The Clar Structure of Fullerenes

A fullerene is a 3-regular plane graph consisting only of pentagonal and hexagonal faces. Fullerenes are designed to model carbon molecules. The Clar number and Fries number are two parameters that are related to the stability of carbon molecules. We introduce chain decompositions, a new method to find lower bounds for the Clar and Fries numbers. In Chapter 3, we define the Clar structure for a fullerene, a less general decomposition designed to compute the Clar number for classes of fullerenes. We use these new decompositions to understand the structure of fullerenes and achieve several results. In Chapter 4, we classify and give a construction for all fullerenes on |V| vertices that attain the maximum Clar number |V|/6 - 2. In Chapter 5, we settle an open question with a counterexample: we construct an infinite family of fullerenes for which a set of faces attaining the Clar number cannot be a subset of a set of faces that attains the Fries number. We develop a method to calculate the Clar number directly for many infinite families of fullerene

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

The Present and Future of Planetary Nebula Research. A White Paper by the IAU Planetary Nebula Working Group

We present a summary of current research on planetary nebulae and their central stars, and related subjects such as atomic processes in ionized nebulae, AGB and post-AGB evolution. Future advances are discussed that will be essential to substantial improvements in our knowledge in the field.Comment: accepted for publication in RMxAA; 37 page

The Structure of 4-Clusters in Fullerenes

Fullerenes can be considered to be either molecules of pure carbon or the trivalent plane graphs with all hexagonal and (exactly 12) pentagonal faces that models these molecules. Since carbon atoms have valence 4 and our models have valence 3, the edges of a perfect matching are doubled to bring the valence up to 4 at each vertex. The edges in this perfect matching are called a Kekule structure and the hexagonal faces bounded by three Kekule edges are called benzene rings. A maximal independent (disjoint) set of benzene rings for a given Kekule structure is called a Clar set, and the maximum possible size of a Clar set over all Kekule structures is the Clar number of the fullerene. For any patch of hexagonal faces in the fullerene away from all pentagonal faces, there is a perfect Kekule structure: a Kekule structure for which the faces of an independent set of benzene rings are packed together as tightly as possible. Starting with such a patch and extending it as far as possible results in a perfect Kekule structure except for isolated regions, called clusters, containing the pentagonal faces. It has been shown that clusters must contain even numbers of pentagonal faces. It has also been shown that the Kekule structure of the patch can be extended into each of these clusters to give a full Kekule structure. However, these Kekule extensions will not admit as tightly packed benzene rings as in the patch external to the clusters. A basic problem in computing the Clar number of a fullerene is to make these extensions in a way that maximizes the number of benzene rings in each cluster. The simplest case, that of 2-clusters, has been completely solved. This thesis is devoted to developing a complete understanding of the Clar structures of 4-clusters

Study on the Q-Conjugacy Relations for the Janko Groups

In this paper, we consider all the Janko sporadic groups J1, J2, J3 and J4 (with orders 175560, 604800, 50232960 and 86775571046077562880, respectively) with a new concept called the markaracter- and Q-conjugacy character tables, which enables us to discuss marks and characters for a finite group on a common basis of Q-conjugacy relationships between their cyclic subgroups. Then by using GAP (Groups, Algorithms and Programming) package we calculate all their dominant classes enabling us to find all possible Q-conjugacy characters for these sporadic groups. Finally, we prove in a main theorem that all twenty six simple sporadic groups are unmatured

Graphene-Based ‘Hybrids’ as High-Performance Electrodes with Tailored Interfaces for Alternative Energy Applications: Synthesis, Structure and Electrochemical Properties

Technological progress is determined to a great extent by developments of novel materials from new combinations of known substances with different dimensionality and functionality. We investigate the development of 3D ‘hybrid’ nanomaterials by utilizing graphene based systems coupled with transition metal oxides (e.g. manganese oxides MnO2 and Mn3O4). This lays the groundwork for high performance electrochemical electrodes for alternative energy owing to their higher specific capacitance, wide operational window and stability through charge-discharge cycling, environmental benignity, cost effective, easily processed, and reproducible in a larger scale. Thus far, very few people have investigated the potential of combining carbon sheets that can function as a supercapacitor in certain systems with transition metals that have faradaic properties to create electrochemical capacitors. Previous work by Wang et al. has focused on the structural combination of Mn3O4 and graphene based materials,1 and research by Jafta et al. studied the electrochemical properties of MnO2 with GO.2 We find that both physical and chemical attachment of manganese oxide on graphene allows for electrical interplay of the materials as indicated in electrochemical analysis and Raman spectroscopy. Attachment of the two materials is also characterized by scanning electron microscopy and X-ray diffraction

Carbon in the Galaxy: Studies from Earth and Space

Presented here is the text of the invited papers presented during a meeting entitled, Carbon in the Galaxy: Studies from Earth and Space, that was held at NASA Ames Research Center on November 5 and 6, 1987. For completeness, abstracts from all of the poster papers and the text of a paper summarizing what was learned during the course of the meeting are also included. The underlying premise for the meeting was that there is much to be gained by bringing together scientists from very different disciplines, all of whom study carbon in different ways for different reasons. The interchanges took place during the meeting and the contents of the enclosed papers validate that premise