﻿ Pent-Heptagonal纳米管基于度的拓扑指数计算 Degree Based Topological Index Calculation of Pent-Heptagonal Nanotubes

Pure Mathematics
Vol.06 No.03(2016), Article ID:17375,8 pages
10.12677/PM.2016.63021

Degree Based Topological Index Calculation of Pent-Heptagonal Nanotubes

Jing He, Wei Gao

School of Information, Yunnan Normal University, Kunming Yunnan

Received: Apr. 3rd, 2016; accepted: Apr. 18th, 2016; published: Apr. 21st, 2016

ABSTRACT

Topological index is an available numerically way of the molecular structure, it can fully reflect the molecular graphs connection information and chemical environment, can effectively express the structure and properties of the compounds. In this paper, by means of edge set partitioning technique, we obtain several topological indices of Pent-Heptagonal nanotubes.

Keywords:Pent-Heptagonal Nanotube, Molecular Graph, Generalized Randić Index, Generalized Zagreb Index

Pent-Heptagonal纳米管基于度的拓扑指数计算

1. 引言

1975年，美国化学家Milan Randić [2] 提出了最古老的连通性指数，定义为

1988年，Bollobas和Erdos在 [6] 中推广了Randi指数，用任意实数k代替了，从而定义了广义的Randić指数：

Azari和Iranmanesh [13] 引入广义Zagreb指数，定义为

Eliasi和Iranmanesh [14] 定义广义几何代数指数如下：

Gutman [16] 给出分子图的乘法Zagreb指数定义如下

Usha等人 [17] 重新定义了三类Zagreb指数如下：

Furtula和Gutman [18] 定义了F指数如下：

Pent-Heptagonal纳米管是由周期长度为5的(五边形)和周期长度为7的(七边形)相互交替构成的网状化合物。设纳米管的2维拉丁网格中，第一行和列的7边形个数分别为m和n (见图2)。将纳米管记为纳米管的圆柱和2维拉丁网络可参考图1图2

Figure 1. Pent-Heptagonal Nanotubes cylindrical lattice

Figure 2. Pent-Heptagonal two-dimensional lattice of nanotubes

2. 主要结果

F指数为：

M-多项式为

3. 总结

Degree Based Topological Index Calculation of Pent-Heptagonal Nanotubes[J]. 理论数学, 2016, 06(03): 143-150. http://dx.doi.org/10.12677/PM.2016.63021

1. 1. Gutman, L., Ruscic, B., Trinajstic, N. and Wilcox, C.F. (1975) Graph Theory and Molecular Orbitals. Journal of Phys-ical Chemistry, 62, 3399-3406. http://dx.doi.org/10.1063/1.430994

2. 2. Randić, M. (1975) On the Characterization of Molecular Branching. Journal of the American Chemical Society, 97, 6609-6615. http://dx.doi.org/10.1021/ja00856a001

3. 3. Bondy, J.A. and Murty, U.S.R. (1976) Graph Theory with Applica-tions. Macmillan Press, London, 1-40. http://dx.doi.org/10.1007/978-1-349-03521-2

4. 4. Balaban, A.T., Motoc, I., Bonchov, D. and Mekenyan, O. (1983) Topological Indices for Structure-Activity Correlations. Topics in Current Chemistry, 114, 21-55. http://dx.doi.org/10.1007/BFb0111212

5. 5. Farahani, M.R. and Gao, W. (2015) On the Omega Polynomial of a Family of Hydrocarbon Moleculs “Polycyclic Aromatic Hydrocarbons PAHK”. Asian Academic Research Journal of Multidisciplinary, 2, 263-268.

6. 6. Bollobas, B. and Erdos, P. (2015) Graph of Extremal Weights. Ars Combinatoria, 50, 225-233.

7. 7. Gao, W. and Rajesh Kanna, M.R. (2015) The Connective Eccentric Index for an Infinite Family of Dendrimers. Indian Journal of Fundamental and Applied Life Sciences, 5, 766-771.

8. 8. Mohammad, R.F. (2013) Connectivity Indices of Pent-Heptagonal Nanotubes. Advances in Materials and Corrosion, 2, 33-35.

9. 9. Gao, W. and Wang, W.F. (2014) Second Atom-Bond Connectivity Index of Special Chemical Molecular Structures. Journal of Chemistry, Article ID: 906254.

10. 10. Xi, W.F. and Gao, W. (2014) Geometric-Arithmetic Index and Zagreb Indices of Certain Special Molecular Graphs. Journal of Advances in Chemistry, 10, 2254-2261.

11. 11. Farahani, M.R. and Gao, W. (2016) On Multiplicative and Redefined Version of Zagreb Indices of V-Phenylenic Nanotubes and Nanotorus. British Journal of Mathematics & Computer Science, 13, 1-8. http://dx.doi.org/10.9734/BJMCS/2016/22752

12. 12. Farahani, M.R. (2013) Connectivity Indices of Pent-Heptagonal Nanotubes. Advance in Materials and Corrosion, 2, 33-35.

13. 13. Azari, M. and Iranmanesh, A. (2011) Generalized Zagreb Index of Graphs. Studia Universitatis Babes-Bolyai, 56, 59- 70.

14. 14. Eliasi, M. and Iranmanesh, A. (2011) On Ordinary Generalized Geometric-Arithmetic Index. Applied Mathematics Letters, 24, 582-587. http://dx.doi.org/10.1016/j.aml.2010.11.021

15. 15. Estrada, E., Torres, L., Rodrıguez, L. and Gutman, I. (1988) Anatombond Connectivity Index: Modelling the Enthalpy of Formation of Alkanes. Indian Journal of Chemistry A, 37, 849-855.

16. 16. Gutman, I. (2011) Multiplicative Zagreb Indices of Trees. Bulletin of the International Mathematical Virtual Institute, 1, 13-19.

17. 17. Usha, A, Ranjini, P.S. and Lokesha, V. (2014) Zagreb Co-Indices, Augmented Zagreb Index, Redefined Zagreb Indices and Their Polynomials for Phenylene and Hexagonal Squeeze. Proceedings of Inter-national Congress in Honour of Dr. Ravi. P. Agarwal, Uludag University, Bursa.

18. 18. Furtula, B. and Gutman, I. (2015) A Forgotten Topological Index. Journal of Mathematical Chemistry, 53, 1184-1190. http://dx.doi.org/10.1007/s10910-015-0480-z

19. 19. Furtula, B., Graovac, A. and Vukicevi, D. (2010) Augmented Zagreb Index. Journal of Mathematical Chemistry, 48, 370-380. http://dx.doi.org/10.1007/s10910-010-9677-3