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
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.