On topological properties of hexagonal and silicate networks

Abstract

Topological indices are the numerical quantities associated to a simple and finite graphs which represent its structure. There are certain types of topological indices such as degree based topological indices, distance based topological indices and counting related topological indices etc. The degree based topological indices are exploited to estimate the bioactivity of chemical compounds. In this paper, we compute first general Zagreb index, general Randi$\acute{c}$ connectivity index, general sum-connectivity index, atom-bond connectivity index, geometric-arithmetic index, $ABC_{4}$ index and $GA_{5}$ index of the line graphs of silicate networks, chain silicate networks and hexagonal networks by using the subdivision method.

Keywords

• Topological index,Zagreb index,hexagonal network,silicate network

References

1. S. Akhter and M. Imran, On degree based topological descriptors of strong product graphs, Can. J. Chem. 94 (6), 559–565, 2016.
2. A.Q. Baig, M. Imran and H. Ali, Computing Omega, Sadhana and PI polynomials of benzoid carbon nanotubes, Optoelectron. Adv. Mater.-Rapid Commun. 9, 248–255, 2015.
3. A.Q. Baig, M. Imran, H. Ali and S.U. Rehman, Computing topological polynomial of certain nanostructures, Optoelectron. Adv. Mater.-Rapid Commun. 17, 877–883 2015.
4. M. Bača, J. Horváthová, M. Mokrišová and A. Suhányiová, On topological indices of fullerenes, Appl. Math. Comput. 251, 154–161, 2015.
5. E. Estrada, L. Torres, L. Rodriguez and I. Gutman, An atom-bond connectivity index: modelling the enthalpy of formation of alkanes, Ind. J. Chem. 37A, 849–855, 1998.
6. M. Ghorbani and M.A. Hosseinzadeh, Computing $ABC_{4}$ index of nanostar den- drimers, Optoelectron. Adv. Mater.-Rapid Commun. 4 (9), 1419–1422, 2010.
7. A. Graovac, M. Ghorbani and M.A. Hosseinzadeh, Computing fifth geometric- arithmetic index for nanostar dendrimers, J. Math. Nanosci. 1 (1), 33–42, 2011.
8. S. Hayat and M. Imran, Computation of topological indices of certain networks, Appl. Math. Comput. 240, 213–228, 2014.

9. S. Hayat and M. Imran, Computation of certain topological indices of nanotubes cov- ered by $C_5$ and $C_7$, J. Comput. Theor. Nanosci. 12 (4), 533–541, 2015.
10. S. Hayat and M. Imran, On some degree based topological indices of certain nanotubes, J. Comput. Theor. Nanosci. 12 (8), 1599–1605, 2015.
11. M. Imran, S. Hayat and M.Y.H. Malik, On topological indices of certain interconnec- tion networks, Appl. Math. Comput. 244, 936–951, 2014.
12. X. Li and H. Zhao, Trees with the first three smallest and largest generalized topological indices, MATCH Commun. Math. Comput. Chem. 50, 57–62, 2004.
13. L.N. Lester and J. Sandor, Computer graphics on hexagonal grid, J. Computer Graph- ics, 8, 401–409, 1984.
14. B. Lučić, N. Trinajstić and B. Zhou, Comparison between the sum-connectivity index and product-connectivity index for benzenoid hydrocarbons, Chem. Phys. Lett. 475, 146–148, 2009.
15. M.F. Nadeem, S. Zafar and Z. Zahid, On certain topological indices of the line graph of subdivision graphs, Appl. Math. Comput. 271, 790–794, 2015.
16. M.F. Nadeem, S. Zafarb and Z. Zahid, On topological properties of the line graphs of subdivision graphs of certain nanostructures, Appl. Math. Comput. 273, 125–130, 2016.
17. M. Randić, On characterization of molecular branching, J. Am. Chem. Soc. 97, 6609– 6615, 1975.
18. P.S. Ranjini, V. Lokesha and I.N. Cangl, On the Zagreb indices of the line graphs of the subdivision graphs, Appl. Math. Comput. 18, 699–702, 2011.
19. P.S. Ranjini, V. Lokesha and M.A. Rajan, On the Shultz index of the subdivision graphs, Adv. Stud. Contemp. Math. 21 (3), 279–290, 2011.
20. T. Shirai, Spectrum of Infinite Regular Line Graphs, Trans. Amer. Math. Soc. 352 (1), 115–132, 1999.
21. G. Su and L. Xu, Topological indices of the line graph of subdivision graphs and their Schur-bounds, Appl. Math. Comput. 253, 395–401, 2015.
22. G. Sridhara, M.R. Rajesh Kanna and R.S. Indumathi, Computation of topological indices of Graphene, J. Nanomaterials, 2015.
23. D. Vukičević and B. Furtula, Topological index based on the ratios of geometrical and arithmetical mmeans of end-vertex degrees of edges, J. Math. Chem. 46, 1369–1376, 2009.
24. H. Wiener, Structural determination of the paraffin boiling points, J. Am. Chem. Soc. 69, 17–20, 1947.
25. B. Zhou and N. Trinajstić, On a novel connectivity index, J. Math. Chem. 46, 1252– 1270, 2009.
26. B. Zhou and N. Trinajstić, On general sum-connectivity index, J. Math. Chem. 47, 210–218, 2010.