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Application of Molecular Topological Descriptors in Chemistry

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Volume 2021 |Article ID 1078792 | https://doi.org/10.1155/2021/1078792

Peng Xu, Muhammad Numan, Aamra Nawaz, Saad Ihsan Butt, Adnan Aslam, Asfand Fahad, "Computing the Hosoya Polynomial of M-th Level Wheel and Its Subdivision Graph", Journal of Chemistry, vol. 2021, Article ID 1078792, 7 pages, 2021. https://doi.org/10.1155/2021/1078792

Computing the Hosoya Polynomial of M-th Level Wheel and Its Subdivision Graph

Academic Editor: Ajaya Kumar Singh
Received03 Jun 2021
Accepted23 Oct 2021
Published27 Nov 2021

Abstract

The determination of Hosoya polynomial is the latest scheme, and it provides an excellent and superior role in finding the Weiner and hyper-Wiener index. The application of Weiner index ranges from the introduction of the concept of information theoretic analogues of topological indices to the use as major tool in crystal and polymer studies. In this paper, we will compute the Hosoya polynomial for multiwheel graph and uniform subdivision of multiwheel graph. Furthermore, we will derive two well-known topological indices for the abovementioned graphs, first Weiner index, and second hyper-Wiener index.

1. Introduction

Let be a finite connected graph with vertex set and edge set . The distance between is the length of the shortest path joining , . The diameter of is . The terminologies not defined here can be seen in [1, 2]. The Weiner index was first put forward in chemistry by Harold Weiner to compute the cardinality of the carbon-carbon bonds among all pairs of carbon atoms in alkane. For a molecular graph , it is defined as

To read more about the chemical application of Weiner index, see [36], and for its mathematical properties, see [7, 8].

Milan Randic coined the term hyper-Wiener index of [9] as

To read more the properties of hyper-Weiner index, see [912]. Hosoya polynomial was first introduced by Hosoya [13] and it received the attention of a lot of researchers. The same notion was independently put forward by Sagan et al. [14] as Weiner polynomial . The Hosoya polynomial of is defined as

Let be the number of ordered pair in with . Then, the above definition of Hosoya polynomial can be expressed as

The Hosoya polynomial has been investigated on polycyclic aromatic hydrocarbons [15], benzenoid chains [16], Fibonacci and Lucas cubes [17], zigzag polyhexnanotorus [9], carbon nanotubes [18], Hanoi graphs [19], and circumcoronene series [20]. A significant importance of is that some distance-based topological indices (TIs) such as and of can be computed from the Hosoya polynomial as

The readers can see the following papers [2125] for the results on distance-based TIs.

2. Hosoya Polynomial of M-th Level Wheel Graph

For , the join is called a wheel graph denoted by . The vertex that comes from the graph is called the core and is denoted by . It has order and size . A level wheel graph denoted by is the graph obtained by taking copies of the cycle and one copy of , such that all the vertices of each copy of are adjacent with the core vertex . The graph of is depicted in Figure 1. Note that has vertices and edges. If we label the vertices of cycle at the m-th level by , then the and the can be written as

Next, the theorem gives the expression for the Hosoya polynomial of .

Theorem 1. Let, thenis of the form

Proof. It is easy to observe that the diameter of is 2. In order to derive the , we compute the coefficients for . By definition, we have and . To compute , we use the following notation:The cardinality of order pairs in with distance can be characterized by the following two sets:The cardinality of the above sets is and and hence the coefficient is equal to . Now, using the values of , , and , we get the desired result.

Corollary 1. Let, thenandare given as

3. Hosoya Polynomial of Subdivision of M-th Level Wheel Graph

The subdivision graph of is constructed from by adding a vertex into each edge of . In other words, we replace each edge of by a path of length . The graph of is depicted in Figure 2. If we label the new vertices that we insert in the cycle at the j-th level by for , then the vertex set and edge set of can be written as

It is easy to observe that order and size of are and , respectively. In the next theorem, we give the analytic formula to derive the .

Theorem 2. Let, then theis of the form

Proof. It is easy to observe that the diameter of is 6. In order to derive the , we find the coefficients for . By definition, we have and . To compute for , we use the following notation:The cardinality of order pairs in at distance can be characterized by the following sets:The cardinality of the above sets is , , and henceThe cardinality of order pairs in at distance can be characterized by the following sets:The cardinality of the above sets is , and henceThe cardinality of order pairs in at distance can be characterized by the following sets:The cardinality of the above sets is , and henceThe cardinality of order pairs in at distance can be characterized by the following sets:The cardinality of the above sets is , and henceThe cardinality of order pairs in at distance can be characterized by the following sets:The cardinality of the above sets is , and henceNow, using the values of , , , , , , and , we get the desired result.

Corollary 2. Let, then theandare

4. Conclusion

We examined the Hosoya polynomial and two vastly studied TIs and for multiwheel graph and subdivision of multiwheel graph .

Data Availability

No data were used for this study.

Disclosure

Mathematics subject classification: 05C09, 05C92, 92E10.

Conflicts of Interest

The authors hereby declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China, No. 62002079.

References

  1. G. Chartrand, Introduction to Graph Theory, Tata McGraw-Hill Education, New York, NY, USA, 2006.
  2. R. Diestel, Graph Theory, Springer-Verlag, Berlin, Germany, 1997.
  3. Z. Mihalic and N. Trinajstic, “A graph theoretical approach to structure property relationships,” Journal of Chemical Education, vol. 69, pp. 701–712, 1992. View at: Google Scholar
  4. I. Gutman, Y. N. Yeh, S. L. Lee, and Y. L. Luo, “Some recent results in the theory of the wiener number,” Indian Journal of Chemistry, vol. 32A, pp. 651–661, 1993. View at: Google Scholar
  5. I. Gutman and J. H. Potgieter, “Wiener index and intermolecular forces,” Journal of the Serbian Chemical Society, vol. 62, pp. 185–192, 1997. View at: Google Scholar
  6. D. H. Rouvray, “The rich legacy of half a century of the wiener index,” in Topology in Chemistry Discrete Mathematics of Molecules, D. H. Rouvray and R. B. King, Eds., pp. 16–37, Horwood, Chichester, UK, 2002. View at: Publisher Site | Google Scholar
  7. A. A. Dobrynin, R. Entringer, and I. Gutman, “Wiener index of trees: theory and applications,” Acta Applicandae Mathematica, vol. 66, no. 3, pp. 211–249, 2001. View at: Publisher Site | Google Scholar
  8. A. A. Dobrynin, I. Gutman, S. Klavžar, and P. Žigert, “Wiener index of hexagonal systems,” Acta Applicandae Mathematica, vol. 72, no. 3, pp. 247–294, 2002. View at: Publisher Site | Google Scholar
  9. M. Randic, “Novel molecular descriptor for structure property studies,” Chemical Physics Letters, vol. 211, pp. 478–483, 1993. View at: Google Scholar
  10. D. J. Klein, I. Lukovits, and I. Gutman, “On the definition of the hyper-wiener index for cycle-containing structures,” Journal of Chemical Information and Computer Sciences, vol. 35, no. 1, pp. 50–52, 1995. View at: Publisher Site | Google Scholar
  11. I. Gutman, “Relation between hyper-wiener and wiener index,” Chemical Physics Letters, vol. 364, no. 3-4, pp. 352–356, 2002. View at: Publisher Site | Google Scholar
  12. I. Gutman and B. Furtula, “Hyper-wiener index vs. wiener index. two highly correlated structure-descriptors,” Monatshefte für Chemie—Chemical Monthly, vol. 134, no. 7, pp. 975–981, 2003. View at: Publisher Site | Google Scholar
  13. H. Hosoya, “On some counting polynomials in chemistry,” Discrete Applied Mathematics, vol. 19, no. 1–3, pp. 239–257, 1988. View at: Publisher Site | Google Scholar
  14. B. E. Sagan, Y.-N. Yeh, and P. Zhang, “The wiener polynomial of a graph,” International Journal of Quantum Chemistry, vol. 60, no. 5, pp. 959–969, 1996. View at: Publisher Site | Google Scholar
  15. S. Nikolic, N. Trinajstic, and Z. Mihalic, “The Wiener index: development and applications,” Croatica Chemica Acta, vol. 68, pp. 105–129, 1995. View at: Google Scholar
  16. W. Gao, Y. Wang, B. Jamil, and M. Kamran, “Characteristics studies of molecular structures in drugs,” Saudi Pharmaceutical Journal, vol. 25, no. 4, pp. 580–586, 2017. View at: Publisher Site | Google Scholar
  17. N. PrabhakaraRao and A. LaxmiPrasanna, “On the Wiener index of pentachains,” Applied Mathematical Sciences, vol. 2, pp. 2443–2457, 2008. View at: Google Scholar
  18. A. A. Ali and A. M. Ali, “Hosoya polynomials of pentachains,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 65, pp. 807–819, 2011. View at: Google Scholar
  19. I. Gutman, S. Klavzar, M. Petkovsek, and P. Zigert, “On Hosoya polynomials of benzenoid graphs,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 43, pp. 49–66, 2001. View at: Google Scholar
  20. M. Eliasi and B. Taeri, “Hosoya polynomial of zigzag polyhex nanotorus,” Journal of the Serbian Chemical Society, vol. 73, no. 3, pp. 311–319, 2008. View at: Publisher Site | Google Scholar
  21. M. Arockiaraj, S. R. J. Kavitha, K. Balasubramanian, and I. Gutman, “Hyper-Wiener and Wiener polarity indices of silicate and oxide frameworks,” Journal of Mathematical Chemistry, vol. 56, no. 5, pp. 1493–1510, 2018. View at: Publisher Site | Google Scholar
  22. W. Gao, Z. Iqbal, M. Ishaq, R. Sarfraz, M. Aamir, and A. Aslam, “On eccentricity-based topological indices study of a class of porphyrin-cored dendrimers,” Biomolecules, vol. 8, no. 3, p. 71, 2018. View at: Publisher Site | Google Scholar
  23. Z. Iqbal, M. Ishaq, A. Aslam, and W. Gao, “On eccentricity based topological descriptors of water soluble dendrimers,” Zeitschrift für Naturforschung C, vol. 74, 2018. View at: Publisher Site | Google Scholar
  24. W. Gao, Z. Iqbal, M. Ishaq, A. Aslam, and R. Sarfraz, “Topological aspects of dendrimers via distance based descriptors,” IEEE Access, vol. 7, 2019. View at: Publisher Site | Google Scholar
  25. Y. Rao, A. Aslam, M. U. Noor, A. Othman Almatroud, and Z. Shao, “Bond incident degree indices of catacondensed pentagonal systems,” Complexity, vol. 2020, Article ID 4935760, 7 pages, 2020. View at: Publisher Site | Google Scholar

Copyright © 2021 Peng Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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