The topology of fullerenes

Fullerenes are carbon molecules that form polyhedral cages. Their bond structures are exactly the planar cubic graphs that have only pentagon and hexagon faces. Strikingly, a number of chemical properties of a fullerene can be derived from its graph structure. A rich mathematics of cubic planar graphs and fullerene graphs has grown since they were studied by Goldberg, Coxeter, and others in the early 20th century, and many mathematical properties of fullerenes have found simple and beautiful solutions. Yet many interesting chemical and mathematical problems in the field remain open. In this paper, we present a general overview of recent topological and graph theoretical developments in fullerene research over the past two decades, describing both solved and open problems. WIREs Comput Mol Sci 2015, 5:96–145. doi: 10.1002/wcms.1207 Conflict of interest: The authors have declared no conflicts of interest for this article. For further resources related to this article, please visit the WIREs website

Leapfrog and Related Operations on Toroidal Fullerenes

A 4-valent square tiled toroid is transformed into 3-valent hexagonal (and other polygonal) lattices either by simple cutting procedures or by some more elaborated operations such as leapfrog and related transformations. Tiling or embedding isomers can be interchanged by means of such operations on toroidal maps, for which rigorous definitions and some theorems are given. Parents and products of most important operations are illustrated

ENERGY AND LAPLACIAN SPECTRUM OF C 4 C 8 (S) NANOTORI AND NANOTUBE

The spectrum of a finite graph is by definition the spectrum of the adjacency matrix, that is, its set of eigenvalues together with their multiplicities. The sum of the absolutes of these eigenvalues is the energy of graph. The Laplace spectrum of a finite undirected graph without loops is the spectrum of the Laplace matrix. There are some topological indices related the Laplacian spectrum. In this paper, using a mathematical model for C 4 C 8 (S) that introduced in Ref. [26], we write a MATHEMATICA program to compute the energy and Laplacian spectrum of molecular graph of arbitrary C 4 C 8 (S) nanotori and nanotube

Advanced Applications in Nanophotonics

Nanophotonics is a fast-growing area of both scientific significance and practical value for applications. Nanophotonics studies the interaction between light and electronic systems in nanomaterials and nanostructures as well as the behavior of light in nanometer scales. It covers many hot topics such as plasmonics, two-dimensional materials, and silicon photonics. Increasing attention is given to the area and nanophotonics is expected to have significant impact on future technology advances. This thesis work focuses on three aspects of nanophotonics. The first aspect is in exploring the nonlocal effect and surface correction for nanometer-length-scale plasmonic structures. Plasmonics is the study of the interaction between electromagnetic fields and free electrons in a metal. It exploits the unique optical properties of metallic nanostructures to enable routing and manipulation of light at the nanoscale, where nonlocal effect becomes important. Here we introduce a new surface hydrodynamic model for plasmon propagation at interfaces, which incorporates both nonlocality and surface contributions. This surface correction is calculated via a discontinuity in the normal component of the electric displacement in conjunction with Feibelman's d-parameters, thus enabling rapid numerical calculation of nanostructures without requiring a full quantum calculation because of its large computational requirement. We examine numerical calculations of surface plasmon polaritons propagation at a single interface structure, and then for a more complex thin-film structures. The second aspect is investigating the third-harmonic generation in thick multilayer graphene. Graphene is the first two-dimensional material to be discovered and has attracted much interest because of its remarkable two-dimensional electronic, optical, mechanical, and thermal properties. Multilayer graphene, can be seen as stacking of monolayer graphene, and it offers an array of properties that are of interest for optical physics and devices. We describe the layer-dependent for third-harmonic generation in thick multilayer graphene on quartz substrate. The third harmonic signal of multilayer graphene exhibits a complex dependence on its layer number showing that the optimal third harmonic signal at 24 layers, in good agreement with two theoretical models. The third aspect is an exploration in silicon photonics of design and demonstration of a differential phase shift keying demodulator based on coherent perfect absorption effect. Silicon photonics is considered a potential future communication system mainly due to its compact footprint, dense integration, and compatibility with mature silicon integrated circuit manufacturing. Differential phase shift keying based system offers advantages, e.g., dispersion tolerance, improved sensitivity, and does not require coherent detection. Coherent perfect absorption uses a ring resonator works for the critical coupling condition at resonance frequency. This work shows a new compact demodulator circuit can be integrated in all optical-system