Study of Fullerenes by Some New Topological Index
A molecular graph is a graph such that its vertices correspond to the atoms and the edges to the bonds of a given molecule. Fullerenes are molecules in the form of polyhedral closed cages made up entirely of n three-coordinate carbon atoms and having 12 p
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Study of Fullerenes by Some New Topological Index Ali Reza Ashrafi, Mohammad Ali Iranmanesh, and Zahra Yarahmadi
Abstract A molecular graph is a graph such that its vertices correspond to the atoms and the edges to the bonds of a given molecule. Fullerenes are molecules in the form of polyhedral closed cages made up entirely of n three-coordinate carbon atoms and having 12 pentagonal and (n/2–10) hexagonal faces, where n is equal or greater than 20. The molecular graph of a fullerene is called fullerene graph. In this chapter, the fullerene graphs under two new distance-based topological indices are investigated. Some open questions are also presented.
14.1 Introduction Graph theory successfully provided the chemist with a variety of tools as molecular graph and topological index. Molecular graphs represent the constitution of molecules. They are generated using the following rule: vertices stand for atoms and edges for bonds. It is clear that the degree of each vertex in a molecular graph is
A.R. Ashrafi () Department of Mathematics, Faculty of Mathematics, Statistics and Computer Science, University of Kashan, Kashan 87317-51167, Islamic Republic of Iran e-mail: [email protected] M.A. Iranmanesh Department of Mathematics, Yazd University, Yazd 89175-741, Islamic Republic of Iran e-mail: [email protected] Z. Yarahmadi Department of Mathematics, Faculty of Science, Khorramabad Branch, Islamic Azad University, Khorramabad, Islamic Republic of Iran A.R. Ashrafi et al. (eds.), Topological Modelling of Nanostructures and Extended Systems, Carbon Materials: Chemistry and Physics 7, DOI 10.1007/978-94-007-6413-2 14, © Springer ScienceCBusiness Media Dordrecht 2013
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at most four. A topological index is a numeric quantity for the molecular graph of a molecule. This number must be invariant under topological symmetry of molecules under consideration. A fullerene graph is the molecular graph of a fullerene molecule. Fullerenes are molecules in the form of polyhedral closed cages made up entirely of n carbon atoms that are bonded in a nearly spherically symmetric configuration. The well-known fullerene, the C60 molecule, is a closed-cage carbon molecule with carbon atoms tiling the spherical or nearly spherical surface with a truncated icosahedral structure formed by 20 hexagonal and 12 pentagonal rings (Kroto et al. 1985, 1993). It is well known that C20 is the unique fullerene constructed fully from pentagons, and by Euler’s theorem, there is no fullerene without pentagons. Suppose p, h, n, and m are the numbers of pentagons, hexagons, carbon atoms, and bonds between them, in a given fullerene F. Since each atom lies in exactly three faces and each edge lies in two faces, the number of atoms is n D (5p C 6 h)/3, the number of edges is m D 3/2n D (5p C 6 h)/2, and the number of faces is f D p C h. By Euler’s formula n m C f D 2, one can deduce that (5p C 6 h)/3 (5p C 6 h)/2 C p C h D 2, and therefore, p D 12, v D 2 h C 20, and e D 3 h C 30. This implies that such molecules m
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