Ante Graovac

The mathematics and topology of fullerenes

Harry KROTO: Foreword.- 1. Ali Reza ASHRAFI/Mircea DIUDEA/Ante GRAOVAC: Omega Polynomials of Fullerenes and Nanotubes.- 2. Ali Reza ASHRAFI: Wiener Index of Nanotubes, Toroidal Fullerenes and Nanostars.- 3. Mircea DIUDEA/A. E. Vizitiu: C60 Structural Relatives. An Omega-aided Topological Study.- 4. Istvan LASZLO/Reti TAMAS/Ante GRAOVAC: Local Combinatorial Characterization of Fullerenes.- 5. Ali IRANMANESH: Computation of some Topological Indices of C60 and C80 Fullerenes by GAP Program.- 6. Mathieu Dutour SIKIRIC/Michel DEZA: 4-Regular and Self-Dual Analogs of Fullerenes.- 7. Helena DODZIUK: Endohedral fullerene complexes and in- out isomerism in perhydrogenated fullerenes. Why the carbon cages cannot be used as the hydrogen containers?.- 8. Damir VUKICEVIC/Milan RANDIC: Detailed Atlas of Kekule Structures of the Buckminsterfullerene.- 9. Ernesto ESTRADA: A Graph Theoretic Approach to Atomic Displacements in Fullerenes.- 10.E.C. KIRBY/R.B. MALLION/P. POLLAK: Counting Spanning Trees in Toroidal Fullerenes.- 11. O.ORI, F.CATALDO, A.GRAOVAC: Topological Determination of 13C-NMR Spectra of C66 Fullerenes.- 12. Giorgio BENEDEK: The Topological Background of Schwarzite Physics.- 13. Haruo HOSOYA: High pi-Electronic Stability of Soccer Ball Fullerene C60 and Truncated Octahedron C24 among Spherically Polyhedral Networks.- 14. Patrick W. FOWLER/A.GRAOVAC: The Estrada Index and Fullerene Isomerism.

Some inequalities for the atom-bond connectivity index of graph operations

The atom-bond connectivity index is a useful topological index in studying the stability of alkanes and the strain energy of cycloalkanes. In this paper some inequalities for the atom-bond connectivity index of a series of graph operations are presented. We also prove our bounds are tight. As an application, the ABC indices of C4 nanotubes and nanotori are computed.

Nanostructures and Eigenvectors of Matrices

Very often the basic information about a nanostructure is a topological one. Based on this topological information, we have to determine the Descartes coordinates of the atoms. For fullerenes, nanotubes, and nanotori, the topological coordinate method supplies the necessary information. With the help of the bi-lobal eigenvectors of the Laplacian matrix, the position of the atoms can be generated easily. This method fails, however, for nanotube junctions and coils and other nanostructures. We have found recently a matrix W which could generate the Descartes coordinates not only of fullerenes, nanotubes, and nanotori but also of nanotube junctions and coils. Solving namely the eigenvalue problem of this matrix W, its eigenvectors with zero eigenvalue give the Descartes coordinates. There are nanostructures, however, whose W matrices have more eigenvectors with zero eigenvalues than it is needed for determining the positions of the atoms in 3D space. In such cases the geometry of nanostructure can be obtained with the help of a projection from a higher-dimensional space in a similar way as the quasicrystals are obtained.

Topological Invariants of Möbius-Like Graphenic Nanostructures

Topological invariants are computed for some carbon zigzag nanoribbons in the limit of infinite carbon atoms N by applying standard and Mobius-like periodicity. Topological modeling considerations allow then to assign to the half-twisted molecules a certain grade of chemical stability based on the actions of two basic topological properties, compactness and efficiency, which represent the influence that long-range connectivity has on lattice stability. Conclusions about Mobius-nanoribbon topological dimensionality are also presented.

Introducing “Colored” Molecular Topology by Reactivity Indices of Electronegativity and Chemical Hardness

Within the context of conceptual density functional theory chemical reactivity definitions of electronegativity (EN) and chemical hardness (HD), nine forms of their finite difference are expressed in order to consider the global “coloring” of the molecular topology with respect to their symmetry centers (atomic centers or bonding centers), according to the so-called Timisoara–Parma rule. The resulting parabolic-reactive energy in terms of EN and HD is compared with the bond topological Wiener index for short list of PAH (poly-aromatic hydrocarbons) selected as paradigmatic structures for validating the new reactivity descriptors based on topological quantities.

Drawing Diamond Structures with Eigenvectors

Very often the basic information about a nanostructure is a topological one. Based on this topological information, we have to determine the Descartes coordinates of the atoms. For fullerenes, nanotubes and nanotori, the topological coordinate method supplies the necessary information. With the help of the bi-lobal eigenvectors of the Laplacian matrix, the position of the atoms can be generated easily. This method fails, however, for nanotube junctions and coils and other nanostructures. We have found recently a matrix W which could generate the Descartes coordinates for fullerenes, nanotubes and nanotori and also for nanotube junctions and coils as well. Solving, namely, the eigenvalue problem of this matrix W, its eigenvectors with zero eigenvalue give the Descartes coordinates. There are nanostructures however, whose W matrices have more eigenvectors with zero eigenvalues than it is needed for determining the positions of the atoms in 3D space. In this chapter, we have studied this problem in the case of diamond structures. We have found that this extra degeneracy is due to the fact that the first and second neighbour interactions do not determine the geometry of the structure. It was found that including the third neighbour interaction as well, diamond structures were described properly.