Graph-Theoretic Analysis of Nanocarbon Structures
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Graph-Theoretic Analysis of Nanocarbon Structures Erica Fagnan1 and Robert Cormia2, 3 1
University of California at Berkeley, Berkeley, CA 94720, USA
2
UCSC/NASA-ARC Advanced Studies Laboratories (ASL), Moffett Field, Mountain View, CA,
94040, USA 3
Engineering, Foothill College Faculty, Los Altos Hills, CA, 94022, USA
ABSTRACT Nanostructures tend to comprise distinct and measurable forms, which can be referred to in this context as nanopatterns. Far from being random, these patterns reflect the order of wellunderstood chemical and physical laws. Under the aegis of said physical and chemical laws, atoms and molecules coalesce and form discrete and measurable geometric structures ranging from repeating lattices to complicated polygons. Rules from several areas of pure mathematics such as graph theory can be used to analyze and predict properties from these well-defined structures. Nanocarbons have several distinct allotropes that build upon the basic honeycomb lattice of graphene. Because these allotropes have clear commonalities with respect to geometric properties, this paper reviews some approaches to the use of graph theory to enumerate structures and potential properties of nanocarbons. Graph theoretic treatment of the honeycomb lattice that forms the foundation of graphene is completed, and parameters for further analysis of this structure are analyzed. Analogues for modelling graphene and potentially other carbon allotropes are presented. INTRODUCTION Graph theory has its foundations in pure mathematics and theoretical computer science. The discipline has been used to analyze data structures as well as generate frameworks to solve search problems. Somewhat limited research on the intersection between graph theory and other fields has been completed. For example, research has shown that structural formulae of compounds can be viewed and potentially modelled as graphs (covalently bonded compounds) [1]. Additional work has been completed on the intersection between graph theory and physics, including applications to tight bonding as well as the Hubbard model [2]. Extending these types of representations into materials science and combining them with graph labelings provides a rigorous framework for modeling of discrete objects. However, research into the intersection between graph theory and materials science has been sparse. The following paper will explore a few ways in which graph theoretic concepts may be used to analyze patterns in materials science with a focus on graphene as a prime example of a basic nanocarbon structure [3].
1761 Downloaded from https://www.cambridge.org/core. Faculty of Classics, University of Cambridge, on 16 Sep 2017 at 14:32:53, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1557/adv.2016.113
THEORY Graph theory can be simply described as the study of points and lines. As such, it is important to note that the graphs being considered in this context are not the familiar coordinate plots that are used to show
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