×

Toroidal fullerenes with the Cayley graph structures. (English) Zbl 1238.05068

Summary: We classify all possible structures of fullerene Cayley graphs. We give each one a geometric model and compute the spectra of its finite quotients. Moreover, we give a quick and simple estimation for a given toroidal fullerene. Finally, we provide a realization of those families in three-dimensional space.

MSC:

05C10 Planar graphs; geometric and topological aspects of graph theory
05C75 Structural characterization of families of graphs
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C90 Applications of graph theory
92E10 Molecular structure (graph-theoretic methods, methods of differential topology, etc.)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Altshuler, A., Construction and enumeration of regular maps on the torus, Discrete Math., 4, 201-217 (1973) · Zbl 0253.05117
[2] Avron, J. E.; Berger, J., Tiling rules for toroidal molecules, Phys. Rev. A, 51, 2, 1146-1149 (1955)
[3] Berger, J.; Avron, J. E., Classification scheme for toroidal molecules, J. Chem. Soc. Faraday Trans., 91, 22, 4037-4045 (1955)
[4] Bovin, S. A.; Chibotaru, L. F.; Ceulemans, A., The quantum structure of carbon tori, J. Molecular Catalysis A: Chem., 166, 47-52 (2001)
[5] Ceulemans, A.; Chibotaru, L. F.; Bovin, S. A.; Fowler, P. W., The electronic structure of polyhex carbon tori, J. Chem. Phys., 11, 9, 4271-4278 (2000)
[6] Chuang, C.; Jin, B.-Y., Systematics of High-Genus Fullerenes, J. Chem. Inf. Model., 49, 7, 1664-1668 (2009)
[7] F.R.K. Chung, B. Kostant, S. Sternberg, Groups and the Buckyball, Lie Theory and Geometry: In honor of Bertram Kostant, 1994, pp. 97-126.; F.R.K. Chung, B. Kostant, S. Sternberg, Groups and the Buckyball, Lie Theory and Geometry: In honor of Bertram Kostant, 1994, pp. 97-126. · Zbl 0844.20031
[8] Coxeter, H. S.M.; Moser, W. O.J, Generators and Relations for Discrete Groups (1980), Springer-Verlag: Springer-Verlag New York · Zbl 0422.20001
[9] DeVos, M.; Goddyn, L.; Mohar, B.; Samal, R., Cayley sum graphs and eigenvalues of (3, 6)-fullerenes, J. Comb. Theory Series B, 99, 2, 358-369 (2009) · Zbl 1217.05140
[10] Diudea, M. V.; John, P. E.; Graovac, A.; Primorac, M.; Pisanskie, T., Leapfrog and related operations on toroidal fullerenes, Croatica Chemica Acta, 76, 2, 153-159 (2003)
[11] N. Elkies, Shimura curve computations, in: Algorithmic Number Theory, in: Lecture Notes in Computer Science, 1998, pp. 1-47.; N. Elkies, Shimura curve computations, in: Algorithmic Number Theory, in: Lecture Notes in Computer Science, 1998, pp. 1-47. · Zbl 1010.11030
[12] Fowler, W.; Hansen, P.; Caporossi, G.; Soncini, A., Poylynes with maximal HOMO-LUMO gap, Chem. Phys. Lett., 342, 105-112 (2001)
[13] Fowler, P. W.; John, P. E.; Sachs, H., (3, 6)-cages, hexagonal toroidal cages, and their spectra, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 51, 139-174 (2000) · Zbl 0972.05032
[14] Fowler, W.; Steer, I., The leapfrog principle: a rule for electron counts of carbon clusters, J. Chem. Soc., Chem. Commun., 52, 1403-1405 (1987)
[15] John, P. E.; Sach, H., Spectra of toroidal graphs, Discrete Math., 309, 9, 2663-2681 (2009) · Zbl 1204.05055
[16] Kirby, E. C.; Mallion, R. B.; Pollak, P., Toroidal polyhexes, J. Chem. Soc. Faraday Trans., 89, 12, 1945-1953 (1993)
[17] Manolopoulus, E.; Woodall, R.; Fowler, W., Electronic stability of fullerenes: eigenvalue theorems for leapfrog carbon clusters, J. Chem. Soc. Faraday Trans., 88, 17, 2427-2435 (1992)
[18] Marušič, D.; Pisanski, T., Symmetries of hexagonal molecular graphs on the torus, Croatica Chemica Acta, 73, 969-981 (2000)
[19] Negami, Seiya, Uniqueness and faithfulness of embedding of toroidal graphs, Discrete Math., 44, 2, 161-180 (1983) · Zbl 0508.05033
[20] Nguyen, P.; Stehlé, D., Low-dimensional lattice basis reduction revisited, Trans. Algorithms, 5, 4 (2009) · Zbl 1300.11133
[21] Serre, J.-P., Linear Representations of Finite Groups (1997), Springer-Verlag
[22] Smith, W. B., Introduction to Theoretical Organic Chemistry and Molecular Modelling (1996), Wiley-VCH
[23] Tamura, R.; Ikuta, M.; Hirahara, T.; Tsukada, M., Positive magnetic susceptibility in polygonal nanotube tori, Phys. Rev. B, 74, 4, 45418-45424 (2005)
[24] Thomassen, C., Tilings of the torus and the klein bottle and vertex-transitive graphs on a fixed surface, Trans. Am. Math. Soc., 323, 605-635 (1993) · Zbl 0722.05031
[25] Zhang, F.; An, C., Acyclic molecules with greatest HOMO-LUMO separation, Discrete Appl. Math., 98, 165-171 (1999) · Zbl 0937.05074
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.