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

On Mathematical Properties of Buckminsterfullerenef

Some of the mathematical properties of buckminsterfullerene are considered, that is, geometrical, topological, group-theoretical and graph-theoretical properties. These mathematical properties are used to predict several structural and chemical properties of buckminsterfullerene

Unusual Permutation Groups in Negative Curvature Carbon and Boron Nitride Structures

The concept of symmetry point groups for regular polyhedra can be generalized to special permutation groups to describe negative curvature polygonal networks that can be expanded to possible carbon and boron nitride structures through leapfrog transformations, which triple the number of vertices. Thus a D surface with 24 hep-tagons and 56 hexagons in the unit cell can be generated by a leapfrog transformation from the Klein figure consisting only of the 24 heptagons. The permutational symmetry of the Klein figure can be described by the simple PSL(2,7) (or heptakisoctahedral) group of order 168 with the conjugacy class structure E + 24C7 + 24C73 + 56C3 + 21C2 + 42C4. Analogous methods can be used to generate a D surface with 12 octagons and 32 hexagons by a leapfrog transformation from the Dyck figure consisting only of the 12 octagons. The permutational symmetry of the Dyck figure can be described by a group of order 96 and the conjugacy class structure E + 24S8 + 6C4 + 3C42 + 32C3 + 12C2 + 18S4. This group is not a simple group since it has a normal subgroup chain leading to the trivial group C1 through subgroups of order 48 and 16 not related to the octahedral or tetrahedral groups

Euler Characteristic of Polyhedral Graphs

Euler characteristic is a topological invariant, a number that describes the shape or structure of a topological space, irrespective of the way it is bent. Many operations on topological spaces may be expressed by means of Euler characteristic. Counting polyhedral graph figures is directly related to Euler characteristic. This paper illustrates the Euler characteristic involvement in figure counting of polyhedral graphs designed by operations on maps. This number is also calculated in truncated cubic network and hypercube. Spongy hypercubes are built up by embedding the hypercube in polyhedral graphs, of which figures are calculated combinatorially by a formula that accounts for their spongy character. Euler formula can be useful in chemistry and crystallography to check the consistency of an assumed structure. This work is licensed under a Creative Commons Attribution 4.0 International License

Unusual Permutation Groups in Negative Curvature Carbon and Boron Nitride Structures

The concept of symmetry point groups for regular polyhedra can be generalized to special permutation groups to describe negative curvature polygonal networks that can be expanded to possible carbon and boron nitride structures through leapfrog transformations, which triple the number of vertices. Thus a D surface with 24 hep-tagons and 56 hexagons in the unit cell can be generated by a leapfrog transformation from the Klein figure consisting only of the 24 heptagons. The permutational symmetry of the Klein figure can be described by the simple PSL(2,7) (or heptakisoctahedral) group of order 168 with the conjugacy class structure E + 24C7 + 24C73 + 56C3 + 21C2 + 42C4. Analogous methods can be used to generate a D surface with 12 octagons and 32 hexagons by a leapfrog transformation from the Dyck figure consisting only of the 12 octagons. The permutational symmetry of the Dyck figure can be described by a group of order 96 and the conjugacy class structure E + 24S8 + 6C4 + 3C42 + 32C3 + 12C2 + 18S4. This group is not a simple group since it has a normal subgroup chain leading to the trivial group C1 through subgroups of order 48 and 16 not related to the octahedral or tetrahedral groups

A 3D Unstructured Mesh FDTD Scheme for EM Modelling

The Yee finite difference time domain (FDTD) algorithm is widely used in computational electromagnetics because of its simplicity, low computational costs and divergence free nature. The standard method uses a pair of staggered orthogonal cartesian meshes. However, accuracy losses result when it is used for modelling electromagnetic interactions with objects of arbitrary shape, because of the staircased representation of curved interfaces. For the solution of such problems, we generalise the approach and adopt an unstructured mesh FDTD method. This co-volume method is based upon the use of a Delaunay primal mesh and its high quality Voronoi dual. Computational efficiency is improved by employing a hybrid primal mesh, consisting of tetrahedral elements in the vicinity of curved interfaces and hexahedral elements elsewhere. Difficulties associated with ensuring the necessary quality of the generated meshes will be discussed. The power of the proposed solution approach is demonstrated by considering a range of scattering and/or transmission problems involving perfect electric conductors and isotropic lossy, anisotropic lossy and isotropic frequency dependent chiral materials

Discrete Element Modeling Of Railroad Ballast Under Simulated Train Loading

Ballasted tracks have been widely used in many countries around the world. Ballast layer is the main element in ballasted track. After service, ballast aggregates degrade and deform. Periodical maintenance for ballast layer is required; which is a cost and time expensive activity. Researchers used numerical approaches to understand the behavior of railroad ballast that leads to efficient design and maintenance. The Discrete Element Method (DEM) has been used increasingly to understand the mechanical behavior of railroad ballast, more frequently through box test. Most researches in the literature simulate the train loading as a pure continuous sinusoid based on train speed and axle spacing; unlike the actual loading induced by trains. This study aims to show the influence of simulated train loadings on ballast mechanical behavior using DEM via box test. The study utilizes the theory of Beam on Elastic Foundation to simulate a more realistic train load. The results from the more realistic simulated train load are compared with those from a sinusoidal load. The compared results highlight the influence of the simulated train load on the mechanical behavior of railroad ballast.Author: " I would like to acknowledge Qatar Rail for their sponsorship to this research under a project entitled “Framework for Research on Railway Engineering” with a grant reference number: QUEX-CENG-Rail 17/18.