in 1990 by Buckley and Harary in their book on the con- cept of distance in graph ... the maximum path sums and independently by John.17. âAuthor to whom ...
Copyright © 2011 American Scientific Publishers All rights reserved Printed in the United States of America
Journal of Nanoscience and Nanotechnology Vol. 11, 1–7, 2011
Computation of Detour Index of TUHC62p q Nanotubes for Any p and q Ali Iranmanesh∗ and Yagoob Pakravesh Department of Mathematics, Tarbiat Modares University P.O. Box: 14115-137, Tehran, Iran The detour di j between vertices i and j of a graph is the number of edges of the longest path connecting these vertices. The matrix whose i j-entry is the detour between vertices i and j is called the detour matrix. The half sum D of detours between all pairs of vertices (in a connected graph) is the detour index, i.e., D = 1/2 j i di j. In this paper, we computed the detour index of TUHC6 2p q nanotubes for any p and q.
Keywords: Detour Index, TUHC6 2p q Nanotubes. 1. INTRODUCTION
∗
Author to whom correspondence should be addressed.
J. Nanosci. Nanotechnol. 2011, Vol. 11, No. xx
2. DETOUR INDEX OF ZIG-ZAG POLYHEX NANOTUBE Let us consider a zig-zag polyhex lattice, as illustrated in Figure 3. We choose an arbitrary vertex v, and obtain the detours between the vertex v and other vertices. Let p be the number of vertical lines in first row and q be the number of horizontal “zig-zag” lines. According to Figure 2, we denote the vertices lying on the row 2k − 1 by max vertex and the vertices lying on the row 2k by min vertex when 1 ≤ k ≤ q/2.
1533-4880/2011/11/001/007
doi:10.1166/jnn.2011.3481
1
RESEARCH ARTICLE
Carbon nanotubes were discovered in 1991 by Iijima1 as multi walled structures and in 1993 as single walled carbon nanotubes (briefly denoted SWNT) independently by Iijima’s group2 and Bethune’s group3 from IBM. SWNTs can be seen as a rolled-up graphit sheet in the cylindrical form. Carbon nanotubes show remarkable mechanical properties. Experimental studies have shown that they belong to the stiffest and elastic known materials.4–6 These mechanical characteristics clearly predestinate nanotubes for advanced composities. SWNTs can exhibit either metallic or semiconductor behavior depending only on the diameter and helicity.7 These properties suggest that nanotubes could lead to a new generation of nanoscopic electronic devices. Experiments are under way in several industrial laboratories. The detour matrix, together with the distance matrix, was introduced into the mathematical literature in 1969 by Frank Harary.8 Both matrices were also briefly discussed in 1990 by Buckley and Harary in their book on the concept of distance in graph theory.9 The detour matrix was introduced into the chemical literature in 1994 under the name the maximum path matrix of a molecular graph by Ivanciuc and Balaban10 and independently by Nanad Trianjsti´c in 1995.11 The detour matrix can be used to compute the so-called detour index12 in the same way as the distance matrix13–14 can be employed to generate the Wiener index.15–16 The detour index, which is a Wiener-like index, was also introduced by Ivanciuc and Balaban10 as the half sum of the maximum path sums and independently by John.17
Lukovits, who introduced the term the detour index, is also very active in studying the properties of this index and its uses in structure-property studies. He reported some of his results on the detour matrix and detour index in J. Chem. Inf. Comput. Sci. and else where.12 18 19 Lukovits was also first to use this index in structure-property modeling.12 He also delivered a very stimulating talk on his work on the detour index and it uses at the Rugjer Boškovi´c Institute in Zagreb on February 29, 1996. Harary8 already pointed out that there exists no efficient procedure for finding the entries of this matrix. The same was also stated by Buckley and Harary,9 but they also mentioned that the problem of finding the detour matrix is NP-complete. We will present here our method for constructing the detour index for zigzag polyhex SWNTs. Note that in the construction version of Diudea et al.,20–23 this case of non twisted tubes is named TUHC6 2p q (see Fig. 1). In Refs. [24–27], some authors worked on the detour index. The method for calculating the detour index of zig-zag polyhex nanotube is described in the following section. All notations in this paper are standard. The symbol “int” is the greatest integer function.
Computation of Detour Index of TUHC6 2p q Nanotubes for Any p and q
Iranmanesh and Pakravesh
(iii) If q ≥ p − 1, then sdp q v1M = 4p2 q − 5pq − 1 − 2p + 18 intp+1/4 p−4j+2 −2 i j=2
i=1
intp/4 p−4j+1 −2 i j=2
Fig. 1.
TUHC6 2p q nanotubes.
In Figure 3, we put the number m over the vertices, which means 2pq − m is the detour from vertex v to other vertices. Let v1M be an arbitrary max vertex on level 1. If sdp q v1M is the sum of detours between vertex v1M and all other vertices lying on levels 2 3 q, then for p ≥ 10 we have: (i) If q < p − 3, then sdp q v1M = 4p2 q − 5pq − 1 − 2q + 6 intp+1/4 p−4j+2 intp/4 p−4j+1 −2 i −2 i
RESEARCH ARTICLE
j=2
+2
i=1
j=2
i=1
intp−q+1/4 p−q−4j+2 intp−q+2/4 p−q−4j+3 i +2 i j=2
i=1
j=2
i=1
(ii) If p − 3 ≤ q < p − 1, then
Now we choose an arbitrary max vertex v on the level 2, and obtain the detour between the vertex v and other vertices. In Figure 4, we put the number m over the vertices, which means 2pq − m is the detour from vertex v to other vertices. Let v2M be an arbitrary max vertex on level 2. If sdp q v2M is the sum of detours between vertex v2M and all other vertices lying on levels 3 4 q, then for p ≥ 10 we have: (i) If q < p − 6, then sdp q v2M = 4p2 q − 7pq − 2 + 2 − 2int
q 2
intp−1/4 p−4j intp/4 p−4j+1 −2 i −2 i j=2
i=1
j=2
i=1
intp−q+1/4 p−q−4j+2 intp−q+2/4 p−q−4j+3 +2 i +2 i j=2
i=1
j=2
i=1
(ii) If q = p − 6, then
sdp q v1M = 4p2 q − 5pq − 1 + 2q − 4p + 20 intp+1/4 p−4j+2 intp/4 p−4j+1 −2 i −2 i j=2
+2
i=1
i=1
j=2
intp−q+1/4 p−q−4j+2 i j=2
i=1
i=1
sdp q v2M = 4p2 q − 7pq − 2 + 2 − 2int −2
q 2
intp−1/4 p−4j intp/4 p−4j+1 i −2 i j=2
i=1
j=2
i=1
intp−q+2/4 p−q−4j+3 +2 i j=2
i=1
Row 1 Row 2
(iii) If q ≥ p − 5, then
Row 3 Row 4
sdp q v2M = 4p2 q − 7pq − 2 + 2 − 2int
Row 5 Row 6
Row 7 Row 8
Fig. 2.
2
TUHC6 (Refs. [4, 8]) lattice.
q 2
intp−1/4 p−4j intp/4 p−4j+1 −2 i −2 i j=2
i=1
j=2
i=1
In this section we choose an arbitrary min vertex v on the level 1, and obtain the detour between the vertex v and other vertices. In Figure 5, we put the number m over the J. Nanosci. Nanotechnol. 11, 1–7, 2011
Iranmanesh and Pakravesh
Computation of Detour Index of TUHC6 2p q Nanotubes for Any p and q
An “zig-zag” polyhex lattice, p = 9 q = 6, detours from v to other vertices.
Fig. 4.
Detours between v and other vertices, p = 9, q = 6.
Fig. 5.
Detours between v and other vertices, p = 9, q = 5.
J. Nanosci. Nanotechnol. 11, 1–7, 2011
RESEARCH ARTICLE
Fig. 3.
3
Computation of Detour Index of TUHC6 2p q Nanotubes for Any p and q
vertices, which means 2pq − m is the detour from vertex v to other vertices. Let v1m be an arbitrary min vertex on level 1. If sdp q v1m is the sum of detours between vertex v1m and all other vertices lying on levels 2 3 q (except min vertices on level q), then for p ≥ 10 we have: (i) If q < p − 6, then sdp q v1m = 4p q − 7pq − 2 − 2p q + 3p − 4q intp−1/4 p−4j intp/4 p−4j+1 −2 i −2 i 2
10
j=2
i=1
j=2
j=2
i=1
i=1
sdp q v1m = 4p2 q − 7pq − 2 − 2p2 q + 3p − 4q intp−1/4 p−4j intp/4 p−4j+1 −2 i −2 i j=2
i=1
j=2
3
4
5
6
7
8
9
10
v2,1
3
4 5
and
i=1
sdp q v1m = 4p2 q − 7pq − 2 − 2p2 q − p + 18 intp−1/4 p−4j intp/4 p−4j+1 −2 i −2 / i j=2
RESEARCH ARTICLE
•
2
(iii) If q ≥ p − 4, then
i=1
j=2
i=1
In the above parts, we obtained sdp q v1m , sdp q v1M and sdp q v2M . We can continue this procedure for v2m , v3m vq−1m and v3M , v4M vq−1M . In every level there are p max vertex and p min vertex so we setsdM p q = p q−1 i=2 sdp q viM and sdm p q = p q−1 sdp q v . im i=1 We calculated the sum of detours between a max vertex lying at level 2 and other vertices lying on levels 3 4 q. Now if v is a max vertex on the level 3, then sum of detours between vertex v and other vertices lying at levels 4 5 q is equal to sdp q v3M = sdp q v2M − T lq v2M , where by Figure 6,
T lk v1m =
j=1
p ⎪ ⎪ ⎪ ⎪ ⎩ dv11 vk2j−1 if k = 2m + 1
So we have: (i) If q < p − 6, then sdM p q = sdp q v2M pq − 2p
intp−q+2/4 p−q−4j+3 + p 2q − 3 i j=2
i=1
j=2
i=1
intp−q+1/4 p−q−4j+2
+
i
intp−q+1/4 p−q−4j+1 p−q−k−4j+2 +2 i j=2
dv21 vq2j + dv21 vq2j−1
If v is a max vertex of level 4, then sum of detours between v and other vertices lying at levels 5 6 q is equal to sdp q v4M = sdp q v2M − T lq v2M − T lq−1 v2M . In general sdp q vkM = sdp q v2M − k−3 i=0 T lq−i v2M . k−2 And sdp q vkm = sdp q v1m − i=0 T Lq−i v1m + T lq−i−1 v1m , where according to Figure 7: ⎧p ⎪ ⎪ dv11 vk2j−1 if k = 2m ⎪ ⎪ ⎨j=1 T Lk v1m = p ⎪ ⎪ ⎪ ⎪ ⎩ dv11 vk2j if k = 2m + 1
⎧p ⎪ ⎪ dv11 vk2j if k = 2m ⎪ ⎪ ⎨j=1
j=1
10
j=1
4
2
Fig. 6. Detours between vertex v2 1 and other vertices.
(ii) If p − 6 ≤ q < p − 4, then
p
1
1
2
intp−q+1/4 p−q−4j+2 +4 i
T lq v21 =
Iranmanesh and Pakravesh
1
1
2
3
4
5
i=1
k=0
6
7
8
9
10
v1,1 •
2 3
4 5
Fig. 7. Detours between vertex v1 1 and other vertices.
J. Nanosci. Nanotechnol. 11, 1–7, 2011
Iranmanesh and Pakravesh
Computation of Detour Index of TUHC6 2p q Nanotubes for Any p and q
intp−q+2/4 p−q−4j+2 p−q−k−4j+3 +2 i j=2
−2
i=1
k=0
k=1
i=1
k=0
sdm p q = sdp q v1m pq − p intp−1/4 p−4j−1 p−k−4j + p −4 i
i=1
k=0
q −4 − 4p q − 7pi + 4int 2 i=1
q−3
2
j=2
(ii) If q = p − 6, then
−
−2
q−3
+ 4int
j=2
i=1
intp−1/4 p−4j−1 p−k−4j −2 i j=2
q−3
k=0
i=1
4p2 q − 7pi + 4int
i=1
q −4 2
And for sdm p q we have (i) If q < p − 6, then sdm p q = sdp q v1m pq − p intp−q+1/4 p−q−4j+2 + p 4q − 2 i j=2
i=1
intp−q+1/4 p−q−4j+1 p−q−k−4j+2 +4 i j=2
i=1
k=0
j=2
k=0
4p2 q − 7pi+2
i=1
J. Nanosci. Nanotechnol. 11, 1–7, 2011
2 p−q−k−3 k=1
k=1
i
i=1
Now we define the following notations. ∗ sdM p q viM is the sum of detours between vertex viM and all max vertices lying on same level (level i) and ∗ ∗ p q = p/2 × sdM p q viM . sdiM sdm∗ p q vim is the sum of detours between vertex vim and all min vertices lying on the same level (level i) and ∗ p q = p/2 × sdm∗ p q vim . sdim sdm∗ p q viM is the sum of detours between vertex viM and all min vertices lying on the same level (level i) and sdi∗ p q = p × sdm∗ p q viM . Let p ≥ 10, then we have (i) ∗ ∗ sd1M p q + sdqm p q = 2p3 q − 2p2 q − 2p2 + 2p p−1 2 + 6p − 2p int 4 2 p−1 + 4p int 4
(ii)
p−2 + 16p − 4p int 4 2 p−2 + 8p int 4
2
i=1
2 p−k−5
sd1∗ p q + sdq∗ p q = 4p3 q − 6p2 − 8p
intp−1/4 p−4j−1 p−k−4j −4 i −
4p2 q − 7pi − 2
i=1
sdM p q = sdp q v2M pq − 2p intp−2/4 p−4j−2 p−k−4j−1 + p −2 i
q−2
q−2
i=1
k=0
i=1
i
5
RESEARCH ARTICLE
(iii) If q > p − 6, then
−
i
i=1
sdm p q = sdp q v1m pq − p intp−1/4 p−4j−1 p−k−4j + p −4 i −
k=0
i=1
k=1
i
(iii) If q ≥ p − 4, then
q + 2p − 20 2
j=2
2 p−q−k−3
+ 2q − 22p − 2q − 9
4p2 q − 7pi
i=1
k=1
i=1
k=0
i=1
k=0
4p2 q − 7pi+2 2 p−k−5
−2
intp−1/4 p−4j−1 p−k−4j i j=2
−
i=1
k=0
q−2 i=1
sdM p q = sdp q v2M pq − 2p intp−2/4 p−4j−2 p−k−4j−1 + p −2 i j=2
i
i=1
(ii) If p − 6 ≤ p − 4, then
intp−1/4 p−4j−1 p−k−4j −2 i j=2
+ 2q − 22p − 2q − 9
intp−2/4 p−4j−2 p−k−4j−1 −2 i j=2
2 p−k−5
Computation of Detour Index of TUHC6 2p q Nanotubes for Any p and q Table II.
Let p ≥ 10, then we have (i)
q−1
∗ sdiM pq = p3 q 2 −2p3 q −p2 q 2 +p2 q −pq +2p2
i=2
Numerical values of detour index, 3 ≤ p ≤ 9 q ≥ 8.
p
D
3
108q − 81q + 36q + 12q − 285 3
p +1 +2p +pq −p q −2p +2p int 4 2 p +1 +2pq −4p int 4
2
q 2 q 500q 3 − 225q 2 + 100q − 90 + 40 − 10qint 2 q 864q 3 − 324q 2 + 120q − 168 + 48 − 12qint 2 q 3 2 1372q − 441q + 126q − 280 + 56 − 14qint 2 q 3 2 2048q − 576q + 80q − 896 + 64 − 16qint 2 q 2916q 3 − 729q 2 − 1890q − 594 + 72 − 18qint 2 256q − 144q 2 + 72q − 40 + 32 − 8qint
5 6 7 8
(ii)
9
q−1
sdi∗ p q = 2p3 q 2 − 4p3 q − 3p2 q + 6p2
i=2
p 2 2 + 4pq − 2p q − 8p + 4p int 4 2 p + 4pq − 8p int 4
RESEARCH ARTICLE
∗ + sdqM p q +
q−1
∗ sdiM p q +
i=2
+
q−1
q−1
sdi∗ p q
i=2
∗ sdim p q
i=2
References and Notes
Let p ≥ 10, then we have ∗ ∗ p q + sdqM p q = 2p3 q − 2p2 − 2p2 q − 2p sd1m p+3 − 2p + p2 int 4 p+3 2 + 4p int 4
Remark. If we rotate the Figure 3 by 180 degree, any ∗ p q = max vertex becomes a min vertex, therefore sdiM ∗ sdim p q, for each 1 < i < q. Table I and Table II includes detour index of zig-zag polyhex nanotube for small value of p, q. Detour index of zig-zag polyhex nanotube in the case p ≥ 10 is as follows: D = p × sdp q v1M + sdM p q + sdm p q ∗ ∗ p q + sdqm p q + p × sdmax p q v1M + sd1M ∗ p q + sd1∗ p q + sdq∗ p q + sd1m
Table I. Numerical values of detour index, 3 ≤ p ≤ 9 3 ≤ q ≤ 7. p q
3
4
5
6
7
8
9
3 4 5 6 7
2046 5523 1643 20415 33126
5800 14328 28704 50456 81144
11685 28710 57265 100350 160995
20484 50304 100308 175440 281040
33243 73094 147539 280862 449463
49284 117568 236560 421104 673760
70416 186813 361316 631188 1001412
6
2
3
4
2
Iranmanesh and Pakravesh
1. S. Iijima, Nature, 354, 56 (1991). 2. S. Iijima and T. Ichlhashi, Nature 363, 603 (1993). 3. D. S. Bethune, C. H. Kiang, M. S. Devries, G. Gorman, R. Savoy, J. Vazquez, and R. Beyers, Nature 363, 605 (1993). 4. M. Endo, S. Iijima, and M. S. Dresselhaus, Carbon Nanotubes, Pergamon (1996). 5. E. W. Wong, P. E. Sheehan, and C. M. Lieber, Science 277, 1971 (1997). 6. B. I. Yakobson, C. J. Brabec, and J. Bernholc, Phys. Rev. Lett. 76, 2511 (1996). 7. T. W. Odom, J. L. Huang, Ph. Kim, and Ch. M. Lieber, J. Phys. Chem. B 104, 2794 (2000). 8. F. Harary, Graph Theory, Addison-Wesley, Reading, MA (1969), p. 203. 9. F. Buckley, F. Harary, Distance in Graphs, Addison-Wesley, Reading, MA (1990), pp. 213–214. 10. O. Ivanciuc and A. T. Balaban, MATCH Commun. Math. Comput. Chem. 30, 141 (1994). 11. D. Ami´c and N. Trianjsti´c, Croat. Chem. Acta 68, 53 (1995). 12. I. Lukovits, The Detour Index, Croat. Chem. Acta 69, 873 (1996). 13. Z. Mihali´c, D. Veljan, D. Ami´c, S. Nikoli´c, D. Palvši´c, and N. Trianjsti´c, J. Math. Chem. 11, 223 (1992). 14. Z. Mihali´c, S. Nikoli´c, and N. Trianjsti´c, J. Chem. Inf. Comput. Sci. 36, 65 (1996). 15. H. Hosoya, Bull. Chem. Soc. Jpn. 44, 2332 (1971). 16. S. Nikoli´c, N. Trianjsti´c, and Z. Mihali´c, Croat. Chem. Acta 68, 105 (1995). 17. P. E. John, MATCH Commun. Math. Comput. Chem. 32, 207 (1995). 18. I. Lukovits, J. Chem. Inf. Comput. Sci. 36, 65 (1996). 19. H. Y. Zhu, D. J. Klein, and I. Lukovits, J. Chem. Inf. Comput. Sci. 36, 420 (1996). 20. M. V. Diudea and A. Graovac, MATCH Commun. Math. Comput. Chem. 44, 93 (2001). 21. M. V. Diudea, I. Silaghi-Dumitrescu, and B. Parv, MATCH Commun. Math. Comput. Chem. 44, 117 (2001).
J. Nanosci. Nanotechnol. 11, 1–7, 2011
Iranmanesh and Pakravesh
Computation of Detour Index of TUHC6 2p q Nanotubes for Any p and q
22. M. V. Diudea and P. E. John, MATCH Commun. Math. Comput. Chem. 44, 103 (2001). 23. M. V. Diudea, Bull. Chem. Soc. Japan 75, 487 (2002). 24. M. V. Diudea, G. Katona, I. Lukovits, and N. Trinajsti´c, Croatica Chemica Acta 71, 459 (1998).
25. M. Randi´c, L. M. DeAlba, and F. E. Harris, Croatica Chemica Acta 71, 53 (1998). 26. F. Yeh, S. Tang, Y. Wang, and T. Ho, European Journal of Operational Research 155, 631 (2004). 27. A. E. Baumann, R. Klein, E. Langetepe, and A. Lingas, Computational Geometry 27, 123 (2004).
Received: xx Xxxx xxxx. Accepted: xx Xxxx xxxx.
RESEARCH ARTICLE
J. Nanosci. Nanotechnol. 11, 1–7, 2011
7
