# Free Topology Of Fullerenes Dissertation Example

TOPOLOGY OF FULLERENES

Introduction

Fullerenes are defined as carbon molecules that form polyhedral cages. There exists a close connection between structural chemistry and the graph theory in pursuing the topology of fullerene. The proposal report seeks to expound on the importance and the objective of the study of the topology of fullerene. The report consists of a brief preview of the previous studies of the study topic. The scope that the study will be able to achieve with the time constraints is also stipulated in the proposal report.

The most considered direct graphical description is a sequence of numbers, as a connection table, sometimes topological index (single derived number), a polynomial or a matrix. In the graph theory, the finite series such as the series of numbers of independent k edge sets or the distance degree sequence or are described by polynomials counting (Cataldo et al., 2011)

PG,x=kpG,k .xk

In this case: pG,x is the occurence frequency of the property G partions k-is the length x is the k holding parameterFullerenes are defined as polyhedral cages forming carbon molecules. Their structure of bond exhibit cubic planar graphs that contain only hexagon and pentagon faces. This dissertation will focus on the planar graphs mathematics that has been used to present the fullerene graphs as studied by Goldberg and Coxeter (Cataldo et al., 2011). In this paper, there will be a wide description of the previous topological and graph theories which have been used in the research of fullerenes.

The study will focus and restrict to isolated pentagon fullerenes which produce better stability. The focus allows the production of fullerene graphs that can sort all the information we need to have a conclusion that can efficiently compute topological indicators. The previous studies have shown fullerenes has a fine property such that the graphs that result are both planar, cubic and connected (thrice) in which all have ether pentagon or hexagon faces. This makes their mathematical description elegant and straightforward such that it’s possible to have their topological properties derive properties, surfaces, spatial shapes and their chemical properties derived from their graphs directly (Putz, 2015).

Objectives

The primary aim of this study is to explore and fill the gap that exists between the graph theory in mathematics and the topology of fullerenes in structural chemistry. The aim is to achieve new principles in topological and graph theoretical treatment of fullerenes.

Secondary objectives

Preview various studies in the topology of fullerenes.

Extensive study of graph theory with reference to topology of fullerenes

Provide a conclusive recommendation of future work to be undertaken as pertains to this study.

Critic previous studies and provide conclusive results from the study.

Statement of scope

The topic that is to be explored is very wide due to the variety of properties of fullerenes. The time constraints will limit the study to a few critical area of the topic. Closer concern shall be placed on the concepts of the graph theory and how they relate to the topology of fullerene. The various geometries that can be used to establish the physical properties shall be evaluated. The study will also preview various research papers on the topic.

Literature review

Background study

Extensive studies over the past 20 years have been carried out in the topological field and theoretical molecular descriptions such as of fullerenes. This sufficiently characterizes a significant sub-discipline within the analytical chemistry. This report will further interrogate the topic. Further investigation has shown that the use of graph theory has not been a comprehensively reviewed (Putz).

Their bond structures of fullerenes exhibit planar cubic graphs that contain only hexagon and pentagon faces. This dissertation will focus on the planar graphs mathematics that has been used to present the fullerene graphs as studied by Goldberg and Coxeter. In this paper, there will be a wide description of the previous topological and graph theories which have been used in the research of fullerenes (Cataldo et al., 2011).

Fullerene graphs using planar embed-mentation

Embedding of planar can be represented on paper to allow for visualization of structures of planar graphs. In the study of the topology of fullerene, the choice of embedding is very critical since not all choices are equally informative. Studies have relied on simple and uncluttered drawings that can allow the exposition of the graphs symmetry and structure. Small graphs can be hand sketched while the large graphs are drawn using computer algorithm and a drawing program (Avis et al., 2006.).

Geometry of fullerenes

The Gaussian curvature is one of the most important quantities for geometry understanding and shape description of fullerenes. Gaussian curvature can be defined by two principal curvatures. The principal exacts that for each surface point is either the minimal or maximal curvatures through that point in any direction. The gaussian curvature at a point can further be shown to be the difference between 2π and the angle required to make a circle in the surrounding of the point (Avis et al., 2006) . Depending on the surface a zero negative and positive Gaussian curvature can be illustrated. In the case where the Gaussian curvature in a point is zero the bending of the surface around that point is in one direction. A surface of this nature, which has a Gaussian curvature of zero at every point is observed as flat. This means that it can be unwound without tearing onto a plane. A surface with Gaussian curvature that is positive can be opened by cutting and unwrapped onto a plane. Further a surface with a curvature around a point that is negative is considered a saddle point, producing a surface that is pringle-like and wobbly (Cataldo et al., 2011). These curvatures that are negative can’t be unwrapped.

The topology of the surface is a direct interdependent of the Gaussian curvature. The Gaussian curvature is isometrically independent of how it is embedded in space. In the analysis of the surface properties of fullerenes, the 3D embedding is not required. This is because the embedding can evaluate the ways possible in which the surface can be isometrically embedded into space and its subsequent 3-D dimensional shape.

Topological indicators

The topological indicator can be defined as a map τl from the graph G into a finite sequence of the values. The fullerene graph G is plotted by connecting the graph for the theoretical properties with the physical properties of fullerene (Ashrafi et al., 2013). The topological index with chemical property relation is called a chemical index. In studies crude chemical bonding models have been placed in the same category as topological indicators (Ashrafi et al., 2013).

Methodology

The study will rely on the preview of the past literature on this topic to offer a conclusive solution to the summarised problems. A critical analysis of the data collected from the previewed research papers will be carried out. However, the group fullerenes elements are too wide, and thus a statistical method will be employed to choose random elements to be put under analysis. Possible interviews and recommendation from previous researchers will also be applied to broaden the study.

Conclusion

The main objectives as stipulated in this proposal will achieve the review of several milestones in the graph theory of fullerenes and its topology. In the preliminary study, a few important concepts have been outlined for the initial roadmap of the study. The study will show the close connection that can exist between mathematics and structural chemistry.

References

Ashrafi, A.R., Cataldo, F., Iranmanesh, A. and Ori, O. eds., 2013. Topological modelling of

nanostructures and extended systems (Vol. 7). Verlag, Berlin: Springer.

Avis D. et al., 2006. Graph Theory, And Combinatorial Optimization. Springer-Verlag

New York Inc.

Cataldo, F., Graovac, A. and Ori, O. eds., 2011. The mathematics and topology of

fullerenes (Vol. 4). Springer Science & Business Media.

Putz, M., 2015. Editorial (Thematic Issue: Graph Theory and Molecular Topology in Organic

Chemistry~ Part 1~). Current Organic Chemistry, 19(3), pp.204-204.

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