5 Shape Analysis of Carbon Nanotubes, Nanotori and Nanotube Junctions
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106
I. La´szlo´ et al.
Fig. 6.5 Drawing of nanotube using three bi-lobal eigenvectors of the adjacency matrix (a) and of
the Laplacian (b)
Fig. 6.6 Top and side view of nanotube junctions obtained by three bi-lobal eigenvectors of the
adjacency matrix (a) and of the Laplacian (b)
graphs describing nanotubes nanotori and nanotube junctions. In each cases we
can note some kind of flattening. The nanotube ends are spiked and they are
turning back in the case of adjacency matrices. This phenomenon shows that for
non-regular graphs the eigenvectors of the Laplacian matrix give better drawings
than those of the adjacency matrix.
Let us suppose that we have calculated the Descartes coordinates ðxi ; yi ; zi Þ of the
atoms in a non-spherical structure of n atoms. Thus X, Y and Z are n-dimensional
vectors containing the x, y and z coordinates of the atoms in order. Let us suppose
further that the centre of mass of the molecule is in the origin of the coordinate
system and the eigenvectors of its tensor of inertia are showing to the direction of
the x, y and z axis.
6 Graph Drawing with Eigenvectors
107
With the help of the following scalar products
aXk ¼ Xck ; aYk ¼ Yck and aZk ¼ Zck
(6.34)
the atomic coordinates can be written as
X¼
n
X
aXk ck ; Y ¼
k¼1
n
X
aYk ck and Z ¼
k¼1
n
X
aZk ck :
(6.35)
k¼1
Here ck is the eigenvector of the Laplacian L and the corresponding eigenvalues lk
are ordered in increasing order. We say that the weights of the eigenvector ck in X,
Y and Z are in order jaXk j2 , jaYk j2 and jaZk j2 .
The measure of the convergence using only m n terms in the summation can
be described with the following notations:
mị
Xi
ẳ
m
X
kẳ1
mị
aXk cki ; Yi
ẳ
m
X
mị
aYk cki and Zi
ẳ
kẳ1
m
X
aZk cki
(6.36)
kẳ1
and
R ẳ X; Y; Zị; Rmị ẳ Xmị ; Ymị ; Zmị :
(6.37)
The convergence of the structure is quantified as follows:
n
2
X
ðmÞ
X À XðmÞ ẳ 1
X i Xi
n iẳ1
!12
n
2
X
mị
Y Ymị ẳ 1
Yi Yi
n iẳ1
n
2
X
mị
Z Zmị ¼ 1
Zi À Zi
n i¼1
(6.38)
!12
(6.39)
!12
n
2
2
2 12
X
ðmÞ
ðmÞ
ðmÞ
R À Rmị ẳ 1
Xi X i
ỵ Y Yi
ỵ Z À Zi
n i¼1
(6.40)
(6.41)
In Graovac et al. (2008b) we obtained that only few of the coefficients jaXk j2 , jaYk j2
and jaZk j2 are significantly greater than the others. Thus we made such kind of
summations similar to the Eq. 6.36 which contained only the significant jaXk j2 ,
jaYk j2 and jaZk j2 terms. In these summations we have found that three bi-lobal
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I. La´szlo´ et al.
Fig. 6.7 Drawing of
nanotorus using ten
eigenvectors of the Laplacian.
Top view and side view
eigenvectors of the Laplacian reproduce well the nanotubes without any spiky
phenomenon at the ends (Fig. 6.5b). For nanotori we had to use 8–10 eigenvectors
of the Laplacian and most of them were bi-lobal. One representative structure is
in the Fig. 6.7.
The details of our nanotube junction analysis can be found in Graovac et al.
(2008a).
We have found that the greatest absolute values of aX2 , aY3 and aZ7 are attributed
to the three bi-lobal eigenvectors . If the number of eigenfunctions (m) in Eq. 6.36 is
smaller than 7 the picture of the structure can be described as a planar or a curved
two dimensional surface (Fig. 6.8). The eigenvectors c4 , c5 and c6 are 4, 4 and
5 –lobal but they have relatively small weight in Z. Although the eigenvectors
c8 , c9 , c10 , c11 and c12 have relatively small weight in Z and their lobality is from 3
to 5 they are important in eliminating the spiky features at the tube ends (Fig. 6.8).
We can obtain no flattened structure only if m is greater than the serial number of
any bi-lobal eigenfunction. The weight of the eigenvectors c13 and c14 is also
significant in the vector Z. If m is greater than 16, practically there are no
appreciated changing in the picture of the structure.
In Graovac et al. (2008a) we have investigated altogether 11 junctions and
obtained three-dimensional structures only if all the three bi-lobal eigenfunctions
were included in the summations. We had to take into consideration some other
eigenvectors for eliminating the spiky behaviours at the end of the tubes. We have
found in each case a value for m which gave satisfactory coordinates that can
be used for initial coordinates in molecular mechanics calculations.
6 Graph Drawing with Eigenvectors
109
Fig. 6.8 Shape analysis of a nanotube junction. Top and side view of the structure obtained by
3 (a), 9 (b) and 16 (c) eigenvectors of the Laplacian
6.6
6.6.1
Toward Topological Coordinates of any
Molecular Arrangement
Embedding with Three Eigenvectors into R3
In the previous paragraph we have seen that the applicability of three eigenvectors
is restricted only to spherical structures and nanotubes. Here we shall show that
there exist such kind of matrix or matrices which can reproduce practically exactly
the ðxi ; yi ; zi Þ Descartes-coordinates with the help of three eigenvectors. Let us
suppose that we have the exact ðxi ; yi ; zi Þ coordinates of the atoms. In this case,
as the number of atoms is n, the values of X, Y and Z are n-dimensional vectors
containing the x, y and z coordinates of the atoms in order. We suppose further,
as before that the centre of mass of the molecule is in the origin and the molecule
is directed in such a way that the eigenvectors of its tensor of inertia are showing to
the directions of the x, y and z axis. Thus we have the following relations:
n
X
xi ¼ 0;
i¼1
n
X
i¼1
yi ¼ 0;
n
X
zi ¼ 0
(6.42)
i¼1
and
XY ¼
n
X
i¼1
xi yi ¼ 0; YZ ¼
n
X
yi zi ¼ 0; ZX ¼
i¼1
n
X
z i xi ¼ 0
(6.43)
i¼1
1
If we define the vector U of n dimension with the relations ui ¼ pﬃﬃﬃ from
n
Eq. 6.42 follows that
UX ¼ 0; UY ¼ 0; UZ ¼ 0
(6.44)
I. La´szlo´ et al.
110
Equations 6.43 come from the conditions that the off-diagonal matrix elements
of the tensor of inertia are zero because of the special position of the molecule.
Let us suppose that the total energy
Erị ẳ E r12 ; r21 ; :::rij ; rji :::
(6.45)
depends only on the inter-atomic distances
rij ẳ
xi xj
2
2
2 12
ỵ yi yj ỵ zi À zj
(6.46)
Here we suppose further that total energy EðrÞ depends on rij and rji in a
symmetric way. If this is not the case, we substitute rij or rji by the value
rij ẳ rji ẳ rij ỵ rij ị=2.
In the followings we shall use the relations:
drij xi À xj drij yi À yj drij zi À zj
¼
;
¼
;
¼
dxi
rij
dyi
rij
dzi
rij
(6.47)
drij xj À xi drij yj À yi drij zj À zi
¼
;
¼
;
¼
dxj
rij
dyj
rij
dzj
rij
(6.48)
À
Á
As the gradients give the forces Fi ¼ Fxi ; Fyi ; Fzi acting on the i-th atom, we have:
@Erị
@Erị
@Erị
ẳ Fxi ;
ẳ Fyi ;
ẳ ÀFzi
@xi
@yi
@zi
(6.49)
Applying the above mentioned relations we obtain.
n
@E X
¼
wij xj ¼ Fxi
@xi
jẳ1
(6.50)
n
@Erị X
ẳ
wij yj ẳ Fyi
@yi
jẳ1
(6.51)
n
@E X
ẳ
wij zj ẳ Fzi
@zi
jẳ1
(6.52)
@Erị @Erị
rij @rij rji @rji
(6.53)
where
wij ¼ À
6 Graph Drawing with Eigenvectors
111
and
wii ¼
n
X
@Erị
j6ẳi
@Erị
ỵ
rij @rij rji @rji
ẳ
n
X
wij
(6.54)
j6ẳi
That is
WX ẳ Fx ; WY ẳ ÀFy WZ ¼ ÀFz
(6.55)
If the atoms are in the equilibrium positions, X, Y and Z can be seen as
eigenvectors of W with zero eigenvalue, that is
WX ¼ 0; WY = 0; WZ = 0
(6.56)
From the relations of Eqs. 6.53 and 6.54 follows that
WU ¼ 0
(6.57)
1
with ui ¼ pﬃﬃﬃ .
n
If the centre of mass of the molecule is in the origin and the molecule is directed
in such a way that the eigenvectors of its tensor of inertia are showing to the
directions of the x, y and z axis, from the Eqs. 6.43, 6.44, 6.56, 6.57 follows that X,
Y, Z and U are orthogonal eigenvectors of the matrix W. That is
X ¼ Sx Cx ; Y ¼ Sy Cy ; Z ¼ Sz Cz
(6.58)
where Cx , Cy , Cz and U are orthogonal and normalized eigenvectors of W with zero
eigenvalue and Sx , Sy and Sz are appropriate scaling factors.
The question arises if we have any orthogonal and normalized eigenvectors Ax ,
y
A , Az and U of W with zero eigenvalue are there any appropriate scaling factors
Sx , Sy and Sz for obtaining the Descartes coordinates with a relation,
X ¼ Sx Ax ; Y ¼ Sy Ay ; Z ¼ Sz Az
(6.59)
If the number of eigenvectors with zero eigenvalue is four, the answer is yes, but
in Eq. 6.59 we obtain a rotation of the molecule as the vectors Ax , Ay , Az can be
obtained as linear combination of the vectors Cx , Cy , Cz . If the vectors Cx , Cy , Cz
are mixed with the vector U it means arbitrary translation and a rotation of the
molecule. As the vector U is known it can be easily subtracted from the linear
combinations in the case of mixings.
Usually the first neighbour distances in a molecule do not determine the
positions of the atoms but the full structure can be describe if we know the second
neighbour distances as well. From this follows that if the edges of a molecular graph
G ẳ V; Eịcorrespond to the first and second neighbours of a molecule, the
matrix W can be generated from a total energy EðrÞ which depends only on the
I. La´szlo´ et al.
112
Fig. 6.9 Coordinates of a
nanotube junction of the tubes
(9,6), (8,7) and (10,5)
obtained by three
eigenvectors of the matrix W
(top view and side view)
first and second neighbors of the molecule. If the dimension of the null space of W
is four than this null space contains three eigenvectors which give a proper
embedding of G ẳ V; Eị into R3 .
6.6.2
Examples for Embedding with Three Eigenvectors into R3
We tested our ideas for several structures. Here we present our results obtained for
nanotube junctions nano tori and helical nanotubes. We calculated the interatomic
interactions with the help of the Brenner potential (Brenner 1990) and harmonic
potentials as well. In the Brenner potential there are first neighbour and second
neighbour interactions.
The matrix W has non-zero matrix elements only for the first and the second
neighbours. We calculated the equilibrium position of the carbon atoms in a nanotube junction, in a torus and in a helical structure. In each cases we could reconstruct
the original coordinates with the help of the relations of Eq. 6.59, see Figs. 6.9–6.11.
In each cases we used Sx ¼ Sy ¼ Sz . The final values for the parameters Sx , Sy and
Sz can be obtained from scaling three independent distance to given values.
6 Graph Drawing with Eigenvectors
113
Fig. 6.10 Coordinates of a
torus obtained by three
eigenvectors of the matrix W
(top view and side view)
6.7
Conclusions
We have examined the possibilities of drawing graphs with eigenvectors of the
adjacency matrix A the Laplacian L and the Colin de Verdie`re matrix M. We have
suggested the matrix W for drawing not only spherical graphs with eigenvectors.
For these symmetric matrices the off-diagonal matrix element of ðu; vÞ 2
= E are
equals to zero. The absolute value of other off-diagonal matrix element for
matrices A and L are equals to 1. In matrices L and W the diagonal matrix elements
are calculated in such a way that the sum of matrix elements be equal to zero in each
row. The off-diagonal matrix elements of M and W are determined using special
kind of conditions. We have shown that in each cases where the adjacency matrix
was applicable the Laplacian matrix gave good drawings as well. We have found
also that the Laplacian could be used even for nanotubes where the adjacency matrix
was not applicable. It was demonstrated that three eigenvectors of the matrix W
produced good drawing in each cases under study in this paper. The drawback of W
is that at present there is not a simple algorithm for its construction. In this work we
could generate it using an energy minimalization algorithm with the help of a
114
I. La´szlo´ et al.
Fig. 6.11 Coordinates of a
helix obtained by three
eigenvectors of the matrix W
(top view and side view)
conjugate gradient algorithm. There is a hope, however, that using appropriate
approximations for the matrix elements of W a method can be found for constructing
topological coordinates of complicated non-spherical structures as well.
Acknowledgements I. La´szlo´ thanks for the supports of grants TAMOP-4.2.1/B-09/1/KONV2010-0003, TAMOP-4.2.1/B-09/1/KMR-2010-0002 and for the support obtained in the frame
work of bi-lateral agreement between the Croatian Academy of Science and Art and the Hungarian
Academy of Sciences.
References
Biyikoglu T, Hordijk W, Leydold J, Pisanski T, Stadler PF (2004) Linear Algebra Appl
390:155–174
Biyikoglu T, Leydold J, Stadler PF (2007) Laplacian eigenvectors of graphs. Perron-Frobenius and
Faber-Krahn type theorems, LNM1915. Springer, Berlin/Heidelberg
Brenner DW (1990) Phys Rev B 42:9458–9471
6 Graph Drawing with Eigenvectors
115
Colin de Verdie`re Y (1998) Spectres de graphes, cours spe´cialise´s 4. Socie´te´ Mathe´matique de
France, Paris
Di Battista G, Eades P, Tamassia R, Tollis IG (1999) Graph drawing: algorithms for the visualization of graphs. Prentice Hall, Upper Saddle River
Dresselhaus MS, Dresselhaus G, Eklund PC (1996) Science of fullerenes and carbon anotubes:
their properties and applications. Academic, New York/London
Fowler PW, Manolopulos DE (1995) An atlas of fullerenes. Clarendon, Oxford
Fowler PW, Pisanski T, Shaw-Taylor J (1995) In Tamassia R, Tollis EG (eds) Graph drawing.
Lecture notes in computer science, vol 894. Springer-Verlag, Berlin
Godsil CD, Royle GF (2001) Algebraic graph theory. Springer, Heidelberg
Graovac A, Plavsˇic´ D, Kaufman M, Pisanski T, Kirby EC (2000) J Chem Phys 113:1925–1931
Graovac A, La´szlo I, Plavsˇic´ D, Pisanski T (2008a) MATCH Commun Math Comput Chem
60:917–926
Graovac A, La´szlo´ I, Pisanski T, Plavsˇic´ D (2008b) Int J Chem Model 1:355–362
Hall KM (1970) Manage Sci 17:219–229
Kaufmann M, Wagner D (eds) (2001) Drawing graphs. Methods and models, LNCS 2025.
Springer-Verlag, Germany
Koren Y (2005) Comput Math Appl 49:1867–1888
La´szlo´ I (2004a) Carbon 42:983–986
La´szlo´ I (2004b) In Buzaneva E, Scharff P (eds) Frontiers of multifunctional integrated
nanosystems. NATO science series, II. Mathematics, physics and chemistry. Kluwer Academic
Publishers, Dordrecht, Boston, London. Vol 152, 11
La´szlo´ I (2005) In: Diudea MV (ed) Nanostructures: novel architecture. Nova Science, New York,
pp 193–202
La´szlo´ I (2008) In: Blank V, Kulnitskiy B (eds) Carbon nanotubes and related structures. Research
Singpost, Kerala, pp 121–146
La´szlo´ I, Rassat A (2003) J Chem Inf Comput Sci 43:519–524
La´szlo´ I, Rassat A, Fowler PW, Graovac A (2001) Chem Phys Lett 342:369–374
Lova´sz L, Schrijver A (1999) Ann Inst Fourier (Grenoble) 49:1017–1026
Lova´sz L, Vesztergombi K (1999) In Hala´sz L, Lova´sz L, Simonovits M, T So´s V (eds) Paul Erdo˝s
and his mathematics. Bolyai Society – Springer Verlag. Berlin, Heidelberg, New York
Manolopoulos DE, Fowler PW (1992) J Chem Phys 96:7603–7614
Pisanski T, Shawe-Taylor JS (1993) In Technical report CSD-TR-93-20, Royal Holloway,
University of London, Department of Computer Science, Egham, Surrey TW200EX, England
Pisanski T, Shawe-Taylor JS (2000) J Chem Inf Comput Sci 40:567–571
Pisanski T, Plestenjak B, Graovac A (1995) Croat Chim Acta 68:283–292
Rassat A, La´szlo´ I, Fowler PW (2003) Chem Eur J 9:644–650
Stone AJ (1981) Inorg Chem 20:563–571
Trinajstic´ N (1992) Chemical graph theory. CRC Press/ Boca Raton/ Ann Arbor, London/Tokyo
Tutte WT (1963) Proc Lond Math Soc 13:743–768
van der Holst H (1996) Topological and spectral graph characterizations. Ph.D. Thesis, University
of Amsterdam, Amsterdam