On Some Neighbourhood Degree-Based Indices of Graphs Derived From Honeycomb Structure

Abstract

In mathematical chemistry, molecular structure of any chemical substance can be expressed by a numeric number or polynomial or sequence of numbers which represent the whole graph is called topological index. An important branch of graph theory is the chemical graph theory. Because of their worldwide uses, chemical networks have inspired researchers since their development. Determination of the expressions for topological indices of different derived graphs is a new and interesting problem in graph theory. In this article, some graphs which are derived from Honeycomb structure are studied and found their exact results for some neighbourhood degree-based topological indices. Additionally, a comparison is shown graphically among all the derived indices.

Keywords

• Topological indices
• Dominating David Derived network
• M_N
• F_N

References

1. [1] Ali, H.; Sajjad, A. On further results of hex derived networks. Open J. Discret. Appl. Math. 2019, 2(1), 32-40.
2. [2] Ali, H.; Binyamin, M. A.; Shafiq, M. K.; Gao, W. On the Degree-Based Topological Indices of Some Derived Networks. Mathematics 2019, 7, 612.
3. [3] Akhtar, S.; Imran, M. On molecular topological properties of benzenoid structures. Can. J. Chem. 2016, 94, 687–698.
4. [4] Babar, U.; Ali, H.; Arshad, S.H. and Sheikh, U., Multiplicative topological properties of graphs derived from honeycomb structure. AIMS Mathematics, 2020, 5(2), 1562.
5. [5] Baig, A.Q.; Imran M.; Ali, H. On topological indices of poly oxide, poly silicate, DOX, and DSL networks. Can. J. Chem. 2015, 93, 730–739.
6. [6] Baig, A. Q.; Naeem, M.; Gao, W. Revan and hyper-Revan indices of Octahedral and icosahedral networks. Applied Mathematics and Nonlinear Sciences, 2018, 3(1), 33-40.
7. [7] Diudea, M.V.; Gutman, I.; Lorentz, J. Molecular Topology; Babes-Bolyai University: Cluj-Napoca, Romania, 2001.
8. [8] Dustigeer, G., Ali, H., Khan, M. I., & Chu, Y. M. On multiplicative degree based topological indices for planar octahedron networks. Main Group Metal Chemistry, 2020, 43(1), 219-228.

9. [9] Furtula, B.; Gutman, I. Forgotten topological index. J. Math. Chem. 2015, 53, doi:10.1007/s10910-015-0480-z.
10. [10] Gao, W.; Baig, A. Q.; Khalid, W.; Farahani, M. R. Molecular description of copper(II) oxide. Maced. J. Chem. Chem. Eng. 2017, 36, 93–99.
11. [11] Ghorbani, M.; Hosseinzadeh, M.A. The third version of Zagreb index. Discret. Math. Algorithms Appl. 2013, 5, doi:10.1142/S1793830913500390.
12. [12] Gutman, I.; Polansky, O.E. Mathematical Concepts in Organic Chemistry; Springer: Berlin, Germany, 1986.
13. [13] HoneyComb Structure [online]. Available from https://www.sciencedirect.com/topics/materials-science/honeycomb-structure
14. [14] Hu, M., Ali, H., Binyamin, M. A., Ali, B., Liu, J. B., & Fan, C. On Distance-Based Topological Descriptors of Chemical Interconnection Networks. Journal of Mathematics, 2021.
15. [15] Imran, M.; Baig, A. Q.; Ali, H. On topological properties of dominating David derived networks, Can. J. Chem. 94, 137–148, 2016.
16. [16] Liu, J. B.; Chunxiang W.; Shaohui W.; Bing W. Zagreb indices and multiplicative Zagreb indices of eulerian graphs. Bulletin of the Malaysian Mathematical Sciences Society, 2019, 42(1), 67-78.
17. [17] Liu, J. B., Zhao, J., He, H., & Shao, Z. Valency-based topological descriptors and structural property of the generalized sierpinski´ networks. Journal of Statistical Physics, 2019, 177(6), 1131-1147.
18. [18] Liu, J. B., Zhao, J., Min, J., & Cao, J. The hosoya index of graphs formed by a fractal graph. Fractals, 2019, 27(08), 1950135.
19. [19] Liu, J. B.; Zhi Y. S.; Ying H. P.; Jinde C.; M. Abdel-Aty; Udai Al-Juboori. Computing the Laplacian spectrum of linear octagonal-quadrilateral networks and its applications. Polycyclic Aromatic Compounds, 2020, 1-12.
20. [20] Liu, J. B.; Jing Z.; Zheng Q. C. On the generalized adjacency, Laplacian and signless Laplacian spectra of the weighted edge corona networks. Physica A: Statistical Mechanics and Its Applications, 2020, 540, 123073.
21. [21] Liu, J. B.; Jing Z.; Zhongxun Z. On the number of spanning trees and normalized Laplacian of linear octagonal-quadilateral networks. International Journal of Quantum Chemistry, 2019, 119(17), e25971.
22. [22] Mondal, S.; De, N.; Pal, A. The Neighborhood Zagreb index of product graphs and its chemical interest. arXiv 2018, arXiv:1805.05273.
23. [23] Mondal, S., De, N., & Pal, A. (2019). On some new neighborhood degree-based indices for some oxide and silicate networks. J—Multidisciplinary Scientific Journal, 2(3), 384-409.
24. [24] Plavsic, D.; Nokolic, S.; Trinajstic, Mihalic, Z. On the Haray index for the Characterization of Chemical Graphs, J. Math. Chem. 12, 235-250, 1993.
25. [25] Simonraj, F.; George, A. Embedding of poly honeycomb networks and the metric dimension of star of david network, GRAPH-HOC, 4(2012); 11 28.
26. [26] Shirdel, G.H.; Rezapour, H.; Sayadi, A.M. The hyper Zagreb index of graph operations. Iran. J. Math. Chem. 2013, 4, 213–220.
27. [27] Smith K. M. On neighbourhood degree sequences of complex networks. Scientific reports. 2019, 6;9(1), 8340.
28. [28] Trinajstic´, N. Chemical Graph Theory; CRC Press: Boca Raton, FL, USA, 1983.
29. [29] Wei, C.C.; Ali, H.; Binyamin, M.A.; Naeem, M.N.; Liu, J.B. Computing Degree Based Topological Properties of Third Type of Hex-Derived Networks. Mathematics, 2019, 7, 368.
30. [30] Wiener, H. Structural determination of the paraffin boiling points. J. Am. Chem. Soc. 1947, 69, 17–20.