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

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


Introduction
Let G be a finite connected graph with vertex set V(G) � V and edge set E(G) � E.
e distance d u,v between u, v ∈ V(G) is the length of the shortest path joining u, v. e diameter d(G) of G is max (u,v) d u,v . e terminologies not defined here can be seen in [1,2]. e Weiner index W 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 G, it is defined as To read more about the chemical application of Weiner index, see [3][4][5][6], and for its mathematical properties, see [7,8].
Milan Randic coined the term hyper-Wiener index WW(G) of G [9] as To read more the properties of hyper-Weiner index, see [9][10][11][12]. Hosoya polynomial was first introduced by Hosoya [13] and it received the attention of a lot of researchers. e same notion was independently put forward by Sagan et al. [14] as Weiner polynomial G. e Hosoya polynomial H(G, x) of G is defined as Let α(G, k) be the number of ordered pair (u, v) in V with d u,v � k. en, the above definition of Hosoya polynomial can be expressed as e 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 H(G, x) is that some distance-based topological indices (TIs) such as W(G) and WW(G) of G can be computed from the Hosoya polynomial as e readers can see the following papers [21][22][23][24][25] for the results on distance-based TIs.

Hosoya Polynomial of M-th Level Wheel Graph
For n ≥ 2, the join K 1 ∨C n is called a wheel graph denoted by W n+1 . e vertex that comes from the graph K 1 is called the core and is denoted by c. It has order n + 1 and size 2n. A m−level wheel graph denoted by mW n is the graph obtained by taking m copies of the cycle C n and one copy of K 1 , such that all the vertices of each copy of C n are adjacent with the core vertex c. e graph of mW n is depicted in Figure 1. Note that mW n has mn + 1 vertices and 2mn edges. If we label the vertices of cycle at the m-th level by u m 0 , u m 1 , u m 2 , . . . , u m n−1 , then the V(mW n ) and the E(mW n ) can be written as Next, the theorem gives the expression for the Hosoya polynomial of mW n .
Proof. It is easy to observe that the diameter of mW n is 2. In order to derive the H(mW n ; x), we compute the coefficients α(mW n , k) for k � 0, 1, 2. By definition, we have α(mW n , 0) � mn + 1 and α(mW n , 1) � 2mn. To compute α(mW n , 2), we use the following notation: e cardinality of order pairs in V(mW n ) with distance 2 can be characterized by the following two sets: e cardinality of the above sets is α A 1 � mn(n − 3) and α A 2 � (n(m − 1)) 2 (m)/2 and hence the coefficient Now, using the values of α(mW n , 0), α(mW n , 1), and α(mW n , 2), we get the desired result.

Corollary 1.
Let m, n ≥ 1, then W(mW n ) and WW(mW n ) are given as

Hosoya Polynomial of Subdivision of M-th
Level Wheel Graph e subdivision graph S(mW n ) of mW n is constructed from mW n by adding a vertex into each edge of mW n . In other words, we replace each edge of mW n by a path of length 2. e graph of S(mW n ) is depicted in Figure 2. If we label the new vertices that we insert in the cycle at the j-th level by It is easy to observe that order and size of are 3mn + 1 and 4mn, respectively. In the next theorem, we give the analytic formula to derive the H(S(mW n ); x).

Theorem 2. Let m, n ≥ 1, then the H(S(mW n ); x) is of the form
Proof. It is easy to observe that the diameter of mW n is 6. In order to derive the H(S(mW n ); x), we find the coefficients α(S(mW n ), k) for k � 0, 1, 2, . . . , 6. By definition, we have α(S(mW n ), 0) � 3mn + 1 and α(S(mW n ), 1) � 4mn. To compute α(S(mW n ), j) for j � 2, 3, 4, 5, 6, we use the following notation: α A � number of pair of vertices in set A. e cardinality of order pairs in V(S(mW n )) at distance 2 can be characterized by the following sets: e cardinality of the above sets is e cardinality of order pairs in V(S(mW n )) at distance 3 can be characterized by the following sets: e cardinality of the above sets is α C 1 � α C 4 � α C 5 � mn, α C 2 � mn(n − 1), α C 3 � (mn) 2 (m − 1)/2, and hence α S mW n , e cardinality of order pairs in V(S(mW n )) at distance 4 can be characterized by the following sets: e cardinality of order pairs in V(S(mW n )) at distance 5 can be characterized by the following sets: e cardinality of the above sets is α E 1 � mn(n − 4), α E 2 � m((m − 1)n) 2 /2, and hence e cardinality of order pairs in V(S(mW n )) at distance 6 can be characterized by the following sets: e cardinality of the above sets is α F 1 � mn(n − 5), α F 2 � m((m − 1)n) 2 /2, and hence Now, using the values of α(S(mW n ), 0), α(S(mW n ), 1), α(S(mW n ), 2), α(S(mW n ), 3), α(S(mW n ), 4), α(S(mW n ), 5), and α(S(mW n ), 6), we get the desired result.

Conclusion
We examined the Hosoya polynomial and two vastly studied TIs W(G) and WW(G) for multiwheel graph mW n and subdivision of multiwheel graph mS n .

Data Availability
No data were used for this study.

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