Distance, symmetry, and topology in carbon nanomaterials

Ali Reza AshrafiDepartment of Pure Mathematics, Faculty of Mathematical Sciences University of Kashan Kashan, Iran Mircea V.. Ali Reza Ashrafi Department of Nanocomputing, Institute of Nan

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Carbon Materials: Chemistry and Physics 9

with complete coverage of carbon materials and carbon-rich molecules Fromelemental carbon dust in the interstellar medium, to the most specialized industrialapplications of the elemental carbon and derivatives With great emphasis on themost advanced and promising applications ranging from electronics to medicinalchemistry.

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Distance, Symmetry, and Topology in Carbon

Nanomaterials

Ali Reza Ashrafi

Department of Pure Mathematics,

Faculty of Mathematical Sciences

University of Kashan

Kashan, Iran

Mircea V DiudeaDepartment of Chemistry, Faculty of Chemistryand Chemical Engineering

Babes-Bolyai UniversityCluj-Napoca, Romania

ISSN 1875-0745 ISSN 1875-0737 (electronic)

Carbon Materials: Chemistry and Physics

ISBN 978-3-319-31582-9 ISBN 978-3-319-31584-3 (eBook)

DOI 10.1007/978-3-319-31584-3

Library of Congress Control Number: 2016941059

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In 1872, Felix Klein published his pioneering paper on the importance of symmetry,which was later named “Erlanger Programm” for his professorship at the University

of Erlangen, Germany He wrote: “we can say that geometry studies those and onlythose properties of the figure F which are shared by F and all the figures which areequal to F” He continued that the most essential idea required in the study ofsymmetry is that of a group of space transformations Topology is the mathematicalstudy of shapes Distance, Symmetry and Topology in Carbon Nanomaterialsgathers the contributions of some leading experts in a new branch of science that

is recently named “Mathematical Nanoscience”

This volume continues and expands upon the previously published titles TheMathematics and Topology of Fullerenes (Carbon Materials: Chemistry and Phys-ics series, Vol 4, Springer 2011) andTopological Modelling of Nanostructures andExtended Systems (Carbon Materials: Chemistry and Physics series, Vol 7, Springer2013) by presenting the latest research on this topic It introduces a new attractivefield of research like the symmetry-based topological indices, multi-shell clusters,dodecahedron nano-assemblies and generalized fullerenes, which allow the reader

to obtain a better understanding of the physico-chemical properties ofnanomaterials

Topology and symmetry of nanomaterials like fullerenes, generalized fullerenes,multi-shell clusters, graphene derivatives, carbon nanocones, corsu lattices, dia-monds, dendrimers, tetrahedral nanoclusters and cyclic carbon polyynes give someimportant information about the geometry of these new materials that can be usedfor correlating some of their physico-chemical or biological properties

We would like to thank to all the authors for their work and support, also toSpringer for giving us the opportunity to publish this edited book and finally toSpringer people who allowed all our efforts to make this an interesting book

1 Molecular Dynamics Simulation of Carbon Nanostructures:

The Nanotubes 1Istva´n La´szlo and Ibolya Zsoldos

2 Omega Polynomial in Nanostructures 13Mircea V Diudea and Beata Szefler

3 An Algebraic Modification of Wiener and Hyper–Wiener Indicesand Their Calculations for Fullerenes 33Fatemeh Koorepazan-Moftakhar, Ali Reza Ashrafi,

Ottorino Ori, and Mihai V Putz

4 Distance Under Symmetry: (3,6)-Fullerenes 51Ali Reza Ashrafi, Fatemeh Koorepazan Moftakhar, and Mircea V

Diudea

5 Topological Symmetry of Multi-shell Clusters 61Mircea V Diudea, Atena Parvan-Moldovan, Fatemeh

Koorepazan-Moftakhar, and Ali Reza Ashrafi

6 Further Results on Two Families of Nanostructures 83Zahra Yarahmadi and Mircea V Diudea

7 Augmented Eccentric Connectivity Index of Grid Graphs 95Tomislav Dosˇlic´ and Mojgan Mogharrab

8 Cluj Polynomial in Nanostructures 103Mircea V Diudea and Mahboubeh Saheli

9 Graphene Derivatives: Carbon Nanocones and CorSu Lattice:

A Topological Approach 133Farzaneh Gholaminezhad and Mircea V Diudea

10 Hosoya Index of Splices, Bridges, and Necklaces 147Tomislav Dosˇlic´ and Reza Sharafdini

11 The Spectral Moments of a Fullerene Graph and Their

Applications 157G.H Fath-Tabar, F Taghvaee, M Javarsineh, and A Graovac

12 Geometrical and Topological Dimensions of the Diamond 167G.V Zhizhin, Z Khalaj, and M.V Diudea

13 Mathematical Aspects of Omega Polynomial 189Modjtaba Ghorbani and Mircea V Diudea

14 Edge-Wiener Indices of Composite Graphs 217Mahdieh Azari and Ali Iranmanesh

15 Study of the Bipartite Edge Frustration of Graphs 249Zahra Yarahmadi

16 The Hosoya Index and the Merrifield–Simmons Index of Some

Nanostructures 269Asma Hamzeh, Ali Iranmanesh, Samaneh Hossein–Zadeh,

and Mohammad Ali Hosseinzadeh

17 Topological Indices of 3-Generalized Fullerenes 281

Z Mehranian and A.R Ashrafi

18 Study of the Matching Interdiction Problem in Some Molecular

Graphs of Dendrimers 303G.H Shirdel and N Kahkeshani

19 Nullity of Graphs 317Modjtaba Ghorbani and Mahin Songhori

20 Bondonic Chemistry: Spontaneous Symmetry Breaking

of the Topo-reactivity on Graphene 345Mihai V Putz, Ottorino Ori, Mircea V Diudea, Beata Szefler,

and Raluca Pop

21 Counting Distance and Szeged (on Distance) Polynomials

in Dodecahedron Nano-assemblies 391Sorana D Bolboaca˘ and Lorentz Ja¨ntschi

22 Tetrahedral Nanoclusters 409Csaba L Nagy, Katalin Nagy, and Mircea V Diudea

23 Cyclic Carbon Polyynes 423Lorentz Ja¨ntschi, Sorana D Bolboaca˘, and Dusanka Janezic

24 Tiling Fullerene Surfaces 437Ali Asghar Rezaei

25 Enhancing Gauge Symmetries Via the Symplectic Embedding

Approach 447Salman Abarghouei Nejad and Majid Monemzadeh

26 A Lower Bound for Graph Energy of Fullerenes 463Morteza Faghani, Gyula Y Katona, Ali Reza Ashrafi,

and Fatemeh Koorepazan-Moftakhar

Index 473

Ali Reza Ashrafi Department of Nanocomputing, Institute of Nanoscience andNanotechnology, University of Kashan, Kashan, Iran

Department of Pure Mathematics, Faculty of Mathematical Sciences, University ofKashan, Kashan, Iran

Mahdieh Azari Department of Mathematics, Kazerun Branch, Islamic AzadUniversity, Kazerun, Iran

Sorana D Bolboaca˘ University of Agricultural Science and Veterinary MedicineCluj-Napoca, Cluj-Napoca, Romania

Department of Medical Informatics and Biostatistics, Iuliu Hat¸ieganu University ofMedicine and Pharmacy, Cluj-Napoca, Romania

Mircea V Diudea Department of Chemistry, Faculty of Chemistry and ChemicalEngineering, Babes-Bolyai University, Cluj-Napoca Romania

Tomislav Dosˇlic´ Faculty of Civil Engineering, University of Zagreb, Zagreb,Croatia

Morteza Faghani Department of Mathematics, Payam-e Noor University,Tehran, Iran

G H Fath-Tabar Department of Pure Mathematics, Faculty of MathematicalSciences, University of Kashan, Kashan, Iran

Farzaneh Gholaminezhad Department of Pure Mathematics, Faculty ofMathematical Sciences, University of Kashan, Kashan, Iran

Modjtaba Ghorbani Department of Mathematics, Faculty of Science, ShahidRajaee Teacher Training University, Tehran, Iran

A Graovac The Rugjer Boskovic Institute, NMR Center, Zagreb, Croatia

Asma Hamzeh Department of Mathematics, Faculty of Mathematical Sciences,Tarbiat Modares University, Tehran, Iran

Mohammad Ali Hosseinzadeh Department of Mathematics, Faculty ofMathematical Sciences, Tarbiat Modares University, Tehran, Iran

Samaneh Hossein-Zadeh Department of Mathematics, Faculty of MathematicalSciences, Tarbiat Modares University, Tehran, Iran

Ali Iranmanesh Department of Pure Mathematics, Faculty of Mathematical ences, Tarbiat Modares University, Tehran, Iran

Sci-Lorentz Ja¨ntschi Department of Physics and Chemistry, Technical University ofCluj-Napoca, Cluj-Napoca, Romania

Institute for Doctoral Studies, Babes¸-Bolyai University, Cluj-Napoca, RomaniaUniversity of Agricultural Science and Veterinary Medicine Cluj-Napoca,Cluj-Napoca, Romania

Department of Chemistry, University of Oradea, Oradea, Romania

Dusanka Janezic Natural Sciences and Information Technologies, Faculty ofMathematics, University of Primorska, Koper, Slovenia

M Javarsineh Department of Pure Mathematics, Faculty of MathematicalSciences, University of Kashan, Kashan, Iran

N Kahkeshani Department of Mathematics, Faculty of Sciences, University ofQom, Qom, Iran

Gyula Y Katona Department of Computer Science and Information Theory,Budapest University of Technology and Economics, Budapest, Hungary

MTA-ELTE Numerical Analysis and Large Networks, Research Group, Budapest,Hungary

Z Khalaj Department of Physics, Shahr-e-Qods Branch, Islamic Azad University,Tehran, Iran

Fatemeh Koorepazan-Moftakhar Department of Nanocomputing, Institute ofNanoscience and Nanotechnology, University of Kashan, Kashan, Iran

Department of Pure Mathematics, Faculty of Mathematical Sciences, University ofKashan, Kashan, Iran

Istva´n La´szlo Department of Theoretical Physics, Institute of Physics, BudapestUniversity of Technology and Economics, Budapest, Hungary

Z Mehranian Department of Mathematics, University of Qom, Qom, IranMojgan Mogharrab Department of Mathematics, Faculty of Sciences, PersianGulf University, Bushehr, Iran

Majid Monemzadeh Department of Particle Physics and Gravity, Faculty ofPhysics, University of Kashan, Kashan, Iran

Csaba L Nagy Department of Chemistry, Faculty of Chemistry and ChemicalEngineering, University of Babes-Bolyai, Cluj-Napoca, Romania

Katalin Nagy Department of Chemistry, Faculty of Chemistry and ChemicalEngineering, University of Babes-Bolyai, Cluj-Napoca, Romania

Salman Abarghouei Nejad Department of Particle Physics and Gravity, Faculty

of Physics, University of Kashan, Kashan, Iran

Ottorino Ori Actinium Chemical Research, Rome, Italy

Laboratory of Computational and Structural Physical-Chemistry for Nanosciencesand QSAR, Department of Biology-Chemistry, Faculty of Chemistry, Biology,Geography, West University of Timis¸oara, Timis¸oara, Romania

Atena Parvan-Moldovan Department of Chemistry, Faculty of Chemistry andChemical Engineering, Babes-Bolyai University, Cluj-Napoca, Romania

Raluca Pop Faculty of Pharmacy, University of Medicine and Pharmacy “VictorBabes” Timis¸oara, Timis¸oara, Romania

Mihai V Putz Laboratory of Computational and Structural Physical-Chemistryfor Nanosciences and QSAR, Department of Biology-Chemistry, Faculty

of Chemistry, Biology, Geography, West University of Timis¸oara, Timis¸oara,Romania

Laboratory of Renewable Energies-Photovoltaics, R&D National Institute forElectrochemistry and Condensed Matter, Timis¸oara, Romania

Ali Asghar Rezaei Department of Pure Mathematics, Faculty of MathematicalSciences, University of Kashan, Kashan, Iran

Mahboubeh Saheli Department of Pure Mathematics, University of Kashan,Kashan, Iran

Reza Sharafdini Department of Mathematics, Faculty of Basic Sciences, PersianGulf University, Bushehr, Iran

G H Shirdel Department of Mathematics, Faculty of Sciences, University ofQom, Qom, Iran

Mahin Songhori Department of Mathematics, Faculty of Science, Shahid RajaeeTeacher Training University, Tehran, Iran

Beata Szefler Department of Physical Chemistry, Faculty of Pharmacy,Collegium Medicum, Nicolaus Copernicus University, Bydgoszcz, Poland

F Taghvaee Department of Pure Mathematics, Faculty of Mathematical Sciences,University of Kashan, Kashan, Iran

Zahra Yarahmadi Department of Mathematics, Faculty of Sciences,Khorramabad Branch, Islamic Azad University, Khorramabad, Iran

G V Zhizhin Member of “Skolkovo” OOO “Adamant”, Saint-Petersburg, RussiaIbolya Zsoldos Faculty of Technology Sciences, Sze´chenyi Istva´n University,Gyo˝r, Hungary

Molecular Dynamics Simulation of Carbon

Nanostructures: The Nanotubes

Istva´n La´szlo and Ibolya Zsoldos

Abstract Molecular dynamics calculations can reveal the physical and chemicalproperties of various carbon nanostructures or can help to devise the possibleformation pathways In our days the most well-known carbon nanostructures arethe fullerenes, the nanotubes, and the graphene The fullerenes and nanotubes can

be thought of as being formed from graphene sheets, i.e., single layers of carbonatoms arranged in a honeycomb lattice Usually the nature does not follow themathematical constructions Although the first time the C60 and the C70 wereproduced by laser-irradiated graphite, the fullerene formation theories are based

on various fragments of carbon chains and networks of pentagonal and hexagonalrings In the present article, using initial structures cut out from graphene will bepresented in various formation pathways for the armchair (5,5) and zigzag (9,0)nanotubes The interatomic forces in our molecular dynamics simulations will becalculated using tight-binding Hamiltonian

The fullerenes, nanotubes, and graphene are three allotrope families of carbon,and their production in the last 27 years has triggered intensive researches in thefield of carbon structures (Fowler and Manolopulos 1995, Dresselhaus

et al 1996) Each of them marks a breakthrough in the history of science Thefullerenes, the multi-walled carbon nanotubes, the single-walled carbonnanotubes, and the graphene jumped into the center of interest in 1985, 1991,

1993, and in 2004 in order (Kroto et al 1985; Iijima1991; Iijima and Ichihashi

Faculty of Technology Sciences, Sze´chenyi Istva´n University, H-9126 Gyo˝r, Hungary

1993; Bethune et al.1993; Novoselov et al.2004) All of these breakthroughs aregood examples for the fact that a breakthrough can be realized only if the sciencehas a certain level of maturity concerning the combination of a set of favorableconditions, as, for example, having the right materials available, as well as therelated theory, investigation tools, and scientific minds (Monthioux andKuznetsov2006).

In 1965 Schultz studied the geometry of closed cage hydrocarbons and amongthem the truncated icosahedron C60H60 molecule (Schultz 1965) In 1966 Jonesusing the pseudonym Daedalus was speculating about graphite balloon formations

in high-temperature graphite productions (Jones1966) Osawa suggested the C60molecule to be a very aromatic one in a paper written in Japanese language(Osawa1970), and two Russian scientists wrote a paper in their native languageabout the electronic structure of the truncated icosahedron molecule (Bochvar andGalpern1973) In the early 1980s, Orville Chapman wanted to develop syntheticroutes to C60 (Kroto 1992; Baggott 1996) All of these theoretical and experi-mental studies were isolated As we have mentioned, the breakthrough happened

in 1985 (Kroto et al.1985), but before that in the years 1982–1983, Kra¨tschmerand Huffman have found some kind of “junk” in the ultraviolet spectrum ofcarbon soot produced in arc discharge experiment (Baggott 1996; Kra¨tschmer

2011) After calculating the ultraviolet spectrum of the C60molecule (La´szlo andUdvardi1987; Larsson et al.1987), they have realized that the “junk” was due tothe C60fullerene Publishing their results they have supplied a new breakthrough

in fullerene research as they have produced fullerenes in crystal structure(Kra¨tschmer et al.1990)

In the early studies of fullerenes, most of the people knew that they werestudying the C60molecule, but they could not produce it The only exception wasKra¨tschmer and Huffman, namely, they produced it without recognizing it In thehistory of nanotubes, there are many people who produced multi-walled carbonnanotubes, but they, or the scientific community, did not realize its importance(Boehm1997; Monthioux and Kuznetsov2006) The first authors who presentedelectron transmission images of multi-walled carbon nanotubes were perhapsRadushkevich and Lukyanovich in a paper written in Russian language in 1952(Radushkevich and Lukyanovich 1952) But as we have mentioned, it was apublication without breakthrough

The story of graphene is also interesting In 1935 and 1937, Peielrs (1935) andLandau (1937) theoretically showed that strictly two-dimensional crystals werethermodynamically unstable and could not exist Thus the study of electronicstructure of graphene seemed to be a theoretical exercise without any experimentalapplication (Wallace1947) It was also found that this nonexisting material is anexcellent condensed-matter analogue of (2þ 1)-dimensional quantum electrody-namics (Semenoff1984) In 2004 Novoselov et al realized the breakthrough byexperimental study producing the graphene (Novoselov et al.2004) There was notany contradiction between the theory and experiment, because it turned out that theproduced graphene was not ideally two-dimensional It was by gentle crumpling inthe third dimension

Since the experimental production of graphene, many authors working oncarbon nanostructures start their talks with a picture presenting a graphene sheetcontaining various cut out patterns which are turning into fullerenes and nanotubes(Geim and Novoselov 2007) These processes follow the textbook geometricconstruction of fullerenes and nanotubes and explain the fact that the graphene isthe “mother of all graphitic forms” (Geim and Novoselov2007) In our previoustight-binding molecular dynamics calculations, we have shown that such processescan be realized by starting the simulation with some graphene cut out patterns(La´szlo and Zsoldos2012a) We have presented more details for the C60formation

in La´szlo and Zsoldos (2012b) and for the C70formation in La´szlo and Zsoldos(2014) In the present work, we give details for the nanotube productions In thenext paragraph, we describe the initial structures used in our molecular dynamicssimulation and give also the parameters of our calculations Then we describe theresults obtained for the formation processes of armchair (5,5) and zigzag (9,0)nanotubes

This idea can be used for any chiral nanotube as well

We cut out special patterns from the graphene in order to use them as initialstructures in molecular dynamics calculations Depending on the initial struc-ture, we obtain fullerenes and nanotubes in a self-organizing way The informa-tion built in the initial structure determine precisely the structure of thefullerenes and nanotubes in study Thus we cut out the initial patterns fromthe graphene in a way which has the following properties: (1) It contains onlyhexagons (2) There are some fourth (or fifth) neighboring atoms on the perim-eter which can approach each other during their heat motion by constructingnew pentagonal (or hexagonal) faces (3) After the formation of new faces, othercarbon atoms will be in appropriate positions for producing pentagons and/orhexagons Repeating steps 2 and 3, we obtain the structure selected by the initialpattern

This construction of initial patterns is similar to D€urer’s unfolding method(O’Rourke2011) but it has some important differences as well Albrecht D€urer,the famous painter, presented polyhedrons by drawing nets for them The nets wereobtained by unfolding of the surfaces to a planar layout which contained each of thepolygons Our layout does not contain all of the polygons but it contains all of thevertices It must contain all of the vertices (atoms) which are arranged in a specialway for supplying the driving forces for the upfolding

In Fig.1.1a, b, we present initial structures applied for the formation of C60and

C70nanotubes Other structures can be found in references (La´szlo and Zsoldos

2012a,b,2014) In the formation of fullerenes, the driving force comes from thepentagon constructions Only the pentagons can form curved surfaces Beforeforming a new bond, the system must cross an energy barrier In order to have aone-way process, the potential energy must decrease after crossing the energybarriers of the potential landscape The system works like a molecular motor.Mathematically the nanotubes are constructed by rolling up a parallelogram cut outfrom a hexagonal network of carbon atoms These constructions do not havepentagons and the curved surface cannot be produced in a self-organizing way.This is why we are using half of a fullerene as a molecular motor This molecularmotor will roll up the parallelogram and the final structure will be a half cappednanotube Figure 1.1c, dshow the initial structures for (5,5) armchair and (9,0)zigzag carbon nanotube The initial pattern of armchair nanotube contains half of a

C70 fullerene and the molecular motor of the zigzag nanotube is half of a C60fullerene

We calculated the interatomic carbon-carbon interaction with the help of aquantum chemical tight-binding method based on parameters adjusted to ab initiodensity functional calculations (Porezag et al 1995) Verlet algorithm (Verlet

1967) supplied the solution of the equations of motion The applied time step was0.7 fs As we have remarked it before, during the formation process, the systemgoes to lower and lower potential energy states Namely, the decreasing potentialenergy can guarantee the progress of the process to the desired structure Fromthe conservation of the energy follows that the kinetic energy will increase, if we

Fig 1.1 Initial and final structures for the formation of (a) fullerene C60, (b) fullerene C70, (c) armchair nanotube (5,5), and (d) zigzag nanotube (9,0) The upper structures are the initial structures and the lower structures are the final ones

decrease the potential energy This increased kinetic energy can destroy otheralready formed bonds of the structure and yielding a fragmentized structureinstead of the desired one In order to avoid this unfavorable situation, weperformed the simulations with constant environmental temperature Tenv In thepresent calculation, the environmental temperature was controlled with the help

of Nose´-Hoover thermostat (Nose´1984; Hoover1985; Allen and Tildesley1996;Frenkel and Smit 1996; La´szlo 1998) The Nose´-Hoover thermostat made itpossible to make distinction between the temperature of the environment andthe instantaneous temperature of the carbon structure The temperature of theenvironment Tenvwas given by a parameter in the algorithm and the temperature

of the carbon atoms was calculated from the kinetic energy of the atoms Thislatter temperature usually made oscillations around the value of the environmen-tal temperature

As the initial velocity of the atoms in the pattern was not known, we determinedrandomly the initial velocities corresponding to an initial temperature Tinit Whenthe number of atoms was N, we generated 3 N random numbers uniform in therange (0,1) After shifting these random numbers by0.5, we added them to theatomic coordinates of the atoms The desired initial temperature was obtained byscaling of this displacement vector In our simulation the interaction between theNose´-Hoover thermostat and the carbon atoms was so strong that the final structuredid not depend strongly on the initial temperature in the range of

0 K< Tinit< 2000 K This is why we used the initial temperature Tinit¼ 1200 Kpractically in all of the simulations

The initial structure of the (5,5) armchair nanotube can be seen on Fig.1.1c Itcontains half of the initial structure of the fullerene C70(Fig.1.1b) First we havemade a constant energy molecular dynamics simulation Here we supposed that theinitial temperature was 0 K The parameters of this run can be seen in Table1.1inthe line of run 1 The final structure after a simulation of 146 ps is shown inFig 1.2a The fragmentation process has already started and we obtained fourpentagons and one heptagon The formation of the pentagon and the starting offragmentation show that the information built into the initial pattern has alreadybeen lost This is why we made the calculations at constant temperatures Theresults of various runs are in Table1.1and Fig.1.2

In run 2 the initial temperature of the carbon atoms was 1200 K and theenvironmental temperature was 500 K Figure 1.2b shows that during thesimulation time of 15.1 ps, two pentagons were formed, but the structure looked

to be frozen That is, we did not hope further structure changing during a

Table 1.1 Simulation processes performed for the formation of a (5,5) armchair nanotube using the initial structure of Fig 1.1c

Run

Random

Time of simulation

Final structure

500

(0)500(46,2–46.41)800 (46,41–46.62)500 {(58.1–58.31)800(58.31–58.52) 500}

{ } repeated in each 2.1 ps (0)500(46,2–46.41)800 (46,41–46.62)500 {(58.1–58.31)1000 (58.31–58.52)500}

(0)500(46,2–46.41)800 (46,41–46.62)500 (57.75–58.03)100 {(58.1–58.45)600(58.31–58.52) 100}

{ } repeated in each 2.1 ps

(continued)

possible simulation time in a computer If the structure was frozen in the timescale of a simulation, it did not mean that it was frozen also in a realistic timescale as well In the various steps of the simulation, we tried to mimic someimagined structure – environment interaction or an experimental intervention.Increasing and decreasing the environmental temperature correspond to therandom interaction with an environmental particle or it corresponds to anappropriate laser impulse or electron beam (Chuvilin et al 2010; Terrones

et al.2002)

In run 3 we increased the environmental temperature to 800 K but the finalstructure was very similar to that of run 2 Thus in run 4, the environmentaltemperature was 1000 K, and Fig.1.2dshows that the system developed further,but it was also frozen in another structure In run 5 we wanted to avoid this frozenstructure by continuous increasing of the initial environmental temperature of

1000 K to 1300 K in the time range from 28.7 ps to 28.91 ps As fragmentationprocess was starting, we changed the random number generator, and we obtained

a promising structure in run 11 at the environmental temperature of 500 K(Fig.1.2k and Table 1.1) Run 12 was the same as run 11 only the temperaturewas raised from 500 K to 800 K from 46,2 ps to 46.41 ps (Fig.1.2land Table1.2)

In run 13 we increased the initial environmental temperature of 500 K to 800 K

as before, but after we decreased it to 500 K in the time range from 46.41 ps to46.62 ps We call such kind of increasing and decreasing the environmentaltemperature sawtooth changing The final structure is in Fig.1.2m In runs from

14 to 19, we put such kind of sawtooth changing of the environmental ature at various point of times and we obtained the desired nanotube in run

temper-18 (Fig 1.2rand Table 1.1).In this successful run, the changing of the mental temperature was the following The simulation started at 500 K Thisenvironmental temperature was raised to 800 K and decreased to 500 K with asawtooth changing from 46.2 ps to 46.62 ps From 57.75 ps to 58.03 ps, wedecreased the environmental temperature to 100 K From 58.1 ps to 58.52 ps, we

Final structure

(46,41–46.62)500

(57.75–58.03)100 (58.1–58.45)600(58.31–58.52) 100

The Tinit and Tenv initial and environmental temperatures are given in K and the time of simulation is given in ps Under the notation (t1–t2)T in Tenv, we mean increasing the temperature during the time interval (t1–t2) to the temperature T The letters in the column of final structure are shown in Fig 1.2 The parameters for the successful nanotube formation are marked by (*) in the run number Random number is the serial number of random number generator used generating the corresponding Tinit

Fig 1.2 The final structures of the processes in Table 1.1 To the Figures a–s correspond the Final structures of Table 1.1

increased this temperature to 600 K and decreased it to 100 K We repeated thissawtooth changing in each 2.1 ps and we obtained the armchair nanotube at thetime of 70.21 ps We continued such kind of variation of environmental temper-ature until 78.02 ps but the obtained nanotube was not destroyed This shows thestability of the obtained nanotube.

The initial structure of the (9,0) zigzag nanotube is shown on Fig.1.1d It containshalf of the initial structure of the fullerene C60(Fig.1.1a) The final structure of theconstant energy calculation is in Fig.1.3awith the parameters of run 1 in Table1.2.Our strategy of simulation was the same as that of the armchair nanotube Theparameters of various runs are in Table1.2and the final structures are in Fig.1.3

We changed the random distribution of the initial velocities and made calculationswith various environmental temperatures

Table 1.2 Simulation processes performed for the formation of a (9,0) zigzag nanotube using the initial structure of Fig 1.1d

Run

Random

Time of simulation

Final structure

is given in ps Under the notation (t1–t2)T in Tenv, we mean increasing the temperature during the time interval (t1–t2) to the temperature T The letters in the column of final structure are shown in

number Random number is the serial number of random number generator used generating the corresponding Tinit

We obtained the zigzag nanotube in run 14 after the simulation time of 37.57 ps(Fig.1.3nTable1.2) Here we started the simulation at 100 K, producing a sawtoothchanging from 28.7 ps to 29.12 ps, and we increased and decreased the environ-mental temperature from 100 K to 400 K and back to 100 K We repeated thischanging in each 3.5 ps As we can see, the final structure obtained at 37.57 pscontains a pentagon at the open side of the nanotube This pentagon can beeliminated by an extra hexagon in the initial structure of Fig.1.1d (La´szlo andZsoldos2012a).

Fig 1.3 The final structures of the processes in Table 1.2 To the Figures a–n correspond the Final structurers of Fig 1.3

1.4 Conclusions

We have shown that using appropriate environmental temperatures, one can controlthe chirality-dependent formation of carbon nanotubes In two examples wepresented results for an armchair and a zigzag nanotube The same idea can beused for nanotubes of any chirality The final structure is coded in the initial patternand it can be produced in a deterministic way According to our experiences, thebeginning of the formation process is the most critical If the structure survives thefirst tens of picoseconds, the final desired structure can be formed In the cases whenthe structure was frozen in the time scale of computer simulations, we increased ordecreased the environmental temperature for simulating the stronger interactionwith the environment Our days of patterning graphene are not yet at atomicprecision There are however promising experiments in electron-beam lithography(Chuvilin et al 2010; Chen et al 2007; Han et al 2007), scanning tunnelingmicroscope lithography (Tapaszto et al.2008), or the rational chemical synthesis

by polycyclic aromatic hydrocarbons (Boorum et al.2001; Scott et al.2002) Untilthe experimental realization of the initial pattern of our simulation, our results notonly give new insight into the formation processes of fullerenes and nanotubes butthey give ideas for experimental selective production of known and not yet knowncarbon materials

MOP-4.2.2/A-11/1/KONV-2012-0029 project.

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Omega Polynomial in Nanostructures

Mircea V Diudea and Beata Szefler

Abstract Omega polynomial, developed in 2006 in Cluj, Romania, counts thenumber of topologically parallel edges in all the opposite edge stripes of aconnected graph Definitions and relations with other polynomials and well-known topological indices are given Within this chapter, omega polynomial iscomputed in several 3D nanostructures and crystal networks, and analytical for-mulas as well as examples are given This polynomial is viewed as an alternative tothe crystallographic description

is not transitive; a small example in this respect is the complete bipartite graphK2,3

A graph is called aco-graph if the relation co is also transitive and relation co is anequivalence relation

For an edge e2 E Gð Þ, let c eð Þ:¼ f 2E Gf ð Þ; f co eg be the set of edgesco-distant toe in G; the set c(e) is called an orthogonal cut of G, with respect to

e If G is a co-graph then its orthogonal cuts C Gð Þ ¼ c1, c2,::, ckform a partition:

E Gð Þ ¼ c1[ c2[ [ ck,ci\ cj¼∅, i 6¼ j

A subgraphH G is called isometric with G if dHðu; vÞ ¼ dGðu; vÞ, for any u; vð Þ

2 H; it is convex if any shortest path in G between vertices of H belongs to H The cubeQnis the graph whose vertices are all binary strings of lengthn, two stringsbeing adjacent if they differ in exactly one position (Harary1969) The distancefunction in the n-cube is just the Hamming distance: the distance between twovertices ofQnequals the number of positions in which they differ A graphG iscalled apartial cube if there exists an integer n such that G is an isometric subgraph

A graphG is bipartite if and only if, for any edge of G, the opposite semicubesdefine a partition ofG: nuvþ nvu¼ v ¼ V Gj ð Þj The relation co is related to the ~(Djokovic´1973) andΘ (Winkler1984) relations (Klavžar2008a,b):

e Θ f , d u; xð Þ þ d v; yð Þ 6¼ d u; yð Þ þ d v; xð Þ ð2:2Þ

In a connected graph, two edges ofG are in the Djokovic´ relation e ~ f if f joins avertex innuvwith a vertex innvu For more information about the relation ~, see refs(Wilkeit1990; Bresˇar2001)

Lemma 2.1.1 In any connected graph,co¼ ~

Proof Lete¼ (uv) and f ¼ (xy) be two edges of a connected graph G Supposefirst e co f, that is, d v; xð Þ ¼ d v; yð Þ þ 1 ¼ d u; xð Þ þ 1 ¼ d u; yð Þ Since

d x; uð Þ< d x; vð Þ, x 2 nuv, and since d y; vð Þ< d y; uð Þ, y 2 nvu, thus, e ~ f Suppose

e ~ f, with x2 nuv and y2 nvu It follows that d v; xð Þ ¼ d v; uð Þ þ d u; xð Þ ¼ d u; xð Þþ1 and d u; yð Þ ¼ d u; vð Þ þ d v; yð Þ ¼ 1 þ d v; yð Þ

Sinced u; xð Þ ¼ d v; yð Þ, we conclude that e co f, q.e.d

In general graphse  Θand in bipartite graphs e ¼ Θ From this and the abovelemma, it follows (Diudea and Klavžar2010)

Theorem 2.1.2 In a bipartite graph, the following statements are equivalent:(Diudea and Klavžar2010)

(i)G is a co-graph; (ii) G is a partial cube; (iii) all semicubes of G are convex;(iv) relationΘ is transitive

Equivalence between (i) and (ii) was observed in Klavžar (2008a,b), lence between (ii) and (iii) is due to Djokovic´ (1973), while the equivalencebetween (ii) and (iv) was proven by Winkler (1984).

Two edgese and f of a plane graph G are in relation opposite, e op f, if they areopposite edges of an inner face ofG Then e co f holds by assuming the faces areisometric Note that relationco involves distances in the whole graph, while op isdefined only locally (it relates face-opposite edges) A partial cube is also aco-graph but the reciprocal is not always true There are co-graphs which arenon-bipartite (Diudea 2010a, b, c), thus being non-partial cubes (see examplesbelow)

Using the relationop, we can partition the edge set of G into opposite edge stripsops: any two subsequent edges of an ops are in op-relation and any three subsequentedges of such a strip belong to adjacent faces Also note that John et al (2007a,b)implicitly used the “op” relation in defining the Cluj-Ilmenau index CI

Lemma 2.1.3 IfG is a co-graph, then its opposite edge strips ops {sk} superimposeover the orthogonal cutsocs {ck}

Proof Let us remind theco-relation is defined on parallel equidistant edge relation(2.1) The same is true for theop-relation, with the only difference (2.1) limited to asingle face Supposee1, e2are two consecutive edges ofops; by definition, they aretopologically parallel and alsoco-distant (i.e., belong to ocs) By induction, anynewly added edge ofops will be parallel to the previous one and also co-distant.Because, in co-graphs, co-relation is transitive, all the edges of ops will be co-distant, and thusops and ocs will coincide, q.e.d

Corollary In a co-graph, all the edges belonging to an ops are topologicallyparallel

Observe the relationco is a particular case of the edge equidistant eqd relation.The equidistance of two edgese¼ (uv) and f ¼ (xy) of a connected graph G includesconditions for both (i) topologically parallel edges (relation (2.1)) and(ii) topologically perpendicular edges (in the tetrahedron and its extensions –relation (2.3)) (Diudea et al.2008; Ashrafi et al.2008a,b):

e eqd f iið Þ , d u; xð Þ ¼ d u; yð Þ ¼ d v; xð Þ ¼ d v; yð Þ ð2:3ÞRecall that a graph isplanar if it allows an embedding into the plane such that notwo edges cross A planar graph together with its fixed embedding into the plane iscalled aplane graph In chemistry, not only the structure of a chemical graph butalso its geometry is important Most of the chemical graphs are by their natureplanar

Theop-strips can be either cycles (if they start/end in the edges eevenof the sameeven facefeven) or paths (if they start/end in the edgeseoddof the same or differentodd facesfodd) (Diudea and Ilic´2009) In a planar bipartite graph, representing apolyhedron, allop-strips are cycles.

Theops is maximum possible, irrespective of the starting edge The choice isabout the maximum size of face/ring searched, and mode of face/ring counting,which will decide the length of the strip

Lets1, s2, ,skbe theop-strips in a connected graph G; the ops form a partition ofE(G) Define the Omega polynomialΩ(x) as (Diudea2006)

The first derivative (computed atx¼ 1) P0(G,1) of these counting polynomials givesinformation on the counted topological property (Diudea2010a,b,c):

Θ G; 1ð Þ ¼Π G; 1ð Þ ¼Ω0

G; 1

ð Þ ¼E Gð Þ ð2:15ÞThere are graphs with singleops, of length s¼ e ¼ E Gj ð Þj, which is precisely acycle (Diudea and Ilic´2009) Also, there are graphs withs¼ 1, namely, graphs witheither odd rings or with no rings, i.e., tree graphs For such graphs, minimal andmaximal value, respectively, ofCI is calculated (Diudea2010a,b,c):

Ω xð ÞCImin¼ 1  xe; CImin¼ e2 e þ e e  1ð ð ÞÞ ¼ 0 ð2:16Þ

Ω xð ÞCImax¼ e  x1; CImax¼ e2 e þ 0ð Þ ¼ e e  1ð Þ ð2:17ÞAmong the graphs onv vertices, CIminis provided by the complete bipartitegraphsK2,v2(withe¼ 2(v2)), while CImaxis given by the complete graphsKv

(withe¼ v(v1)/2):

CImin ¼ CI Kð 2 , v2Þ ¼ 0 ð2:18Þ

CImax¼ CI Kð vÞ ¼ 1=4ð Þv v  1ð Þ v 2 v  2

ð2:19ÞThe first derivative (inx¼ 1) of Sadhana polynomial equals the Sadhana index(Khadikar et al.2002) and is a multiple of |E(G)|:

et al.2010a,b):

Sd0ðG; 1Þ ¼Ω0

G; 1

ð ÞðΩ G; 1ð Þ  1Þ ð2:21Þ

In words, the first derivative of Sadhana polynomial (inx¼ 1) is the product

of the number of edges e¼ |E(G)| and the number of strips Ω(1) less one Inco-graphs, there is the equality (Diudea and Klavžar 2010; Diudea 2010a,b,c;Djokovic´1973)

CI Gð Þ ¼Π Gð Þ It comes out from their definitions:

si

j j þXk i¼1

si

j j 6¼ cj j, and as a consequence, CI Gk ð Þ 6¼Π Gð Þ

The above equality between the two indices can be considered as a necessarycondition for theco-graph/partial cube status This is, however, not sufficient, andfinally, the transitivity ofocs/ops must be proven

Now consider relations (2.9,2.10, and2.11) in reformulating (2.22) as

distance between the given point and the two end points of that edge (Ashrafi

et al.2006,2008a,b):

d z; eð Þ ¼ min d z; uf ð Þ, d z; vð Þg ð2:24ÞThen, for two edgese¼ (uv) and f ¼ (xy) of G,

e eqd f iiið Þ , d x; eð Þ ¼ d y; eð Þ and d u; fð Þ ¼ d v; fð Þ ð2:25Þ

In bipartite graphs, relations (2.1) and (2.3) superimpose over relations (2.24)and (2.25); in such graphs, the following equality holdsΠ Gð Þ ¼ PIeð Þ In generalGgraphs, this is, however, not true

The problem of equidistance of vertices was firstly put by Gutman whendefining the Szeged index (Gutman1994) Sz(G) of which calculation leaves outthe equidistant vertices Similarly, the index PIe(G) does not account for theequidistant edges According to Ashrafi’s notations (Ashrafi et al.2006), it can bewritten as

where n(e, u) is the number of edges lying closer to the vertex u than to the

v vertex and m(u, v) is the number of equidistant edges from u and v This indexcan be calculated as the first derivative, in x¼ 1, of the polynomial defined byAshrafi as

et al.2003)

The number of structures investigated for transitivity can be reduced by using anorthogonal edge-cutting procedure (see below).

In bipartite graphs, an orthogonal edge-cutting procedure (Diudea 2010a, b, c;Diudea et al.2010a,b; Gutman and Klavžar1995; Klavžar2008a,b) can be used

to generate theops In performing a cut, take a straight line segment, orthogonal tothe edgee, and intersect e and all its parallel edges (in a polygonal plane graph) Theset of these intersections is called anorthogonal cut, with respect to e (Fig.2.1) Toany orthogonal cut ck, two numbers are here associated: (i) the number ofintersected edges ek (or the cutting cardinality |ck|) and (ii) the number of points

vk, lying to the left hand with respect tock(in round brackets, in Fig.2.1) Ingraphs, the exponents in Omega and Theta polynomials are identical, while inΠ(x),they result as the difference betweene¼ |E(G)| and the Omega skexponents Sincethe polyhexes are bipartite graphs (and partial cubes), the triple equalityCI Gð Þ ¼ΠG

co-ð Þ ¼ PIeð Þ holds (DiudeaG 2010a,b,c)

co-2.2.3.1 Tree Graphs

Tree graphs are bipartite and partial cubes but non-co-graphs; Omega polynomialsimply counts the non-equidistant edges or self-equidistant ones, being included inthe term of exponent s¼ 1 In such graphs, CI Gð Þ ¼Π Gð Þ ¼ v  1ð Þ v  2ð Þ(a result known from Khadikar) (Diudea et al 2006) and the Omega and Thetapolynomials show the same expression

(a)a¼ 4; n ¼ odd; bipartite, co-graphs but non-partial cubes

(b) a¼ 4; n ¼ even; bipartite; non-co-graphs and non-partial cubes

(c)a> 4; a ¼ odd; n-all; non-bipartite, non-partial cubes but co-graphs

(d) a> 4; a ¼ even; n-all; bipartite, partial cubes and co-graphs

2.2.3.4 Toroidal Graphs

There are distinct (4,4) tori (Table2.3), which show both degenerateΩ(x) mial and indexCI(G) (rows 1 and 2, in italics), for whichΠ(x) and Θ(x) are distinct(Diudea et al 2008; Diudea 2010a, b, c) Next, there are (4,4) tori which showdegenerateΠ(G) and Θ(G) index values but distinct Π(x) and Θ(x) polynomials(as the tori in Table2.3, rows 1 and 3) The above tori all show CI Gð Þ 6¼Π Gð Þ.Finally, there are excepted (4,4) tori which show CI Gð Þ ¼Π Gð Þ (e.g., the torusT(4,4)[4,4] or that in the fourth row of Table2.3; see below)

polyno-According to the parity of net parameters, non-twisted tori T(4,4)[c,n] are(Diudea2010a,b,c):

(a)c,n-all odd; non-bipartite, non-co-graphs and non-partial cubes

(b) c-odd/even; n-even/odd; non-bipartite; non-partial cubes but co-graphs(c)c,n-all even; bipartite, co-graphs and partial cubes

The only torus which shows CI Gð Þ ¼Π Gð Þ is the simplest T(4,4)[4,4]: Ω xð Þ

¼ 4x8;CI Gð Þ ¼ 768;Π xð Þ ¼ 32x24;Π Gð Þ ¼ 768;Θ xð Þ ¼ 32x8; Θ Gð Þ ¼ 256.The example in Table2.3, entry 4, is a twisted (4,4) torus (Fig.2.2):

It is a 3D M€obius structure, which is a union of 3 co-graphs (for which theidentityf g csk f g is true) and the equality CI Gk ð Þ ¼Π Gð Þ holds (Diudea2010a,b,

c) However, per global, it is a non-bipartite graph, which is shown in

PIvð Þ ¼ 120xx 54, with the exponent lower than the number of vertices (54< 60),the difference being equidistant vertices, nonexisting in bipartite graphs

On the other hand, the Cluj polynomial shows a single term, as in bipartite tori It

is the only found non-bipartite graph which shows the equality Π Gð Þ ¼ PIeð ÞG(usually holding in bipartite graphs)

Tori of other tessellation ((6,3); ((4,8)3), ((5,7)3), etc.) all showCI Gð Þ 6¼Π Gð Þand only exceptionally this relation becomes equality (see Table2.4); all these toriare non-co-graphs and non-partial cubes

The third torus in Fig.2.3and Table2.4is a non-bipartite torus, covered by oddfaces, in a ((5,7)3) tessellation It is included here to illustrate the triple inequality

CI Gð Þ 6¼Π Gð Þ 6¼ PIeð Þ and an extreme case ofG Ω(x), with a single term, and ofexponent 1 (see relation (2.17)) (Diudea2010a,b,c)

2.2.3.5 Cubic Net and Corresponding Cage

The cubic net in Fig.2.4left is bipartite and for sure partial cube and co-graph; itshows the triple equalityCI Gð Þ ¼Π Gð Þ ¼ PIeð Þ Its corresponding cage (Fig.G 2.4,

Table 2.3 Polynomials in (4,4) tori: bipartite graphs for which CI G ð Þ 6¼ Π G ð Þ (rows 1 to 3) and a non-bipartite graph showing CI G ð Þ ¼ Π G ð Þ (row 4); all tori are non-partial cubes

Fig 2.2 Twisted torus TWV3[6,10](4,4); non-bipartite (left) and as the union of 3 co-graphs, one

of them being colored in alternating white red (right); non-partial cube

to each other despite both belonging to the sameops); the graph is non-co-graph andnon-partial cube and showsCI Gð Þ 6¼Π Gð Þ (Diudea2010a,b,c, d).

Numerical data were computed by our software program Nano Studio (Nagy andDiudea2009)

In conclusion, the only clear answer to the question “whenCI Gð Þ ¼Π Gð Þ?” iswhen sk¼ck, which can happen in any graph, irrespective if it is co-graphand/or partial cube In plane, bipartite graphs, which are alsoco-graphs and partialcubes, the above condition is achieved immediately In 3D structures,non-isometric subgraphs often appear, this turning the above equality into aninequality Despiteco-graph/partial cube properties being decisive in planar graphs,

in 3D structures, they seem not so much involved Symmetry plays an importantrole in achieving the above equality Reformulating “which kind of graph shows theabove equality?” will remain as an open question

Fig 2.3 Tori of different tessellation; all are non-co-graphs and non-partial cubes

9

( )x 6x ; (1) 54

Ω = Ω = ; CI =2430; (R[4])

9( )x 54x

e

PI x = x ;PI e(1)=2430

8( )x 6x ; ′(1) 48

Fig 2.4 Cubic net (left) and its corresponding cage (right)