Nanoscale Res Lett (2007) 2:202–206 DOI 10.1007/s11671-007-9051-y
NANO EXPRESS
A new algorithm for computing distance matrix and Wiener index of zig-zag polyhex nanotubes Ali Reza Ashrafi Æ Shahram Yousefi
Received: 31 December 2006 / Accepted: 20 February 2007 / Published online: 10 April 2007 to the authors 2007
Abstract The Wiener index of a graph G is defined as the sum of all distances between distinct vertices of G. In this paper an algorithm for constructing distance matrix of a zig-zag polyhex nanotube is introduced. As a consequence, the Wiener index of this nanotube is computed. Keywords Zig-zag polyhex nanotube Distance matrix Wiener index
Introduction Carbon nanotubes form an interesting class of carbon nanomaterials. These can be imagined as rolled sheets of graphite about different axes. These are three types of nanotubes: armchair, chiral and zigzag structures. Further nanotubes can be categorized as single-walled and multiwalled nanotubes and it is very difficult to produce the former. Graph theory has found considerable use in chemistry, particularly in modeling chemical structure. Graph theory has provided the chemist with a variety of very useful tools, namely, the topological index. A topological index is a numeric quantity that is mathematically derived in a direct and unambiguous manner from the structural graph of a molecule. It has been found that many properties of a
A. R. Ashrafi (&) Institute for Nanoscience and Nanotechnology, University of Kashan, Kashan, Iran e-mail:
[email protected] S. Yousefi Center for Space Studies, Malek-Ashtar University of Technology, Tehran, Iran
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chemical compound are closely related to some topological indices of its molecular graph [1, 2]. Among topological indices, the Wiener index [3] is probably the most important one. This index was introduced by the chemist H. Wiener, about 60 years ago to demonstrate correlations between physico-chemical properties of organic compounds and the topological structure of their molecular graphs. Wiener defined his index as the sum of distances between two carbon atoms in the molecules, in terms of carbon–carbon bonds. Next Hosoya named such graph invariants, topological index [4]. We encourage the reader to consult Refs. [5–7] and references therein, for further study on the topic. The fact that there are good correlations between and a variety of physico-chemical properties of chemical compounds containing boiling point, heat of evaporation, heat of formation, chromatographic retention times, surface tension, vapor pressure and partition coefficients could be rationalized by the assumption that Wiener index is roughly proportional to the van der Waals surface area of the respective molecule [8]. Diudea was the first chemist which considered the problem of computing topological indices of nanostructures [9–15]. The presented authors computed the Wiener index of a polyhex and TUC4C8(R/S) nanotori [16–18]. In this paper, we continue this program to find an algorithm for computing distance matrix of a zig-zag polyhex nanotube. As an easy consequence, the Wiener index of this nanotube is computed. John and Diudea [9] computed the Wiener index of zigzag polyhex nanotube T = T(p, q) = TUHC6[2p,q], for the first times. In this paper, distance matrix of these nanotubes are computed. As an easy consequence of our results, a matrix method for computing the Wiener index of a zig-zag polyhex nanotube is introduced. We also prepare an
Nanoscale Res Lett (2007) 2:202–206
203
algorithm for computing distance matrix of these nanotubes. Throughout this paper, our notation is standard. They are appearing as in the same way as in the following [2, 19].
(1 ,1 )
(a)
x 1 ,3
B ase
x
Main results and discussion In this section, distance matrix and Wiener index of the graph T = TUHC6[m,n], Fig. 1, were computed. Here m is the number of horizontal zig-zags and n is the number of columns. It is obvious that n is even and |V(T)| = mn.
(1 ,1 ) 2 ,2
(b) B ase
An algorithm for constructing distance matrix of TUHC6[m,n] We first choose a base vertex b from the 2-dimensional lattice of T and assume that xij is the (i,j)th vertex of T, Fig. 2. ð1;1Þ ð1;1Þ ð1;1Þ Define Dmn ¼ ½di;j ; where di;j is distance between (1,1) and (i,j), i = 1, 2, ..., m and j = 1, 2, ..., n. By Fig. 2, there are two separates cases for the (1,1)th vertex. For example in the ð1;1Þ ð1;1Þ ð1;1Þ case (a) of Fig. 2, d1;1 ¼ 0; d1;2 ¼ d2;1 ¼ 1 and in case ð1;1Þ ð1;1Þ ð1;1Þ (b), d1;1 ¼ 0; d1;2 ¼ 1; d2;1 ¼ 3: In general, we assume ðp;qÞ that Dmn is distance matrix of T related to the vertex (p,q) ðp;qÞ ðp;qÞ and si is the sum of ith row of Dmn : Then there are two ðp;2k1Þ ðp;1Þ distance matrix related to (p,q) such that si ¼ si ; ðp;2kÞ ðp;2Þ si ¼ si ; 1 k n=2; 1 i m; 1 p m: By Fig. 2 and previous notations, if b varies on a column of T then the sum of entries in the row containing base vertex is equal to the sum of entries in the first row of ð1;1Þ Dmn : On the other hand, one can compute the sum of entries in other rows by distance from the position of base vertex. Therefore, ( ði;jÞ sk
¼ (
ði;jÞ sk
¼
ð1;1Þ
We now describe our algorithm to compute distance matrix of a zig-zag polyhex nanotube. To do this, we define ðaÞ and matrices Amðn=2þ1Þ ¼ ½aij ; Bmðn=2þ1Þ ¼ ½bij ðbÞ Amðn=2þ1Þ ¼ ½cij as follows: a1,1= 0 a1,2 = 1 ai;j ¼ c1,1= 0 c1,2 = 1 ci;j ¼ bi,1 = i–1;
sikþ1 ð1;2Þ skiþ1
1 k i m; 1 j n If 1 i k m; 1 j n
2j(i+j)
ð1;2Þ
1 k i m; 1 j n If 1 i k m; 1 j n
2-(i + j)
sikþ1 ð1;1Þ skiþ1
Fig. 2 Two basically different cases for the vertex b
ai;1 ai;2
2-j ; 2jj
ai;1 ¼ ai1;1 þ 1; ai;2 ¼ ai;1 þ 1; ai;2 ¼ ai1;2 þ 1; ai;1 ¼ ai;2 þ 1;
2ji 2-i
ci;1 ci;2
2-j ; 2jj
ci;2 ¼ ci1;2 þ 1; ci;1 ¼ ci;2 þ 1; ci;1 ¼ ci1;1 þ 1; ci;2 ¼ ci;1 þ 1;
2ji 2-i
bi,j = bi,j–1+1
For computing distance matrix of this nanotube we must ðaÞ ðbÞ compute matrices Dmn ¼ ½dai;j and Dmn ¼ ½dbi;j : But by our calculations, we can see that dai;j
¼
dbi;j ¼
Maxfai;j ; bi;j g
1 j n=2
di;njþ2 j[n=2 þ 1 Maxfai;j ; ci;j g 1 j n=2 di;njþ2
and
j[n=2 þ 1
This completes calculation of distance matrix. Computing Wiener index of TUHC6[m,n] ðp;qÞ
Fig. 1 The zig-zag polyhex nanotube TUHC6[20,n]
In previous section, distance matrix Dmn related to vertex ðp;qÞ (p,q) is computed. Suppose si is the sum of ith row of ðp;qÞ ðp;2k1Þ ðp;1Þ ðp;2kÞ ðp;2Þ Dmn : Then si ¼ si and si ¼ si ; where 1 k n=2; 1 i m; 1 p m: On the other hand, by our calculations in section ‘‘An algorithm for constructing distance matrix of TUHC6[m,n]’’,
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( ð1;2k1Þ si
ð1;2kÞ si
n2 þ (n þ i 2Þði 1Þ 4
¼
i n2 þ 1 inþ1
n (4i 5Þ 2 2 ( 2 n þ (n þ iÞði 1Þ i n þ 1 4 2 n (4i 3Þ inþ1
¼
2
; SðbÞ p
2
n 1k : 2
1 i m;
of dis-
(mn/4)(2mþn2Þþðm=3Þðm1Þðm2Þmn=2þ1 ; (mn/2)(2m3Þþðn=24Þðnþ2Þðnþ4Þ mn=2þ1
ðbÞ
SðbÞ p
Therefore, 2
(mn/4)(2mþn2Þþðm=3Þðm 1Þ m n=2þ1 : (mn/2)(2m1Þþðn=24Þðn2 4Þ m n=2þ1
If p is arbitrary then one can see that: ðaÞ
SðaÞ p ¼ S1 þ
p X
ð1;2Þ
si
ðbÞ
SðbÞ p ¼ S1 þ
p X i¼2
m X
ð1;1Þ
si
i¼mpþ2
i¼2
ð1;1Þ
si
m X
ð1;2Þ
si
Wmn ¼ 8 " ðm1Þ=2 > P ðaÞ ðbÞ > > Si þSi ðn=2Þ > > > i¼1 > > > # > < ðaÞ ðbÞ þð1=2Þ Sðmþ1Þ=2 þSðmþ1Þ=2 > > > > > > > m=2 > P ðaÞ ðbÞ > > Si þSi :ðn=2Þ
: 2-m 2jm
i¼1
i¼mpþ2
Thus it is enough to compute m £ n/2, one can see that: SðaÞ p
n n 2 n 4 þ (m 2pÞð2m þ 1Þ þ 2np2 12 2 n n 2 n þ 8 þ (m 2pÞð2m þ 3Þ þ 2np2 ¼ 12 2
SðaÞ p ¼
ðaÞ S1 ¼
S1 ¼
m > n + 1 and p[ n2 þ 1: In this case,
(III)
ðbÞ Suppose SðaÞ p and Sp are the sum of all entries ðp;qÞ tance matrix Dmn in two cases (a) and (b). Then
mn m (2m þ n þ 2Þ þ ðm2 1Þ 4 3 p m2 þ mn þ n þ p2 (m þ n) mn m (2m þ n þ 2Þ þ (m þ 1Þðm þ 2Þ ¼ 4 3 p m2 þ mn þ n þ 2m þ p2 (m þ n)
SðaÞ p ¼
SðaÞ p
2
and
¼(mn/4)(2m þ n þ 2Þ þ ðm=3Þ m 1 p m2 þ mn þ n þ p2 (m þ n)
SðbÞ p :
When
We now substitute the values of SðaÞ p to compute the Wiener index of T, as follows: ( Wmn ¼
mn2 ð4m2 þ 3mn 4Þ þ m2 n ðm2 1Þ m n þ 1 24 12 2 mn2 ð8m2 þ n2 6Þ n3 ðn2 4Þ m[ n þ 1 : 24 192 2
SðbÞ p ¼(mn/4)(2m þ n þ 2Þ þ ðm=3Þðm þ 1Þðm þ 2Þ p m2 þ mn þ n þ 2m þ p2 (m þ n)
Constructing distance matrices of some nanotubes
To complete our argument, we must investigate the case of m > n/2 + 1. To do this, we consider three cases that p £ n/2 + 1, p £ m – n/2 + 1; m £ n + 1, m – n/2 + 1 < p £ n/2 + 1 and m > n + 1, p > n/2 + 1. (I) p n þ 1 and p m n þ 1: In this case we have:
In this section, distance matrices of TUHC6[8,10] and TUHC6[8,16] together with their Wiener indices are computed. To construct distance matrices of TUHC6[8,10], we ðaÞ ðbÞ must compute matrices A86 ; A86 and B8 · 6. By definition of these matrices, we have:
2
2
mn n ð2mþ1Þþ n2 4 SðaÞ p ¼ 2 24 p 3n p3 þ ð3n2 24mn12n4Þþ p2 þ 12 2 3 mn n ð2mþ3Þþ ðn2Þðn4Þ SðbÞ ¼ p 2 24 p2 p 2 p3 þ 3n 24mn24nþ8 þ ð3n2Þþ 12 2 3 (II)
m £ n + 1 and m n2 þ 1\p n2 þ 1: Therefore,
123
2
ðaÞ
A86
0 6 1 6 6 6 4 6 6 5 6 ¼6 6 8 6 6 9 6 6 4 12 13
1 2
0 1
1 2
0 1
3
4
3
4
6 7
5 8
6 7
5 8
10 9 11 12
10 11
9 12
3 1 2 7 7 7 3 7 7 6 7 7 7; 7 7 7 10 7 7 7 11 5
14 13
14
13
14
Nanoscale Res Lett (2007) 2:202–206
2
0 6 3 6 6 4 6 6 7 ðbÞ A86 ¼ 6 6 8 6 6 11 6 4 12 15 2 0 61 6 62 6 63 B86 ¼ 6 64 6 65 6 46 7
1 0 2 3 5 4 6 7 9 8 10 11 13 12 14 15 1 2 3 4 5 6 7 8
2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10
1 2 5 6 9 10 13 14
0 3 4 7 8 11 12 15
205
3 1 2 7 7 5 7 7 6 7 7 9 7 7 10 7 7 13 5 14 3
2
ðbÞ
A89
1 0 2 3 5 4 6 7 9 8 10 11 13 12 14 15
1 2 5 6 9 10 13 14
0 3 4 7 8 11 12 15
1 2 5 6 9 10 13 14
0 1 3 2 4 5 7 6 8 9 11 10 12 13 15 14
3 0 3 7 7 4 7 7 7 7 7 8 7 7 11 7 7 12 5 15
On the other hand,
4 5 5 6 7 7 6 7 7 7 7 8 7 7: 8 9 7 7 9 10 7 7 10 11 5 11 12 ðaÞ
0 6 3 6 6 4 6 6 7 ¼6 6 8 6 6 11 6 4 12 15
2
B89
ðbÞ
We now compute matrices D810 and D810 : By definition, entries of the first n/2 + 1 columns of these matrices ðaÞ ðbÞ are maximum values of fA86 ; B8 · 6} and fA86 ; B86 g; respectively. Thus, 3 2 0 1 2 3 4 5 4 3 2 1 61 2 3 4 5 6 5 4 3 27 7 6 64 3 4 5 6 7 6 5 4 37 7 6 65 6 5 6 7 8 7 6 5 67 ðaÞ 7; 6 D810 ¼ 6 7 68 7 8 7 8 9 8 7 8 77 6 9 10 9 10 9 10 9 10 9 10 7 7 6 4 12 11 12 11 12 11 12 11 12 11 5 13 14 13 14 13 14 13 14 13 14
0 61 6 62 6 63 ¼6 64 6 65 6 46 7
1 2 3 4 5 6 7 8
2 3 4 5 6 7 8 9
3 4 5 6 7 8 9 10
4 5 6 5 6 7 6 7 8 7 8 9 8 9 10 9 10 11 10 11 12 11 12 13
3 7 8 7 7 9 7 7 10 7 7; 11 7 7 12 7 7 13 5 14
Therefore, 3 0 1 2 3 4 5 6 7 8 7 6 5 4 3 2 1 61 2 3 4 5 6 7 8 9 8 7 6 5 4 3 27 7 6 6 4 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 7 7 6 6 5 6 5 6 7 8 9 10 11 10 9 8 7 6 5 6 7 ðaÞ 7 D816 ¼ 6 6 8 7 8 7 8 9 10 11 12 11 10 9 8 7 8 7 7; 7 6 6 9 10 9 10 9 10 11 12 13 12 11 10 9 10 9 10 7 7 6 4 12 11 12 11 12 11 12 13 14 13 12 11 12 11 12 11 5 13 14 13 14 13 14 13 14 15 14 13 14 13 14 13 14 2
3 0 1 2 3 4 5 4 3 2 1 63 2 3 4 5 6 5 4 3 27 7 6 64 5 4 5 6 7 6 5 4 57 7 6 67 6 7 6 7 8 7 6 7 67 ðbÞ 7 6 D810 ¼ 6 7 68 9 8 9 8 9 8 9 8 97 6 11 10 11 10 11 10 11 10 11 10 7 7 6 4 12 13 12 13 12 13 12 13 12 13 5 15 14 15 14 15 14 15 14 15 14
3 0 1 2 3 4 5 6 7 8 7 6 5 4 3 2 1 63 2 3 4 5 6 7 8 9 8 7 6 5 4 3 27 7 6 6 4 5 4 5 6 7 8 9 10 9 8 7 6 5 4 5 7 7 6 6 7 6 7 6 7 8 9 10 11 10 9 8 7 6 7 6 7 ðbÞ 7: 6 D816 ¼ 6 7 6 8 9 8 9 8 9 10 11 12 11 10 9 8 9 8 9 7 6 11 10 11 10 11 10 11 12 13 12 11 10 11 10 11 10 7 7 6 4 12 13 12 13 12 13 12 13 14 13 12 13 12 13 12 13 5 15 14 15 14 15 14 15 14 15 14 15 14 15 14 15 14
This implies that W(TUHC6[8,10]) = 19,700. To construct distance matrices of TUHC6[8,16], we must compute ðaÞ ðbÞ matrices A89 and A89 : Using a similar argument as above, we have: 3 2 0 1 0 1 0 1 0 1 0 6 1 2 1 2 1 2 1 2 1 7 7 6 6 4 3 4 3 4 3 4 3 4 7 7 6 6 5 6 5 6 5 6 5 6 5 7 ðaÞ 7; A89 ¼ 6 6 8 7 8 7 8 7 8 7 8 7 7 6 6 9 10 9 10 9 10 9 10 9 7 7 6 4 12 11 12 11 12 11 12 11 12 5 13 14 13 14 13 14 13 14 13
By our calculations, it W(TUHC6[8,16]) = 59,648.
2
2
is
easy
to
see
that
Acknowledgements We would like to thank from referees for their helpful remarks and suggestions. This work was partially supported by the Center of Excellence of Algebraic Methods and Applications of the Isfahan University of Technology.
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