Journal of Chemistry
Volume 2021 (2021), Article ID 1078792, 7 pages
https://doi.org/10.1155/2021/1078792

Research Article

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

Peng Xu,1,2 Muhammad Numan,³ Aamra Nawaz,³ Saad Ihsan Butt,⁴ Adnan Aslam,⁵ and Asfand Fahad⁶

¹Institute of Computational Science and Technology, Guangzhou University, Guangzhou, China
²School of Computer Science of Information Technology, Qiannan Normal University for Nationalities, Duyun 558000, China
³Department of Mathematics, COMSATS University Islamabad, Attock Campus, Islamabad, Pakistan
⁴Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Islamabad, Pakistan
⁵Department of Natural Sciences and Humanities, University of Engineering and Technology, Lahore (RCET), Lahore, Pakistan
⁶Department of Mathematics, COMSATS University Islamabad, Vehari 61100, Pakistan

Correspondence should be addressed to Asfand Fahad; asfandfahad1@yahoo.com

Received 3 June 2021; Accepted 23 October 2021; Published 27 November 2021

Academic Editor: Ajaya Kumar Singh

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.

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 [3–6], 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 [9–12]. 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 [21–25] 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

Figure 1: Multiwheel graph .

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

Figure 2: Uniform subdivision of multiwheel graph .

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 hence

The cardinality of order pairs in at distance can be characterized by the following sets:

The cardinality of the above sets is , and hence

The cardinality of order pairs in at distance can be characterized by the following sets:

The cardinality of the above sets is , and hence

The cardinality of order pairs in at distance can be characterized by the following sets:

The cardinality of the above sets is , and hence

The cardinality of order pairs in at distance can be characterized by the following sets:

The cardinality of the above sets is , and hence

Now, 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.

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