Characterization of Graphene Structures Using the Tutte

Polynomial with Maple and Sage.

DAVID ALBERTO BOLÍVAR RUIZ

Logic and Computation Group

Theoretical and Computational Physics Group

Physics Engineering Program

Sciences and Humanity School

EAFIT University

Medellin, Colombia.

dboliva1@eafit.edu.co.

Abstract - Nowadays, some research in quantum computing and nanotechnology in the

world are focused on developing new materials, this in order to store information on

nano-metric scale in increasing amounts, this article shows a study on how to establish

criteria evaluation for the degree of complexity of arrangements physicochemical and

logical structures, in particular graphene molecules, and the interest to analyze the

properties of this material is that it understands its basic organization and this is useful for

the development of electric circuits very small scale. Graphene is a flat sheet structure of

an atom thick, composed of densely packed carbon atoms which allows its manipulation

for the development of nanotechnology devices. The evaluation criterion is proposed, is

to study the logic gates analogous electrical circuits and structures of graphene,

implementing the Tutte polynomials are mathematical objects that provide information on

non-directional flat geometries in graph theory, all these results will obtained using

computational methods, specifically the MAPLE and SAGE software.

Keywords: - Tutte polynomials, characterization of the structures of graphene,

nanotechnology, physical chemistry, logic gates, quantum computation.

Applied Mathematics and Informatics

ISBN: 978-960-474-260-8 195

1. Introduction In recent times man has made great

progress on the applications of

nanotechnology [1] in all research

areas of modern science,

nanotechnology is a branch of

science that studies the production of

nano-scale machines. Possible

technologies that may emerge from

these applications is unlimited due to

the potential of atomic theories, the

implications of these studies ranging

from quantum computing [2] and

information theory to the

construction of any type of device

from new materials.

The main contribution of this paper

is to establish an evaluation criterion

for the complexity of the structures

of grapheme [3,4,5,6], we tried to

apply a theoretical tool such as graph

theory [7] to achieve this objective,

the particular interest of studying the

behavior of graphene lies the fact

that this has properties suitable to be

a useful material in the micro-

fabrication and nanotechnology,

these results highly relevant for

future applications of graphene as a

material for nano-manufacturing.

2. Problem

Carbon is an element of nature for

their physical and chemical

properties is often a major trainer of

compounds, but can also be found

individually in molecular form as a

gas or solid, such as C2 (Gaseous),

diamond, graphite general graphene

structures that are as solid on the

environment, all these attributes

make it a perfect candidate to study

its behavior as nano-manipulated

materials.

The problem that caused the

initiation of this investigation is the

fact that there is no standard of

comparison and evaluation to select

the graphene structures that are

suitable for nano-manufacturing, this

article shows an initial idea to set a

criterion algebraic how to

characterize these structures.

3. Method

The method implemented in the

preparation of the paper first was an

investigation that began with the

study of the applications of graph

theory, particularly the academic

interests have always been inclined

to the implementation of such a

theory on the construction of new

materials to make progress and

applications in computing, after

deciding to tackle the problem of

evaluation criteria for graphene

molecules using graph theory we

proceeded with the following steps:

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ISBN: 978-960-474-260-8 196

3.1. Software SAGE

implementation

The process to create such an

approach began studying how

structures can be described as

graphene, it came to the conclusion

that the graphs are isomorphic to

those molecules which can establish

criteria for the complexity of these

using the Tutte polynomial [8] is

constructed using SAGE [9] software

graphs describing molecules.

3.2. Software MAPLE

implementation.

This software is key to writing this

article because he is the one that

calculates the Tutte polynomial of

each graph, and this is what sets the

criterion of complexity to the graph

and hence for the associated

graphene molecule. The commands

implemented were: 3.2.1. Command

This command initializes the

MAPLE [10] software and loads the

graph theory package. 3.2.2. Command

This command allows you to

calculate the Tutte polynomial of

each graph that in turn represent s

graphene structures. 3.2.3. Command

This command replaces the Tutte

polynomial of the Hamiltonian

associated with the graphene

molecule.

3.2.4. Command

This command sets the equation for

each possible Schödinger graphene

molecule.

4. Results

4.1. Physical results.

Figures 3. Graphene

molecules.

4.1.1. Associated Tutte Polynomial.

The Corresponding Tutte polynomial is

given by equation (1).

4.1.2. The Hamiltonian of this

polynomial is given by equation (2).

4.1.3. Schrödinger equation for

this structure of graphene is given

by the equation (3).

4.1.4. Schrödinger expand

equation for this structure of

graphene is given by the equation

(4).

4.1.5. Implemented substitution (5).

4.1.6. Quantum energy for the

grapheme molecule is given by (6).

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ISBN: 978-960-474-260-8 197

(1)

4. CONCLUSIONS

This work allowed for a study on a

possible approach to implement

algebraic graph theory to

characterize the complex structures

of graphene, after calculating the

energy levels of these molecules by

the equation result, you can

determine which are the most

viable the manufacture of nano-

structured

(2)

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(3)

(4)

(5)

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(6)

devices even if they are under

strain or deformation. Although

the Tutte polynomials are the

most commonly used to describe

graphs, are not completely

universal because the same

polynomial can represent several

graphs, therefore the final

equations may very well

represent the energy of different

molecules but not its geometry.

The implications for future

studies in this area can be quite

useful in fields like quantum

computing and nanotechnology.

5. REFERENCES

[1] Nanotechnology:

http://en.wikipedia.org/wiki/Nanotec

hnology

[2] Quantum Computing:

http://en.wikipedia.org/wiki/Quantu

m_computer

[3] Graphene:

http://en.wikipedia.org/wiki/Graphen

e

[4] A. K. Geim and K.S. Novoselov,

Nature Materials 6, 183 (2007)

[5] P. R. Wallace, Phys. Rev. 71, 622

(1947).

[6] K. S. Novoselov, A. K. Geim, S.

V. Morozov,D. Jiang, Y. Zhang, S.

V. Dubonos, I. V. Gregorieva and

A. A. Forsov, Science, 306, 666

(2004).

[7] Graph Theory:

http://en.wikipedia.org/wiki/Graph_t

heory

[8] Joanna Ellis-Monagham, Criel

Merino, Graph polynomials and their

applications I: The Tutte polynomial,

arXiv:0803.3079

Tutte Polynomial:

http://en.wikipedia.org/wiki/Tutte_p

olynomial

[9] Sage: http://www.sagemath.org/

[10] Maple: www.maplesoft.com

Applied Mathematics and Informatics

ISBN: 978-960-474-260-8 200