# Computing the Hosoya Polynomial ofM-th Level Wheel and Its Subdivision Graph

Authors: Peng Xu; Muhammad Numan; Aamra Nawaz; Saad Ihsan Butt; Adnan Aslam; Asfand Fahad
Journal: Journal of Chemistry
 (2021)

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## 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.


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## Body

## 1. Introduction

LetG be a finite connected graph with vertex set VG=V and edge set EG=E. The distance du,v between u,v∈VG is the length of the shortest path joining u, v. The diameter dG of G is maxu,vdu,v. The terminologies not defined here can be seen in [1, 2]. The 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(1)WG=∑u,v∈Vdu,v.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 indexWWG of G [9] as(2)WWG=12∑u,v∈Vdu,v+du,v2.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 G. The Hosoya polynomial HG,x of G is defined as(3)HG,x=∑u,v∈Vxdu,v.LetαG,k be the number of ordered pair u,v in V with du,v=k. Then, the above definition of Hosoya polynomial can be expressed as(4)HG,x=∑k=0dGαG,kxk.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 HG,x is that some distance-based topological indices (TIs) such as WG and WWG of G can be computed from the Hosoya polynomial as(5)WG=H′G:1,WWG=H′G;1+12H″G;1.The readers can see the following papers [21–25] for the results on distance-based TIs.


## 2. Hosoya Polynomial ofM-th Level Wheel Graph

Forn≥2, the join K1∨Cn is called a wheel graph denoted by Wn+1. The vertex that comes from the graph K1 is called the core and is denoted by c. It has order n+1 and size 2n. A m−level wheel graph denoted by mWn is the graph obtained by taking m copies of the cycle Cn and one copy of K1, such that all the vertices of each copy of Cn are adjacent with the core vertex c. The graph of mWn is depicted in Figure 1. Note that mWn has mn+1 vertices and 2mn edges. If we label the vertices of cycle at the m-th level by u0m,u1m,u2m,…,un−1m, then the VmWn and the EmWn can be written as(6)VmWn=c,uij,0≤i≤n−1,1≤j≤m,EmWn=cuij,0≤i≤n−1,1≤j≤m∪uijui+1j,0≤i≤n−1,1≤j≤m.Figure 1
Multiwheel graphmWn.Next, the theorem gives the expression for the Hosoya polynomial ofmWn.Theorem 1.
Letm,n≥1, thenHmWn;xis of the form(7)HmWn;x=mn+1+2mnx+mnn−3+nm−12m2x2.Proof.
It is easy to observe that the diameter ofmWn is 2. In order to derive the HmWn;x, we compute the coefficients αmWn,k for k=0,1,2. By definition, we have αmWn,0=mn+1 and αmWn,1=2mn. To compute αmWn,2, we use the following notation:(8)αA=number of pair of vertices in setA.
The cardinality of order pairs inVmWn with distance 2 can be characterized by the following two sets:(9)A1=uij,ui+lj,1≤j≤m,0≤i≤n−1,2≤l≤n−2,A2=uij,ulh,0≤i≤n−1,0≤l≤n−1,1≤j≤m−1,j+1≤h≤m.
The cardinality of the above sets isαA1=mnn−3 and αA2=nm−12m/2 and hence the coefficient αmWn,2 is equal to αmWn,2=αA1+αA2=mnn−3+nm−12m/2. Now, using the values of αmWn,0, αmWn,1, and αmWn,2, we get the desired result.Corollary 1.
Letm,n≥1, thenWmWnandWWmWnare given as(10)WmWn=2mn+2mnn−3+nm−12m2,WWmWn=2mn+3mnn−3+nm−12m2.


## 3. Hosoya Polynomial of Subdivision ofM-th Level Wheel Graph

The subdivision graphSmWn of mWn is constructed from mWn by adding a vertex into each edge of mWn. In other words, we replace each edge of mWn by a path of length 2. The graph of SmWn is depicted in Figure 2. If we label the new vertices that we insert in the cycle at the j-th level by x1j,x2j,…,xn−1j for j=1,2,…,m, then the vertex set and edge set of SmWn can be written as(11)VSmWn=c,uij,vij,xij,0≤i≤n−1,1≤j≤m,ESmWn=cvij,vijuij,xijuij,uijxi+1j,0≤i≤n−1,1≤j≤m∪uijui+1j,0≤i≤n−1,1≤j≤m.Figure 2
Uniform subdivision of multiwheel graphmSn.It is easy to observe that order and size of are3mn+1 and 4mn, respectively. In the next theorem, we give the analytic formula to derive the HSmWn;x.Theorem 2.
Letm,n≥1, then theHSmWn;xis of the form(12)HSmWn;x=3mn+1+4mnx+5mn+mnn−1x2+mn2m−12+3mn+mnn−1x3+mm−1n2+mn2n−5+mnx4+mnn−4+mm−1n22x5+mnn−5+mm−1n22x6.Proof.
It is easy to observe that the diameter ofmWn is 6. In order to derive the HSmWn;x, we find the coefficients αSmWn,k for k=0,1,2,…,6. By definition, we have αSmWn,0=3mn+1 and αSmWn,1=4mn. To compute αSmWn,j for j=2,3,4,5,6, we use the following notation:(13)αA=number of pair of vertices in setA.
The cardinality of order pairs inVSmWn at distance 2 can be characterized by the following sets:(14)B1=c,uij,1≤j≤m,0≤i≤n−1,B2=vij,vi+lj,1≤j≤m,0≤i≤n−1,1≤l≤n−1,B3=vij,xi+1j,1≤j≤m,0≤i≤n−1,B4=vij,xij,0≤i≤n−1,1≤j≤m,B5=uij,ui+1j,1≤j≤m,0≤i≤n−1,B6=xij,xi+1j,1≤j≤m,0≤i≤n−1.
The cardinality of the above sets isαB1=αB3=αB4=αB5=αB6=mn, αB2=mnn−1, and hence(15)αSmWn,2=αB1+αB2+αB3+αB4+αB5+αB6=5mn+mnn−1.
The cardinality of order pairs inVSmWn at distance 3 can be characterized by the following sets:(16)C1=c,xij,1≤j≤m,0≤i≤n−1,C2=vij,ui+lj,1≤j≤m,0≤i≤n−1,1≤l≤n−1,C3=vij,u1j+k,1≤j≤m,1≤k≤m−j,0≤i≤n−1,0≤l≤n−1,C4=uij,xi+2j,1≤j≤m,0≤i≤n−1,C5=xij,ui+1j,1≤j≤m,0≤i≤n−1.
The cardinality of the above sets isαC1=αC4=αC5=mn,αC2=mnn−1,αC3=mn2m−1/2, and hence(17)αSmWn,3=αC1+αC2+αC3+αC4+αC5=mn2m−12+3mn+mnn−1.
The cardinality of order pairs inVSmWn at distance 4 can be characterized by the following sets:(18)D1=vij,xi+lj,1≤j≤m,0≤i≤n−1,2≤l≤n−1,D2=vij,xlj+k,1≤j≤m−1,1≤k≤m−j,0≤i≤n−1,0≤l≤n−1,D3=uij,ui+lj,1≤j≤m,0≤i≤n−1,2≤l≤n−2,D4=uij,uij+k,1≤j≤m−1,1≤k≤m−j,0≤i≤n−1,0≤l≤n−1,D5=xij,xi+2j,1≤j≤m,0≤i≤n−1.
The cardinality of the above sets isαD1=mnn−2,αD2=mm−1n2/2,αD3=mnn−3,αD4=mm−1n2/2,αD5=mn, and hence(19)αSmWn,4=αD1+αD2+αD3+αD4+αD5=mm−1n2+mn2n−4.
The cardinality of order pairs inVSmWn at distance 5 can be characterized by the following sets:(20)E1=uij,xi+lj,1≤j≤m,0≤i≤n−1,3≤l≤n−2,E2=uij,xlj+k,1≤j≤m−1,1≤k≤m−j,0≤i≤n−1,0≤l≤n−1.
The cardinality of the above sets isαE1=mnn−4,αE2=mm−1n2/2, and hence(21)αSmWn,5=αE1+αE2=mnn−4+mm−1n22.
The cardinality of order pairs inVSmWn at distance 6 can be characterized by the following sets:(22)F1=xij,xi+lj,1≤j≤m,0≤i≤n−1,3≤l≤n−3,F2=xij,xlj+k,1≤j≤m−1,1≤k≤m−j,0≤i≤n−1,0≤l≤n−1.
The cardinality of the above sets isαF1=mnn−5,αF2=mm−1n2/2, and hence(23)αSmWn,6=αF1+αF2=mnn−5+mm−1n22.
Now, using the values ofαSmWn,0, αSmWn,1, αSmWn,2, αSmWn,3, αSmWn,4, αSmWn,5, and αSmWn,6, we get the desired result.Corollary 2.
Letm,n≥1, then theWmSnandWWmSnare(24)WmSn=4mn+25mn+mnn−1+3mn2m−12+3mn+mnn−1+4mm−1n2+mn2n−5+mn+5mnn−4+mm−1n22+6mnn−5+mm−1n22,WWmSn=4mn+35mn+mnn−1+6mn2m−12+3mn+mnn−1+10mm−1n2+mn2n−5+mn+15mnn−4+mm−1n22+21mnn−5+mm−1n22.


## 4. Conclusion

We examined the Hosoya polynomial and two vastly studied TIsWG and WWG for multiwheel graph mWn and subdivision of multiwheel graph mSn.



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*Source: 1078792-2021-11-27.xml*