference. All 84 000 geometry optimisations reported here used the DFTB model, with checking of the new parameters for C/F and F/F interactions against LDA and GGA full density-functional methods.

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