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Graph-Theoretic Analysis of Nanocarbon Structures

Published online by Cambridge University Press:  09 February 2016

Erica Fagnan  and
Robert Cormia
Erica Fagnan*
Affiliation:
University of California at Berkeley, Berkeley, CA 94720, USA
Robert Cormia
Affiliation:
UCSC/NASA-ARC Advanced Studies Laboratories (ASL), Moffett Field, Mountain View, CA, 94040, USA Engineering, Foothill College Faculty, Los Altos Hills, CA, 94022, USA
*
*(Email: [email protected])
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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 well-understood 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.

Keywords

structuralnanostructureC
Type
Articles
Information
MRS Advances , Volume 1 , Issue 24: Theory, Characterization, and Modeling , 2016 , pp. 1761 - 1766
Copyright
Copyright © Materials Research Society 2016 

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References

REFERENCES

Balaban, A. T., J. Chem. Inf. Comp. Sci. 25, 334 (1985).CrossRefGoogle Scholar
Estrada, E.. Graph and Network Theory in Physics. WWW Document. (http://arxiv.org/abs/1302.4378).Google Scholar
Cormia, R.D., Johnsen, J.N.. Nanotech. IEEE Proc., 11, 942 (2011).Google Scholar
Marr, A. and Wallis, W.D., Magic Graphs, 2nd Ed. (Springer, New York, 2013), pp. 15–23.CrossRefGoogle Scholar
Lee, S-M, Su, H-H, and Wang, Y-C, Cong. US Numer. 193, 49 (2008).Google Scholar
Bloom, G. S. and Golomb, S.W., Lec. Notes in Math. 642, 53 (1978).CrossRefGoogle Scholar
Bloom, G. S. and Golomb, S. W., Proc. IEEE 65, 562 (1977).CrossRefGoogle Scholar
Baca, M.. Disc. Math. 105, 305 (1992).CrossRefGoogle Scholar
Baker, A. and Sawada, J., Lec. Notes Comp. Sci. 5165, 361 (2008).CrossRefGoogle Scholar
Yakovlev, V. S., I Stockman, M., Krausz, F. and Baum, P., Sci. Rep. 5: 14581, (2015).CrossRefGoogle Scholar
U.S. Army Materiel Command, CC BY 2.0. “Scanning probe microscopy image of graphene,” (2012).Google Scholar
Tomruen (Own work), CC BY-SA 4.0. “Hexagonal Lattice,” (2015).Google Scholar
Weisstein, E. W.. “Hexagonal Grid.” WWW Document. (http://mathworld.wolfram.com/HexagonalGrid.html).Google Scholar
Weisstein, E. W.. “Graph.” WWW Document. (http://mathworld.wolfram.com/Graph.html).Google Scholar
Weisstein, E. W.. “Magic Graph.” WWW Document. (http://mathworld.wolfram.com/ MagicGraph.html).Google Scholar
Park, J., He, G., Feenstra, R. M., Li, A.. “Atomic-Scale Mapping of Thermoelectric Power on Graphene: Role of Defects and Boundaries.” Nano Lett. 13, 3269 (2013).CrossRefGoogle Scholar