# Comparison of the Wiener and Kirchhoff Indices of Random Pentachains

Authors: Shouliu Wei; Wai Chee Shiu; Xiaoling Ke; Jianwu Huang
Journal: Journal of Mathematics
 (2021)

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## Abstract

LetG be a connected (molecule) graph. The Wiener index WG and Kirchhoff index KfG of G are defined as the sum of distances and the resistance distances between all unordered pairs of vertices in G, respectively. In this paper, explicit formulae for the expected values of the Wiener and Kirchhoff indices of random pentachains are derived by the difference equation and recursive method. Based on these formulae, we then make comparisons between the expected values of the Wiener index and the Kirchhoff index in random pentachains and present the average values of the Wiener and Kirchhoff indices with respect to the set of all random pentachains with n pentagons.


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## Body

## 1. Introduction

All graphs considered in this paper are simple, undirected, and connected. All notations not defined in this paper can be found in [1]. Let G be a graph with the vertex set VG=v1,v2,…,vn and edge set EG. The traditional distance dvr,vs between vertices vr and vs is the number of edges of a shortest path connecting these vertices in G. In 1993, Klein and Randić [2] conceived the novel concept of resistance distance based on the electric resistance in an electrical network N corresponding to the considered graph G, in which the resistance between any two adjacent nodes is 1 Ohm. Similar to the long recognized shortest path distance, the resistance distance is also intrinsic to the graph, not only with some nice purely mathematical and physical interpretations [2, 3], but also with a substantial potential for chemical applications. Denote by rvi,vj the resistance distance between the vertices vi and vj in G, which is computed by Ohm’s and Kirchhoff’s laws in N.For the resistance distance of a connected graph, a formula for computing it was presented by Klein and Randić [2] in 1993. We may compute the distance of two vertices in a connected graph G of order n from its Laplacian matrix. Precisely, let L be the Laplacian matrix of G and J be the n×n matrix whose entries are 1. Let X=L+1/nJ−1. Then, the resistance distance between vertices i and j of G is(1)ri,j=Xi,i+Xj,j−2Xi,j.The Wiener indexWG [4, 5] of a graph G was first introduced by Wiener in the study of paraffin boiling points and a graph invariant based on distances in 1947. This is the oldest topological index related to molecular branching [6]. It is defined as the sum of distances between all unordered pairs of vertices in G, i.e.,(2)WG=12∑r=1n∑s=1ndvr,vs=12∑r=1ndvr|G,where dvi|G denotes the sum of distances between vi and all other vertices of G, defined by(3)dvi|G=∑j=1nrvi,vj.For a survey of results and bibliography on the chemical applications and the mathematical literature of the Wiener index, see [7–15] and the references cited therein.Similar to the Wiener index, the Kirchhoff indexKfG of a graph G [2, 16] is defined as(4)KfG=12∑i=1n∑j=1nrvi,vj=12∑i=1nrvi|G,where rvi|G denotes the sum of resistance distances between vi and all other vertices of G, defined by(5)rvi|G=∑j=1nrvi,vj.Klein and Randić [2] proved that rvi,vj≤dvi,vj and KfG≤WG with equality if and only if G is a tree. Like many topological indices, the Kirchhoff index based on resistance distance is also a graph invariant and molecular structure descriptor. Unfortunately, it is difficult to implement some algorithms [2, 17, 18] to compute resistant distance and Kirchhoff index in a graph from their computational complexity. Hence, it makes sense to find explicit closed-form formula for the Kirchhoff index. Up to now, the Kirchhoff index has been found to have noteworthy applications in chemistry, such as in assessing cyclicity of polycyclic structures including fullerenes, linear hexagonal chains, and some special molecular graphs such as distance-regular graphs and Möbius ladders [17–21]. Bonchev et al. in [16] considered the polymer science by using it and found that the Kirchhoff index in their approach is especially useful for defining the topological radius Rtop=Kf/n2 of macromolecules containing cyclic fragments. Many mathematical properties have also been established (see [22–41] and the references cited therein).Motivated by the works in [42–44], in this paper, we derive explicit formulae for the expected values of the Kirchhoff indices of random alpha-pentachain [43] and spiro-pentachain as shown in Figure 1 and Wiener index of random spiro-pentachain. Based on these formulae, we then make comparisons between the expected values of the Wiener index and the Kirchhoff index in random pentachains and present the average values of the Wiener and Kirchhoff indices with respect to the set of all random pentachains with n pentagons. In fact, there also exist numerous research results for a general pentachain. For example, Xiao et al. [45] studied a connection between the Kekulé structures of pentachains and the Hosoya index of caterpillar trees, and a general expression for calculating the bond incident degree indices of pentachains was established in [46]. Denote by EΞ the expected value of a random variable.Figure 1
Two types of pentachains considered in this paper: (a) alpha-pentachain; (b) spiro-pentachain.
(a)(b)


## 2. Wiener and Kirchhoff Indices in Alpha-Pentachains

Analpha-pentachain of length n is obtained from a sequence of 5 cycles (pentagons) C1,C2,…,Cn by adding a bridge to each pair of consecutive 5 cycles, which is denoted by Pnα. The cycle Ci will be called the i-th pentagon of Pnα, 1≤i≤n. Figure 2 shows the unique alpha-pentachains for n=1,2 and all the alpha-pentachains for n=3, and Figure 3 shows a general case, where vn−1 is a vertex of Cn−1 in Pn−1α. Note that there are two ways to add a bridge between two consecutive cycles. So, Pnα may not be unique when n≥3.Figure 2
The alpha-pentachains forn=1,2,3.Figure 3
An alpha-pentachainPnα with n pentagons.ConsiderPn−1α for n≥3. Let Cn−1=x1x2x3x4x5x1, and there is a bridge connecting x1 and vn−2 which is a vertex in Cn−2. There are two possible constructions of getting Pna from Pn−1α by the symmetry of the structure of an alpha-pentachain (see Figure 4). Precisely, let Pnα1 and Pnα2 be obtained by adding a bridge connecting a vertex of the extra pentagon Cn with the vertices x2 and x3 of Cn−1, respectively (see Figure 4). We may assume that for getting a Pnα from a fixed Pn−1α is a random process, namely, the probability of getting Pnα1 and Pnα2 from a fixed Pn−1α is p and 1−p, respectively.Figure 4
Two types of local arrangements in an alpha-pentachain.We assume that the probabilityp is constant and independent of n, i.e., the process described above is a zeroth-order Markov process. After associating probabilities, such an alpha-pentachain is called a random alpha-pentachain and denoted by Pn;pα. Gutman et al. [47, 48] considered the Wiener indices of random benzenoid chain graphs in 1990s. Chen and Zhang [49] obtained explicit analytical expressions for the expected value of the Wiener index and the number of perfect matchings in a random phenylene chain. Yang and Zhang [15] studied the Wiener indices of random polyphenyl chain graphs in 2012. Wei et al. [44, 50, 51] obtained simple formulae for the expected value of the Wiener indices of random polygonal chains. Wang et al. [43] studied the expected value of the Wiener indices of random alpha-pentachains.Theorem 1 (see [43]).
LetPn;pα be a random alpha-pentachain. Then,(6)EWPn;pα=5615−5pn3+5+252pn2−52+253pn.In addition, Huang et al. [42] considered the problem about the Kirchhoff index in random alpha-pentachains. Wei and Ke [44] presented a simple formula for the expected value of Kirchhoff index in random generalized polyomino chain. For more details concerning random chain graphs, one may refer to [52–55] and the references cited therein. In this section, we will further consider the Kirchhoff index of random alpha-pentachain and have the following recurrence formula for computing it.Lemma 1 (see [2]).
Lety be a cut vertex of a connected graph G, and let a and b be vertices occurring in different components which arise upon deletion of y. Then,(7)ra,b=ra,y+ry,b.Lemma 2.
LetPnα be an alpha-pentachain of length n, where n≥2. Then,(8)KfPnα=KfPn−1α+5rvn−1|Pn−1α+45n−35.Proof.
Keeping the notation defined in Figure3, note that by (1), the resistance distance matrix of the graph induced by vn−1,x1,x2,x3,x4,x5 is(9)01951151159510456565459545045456511565450456511565454504595456565450.
By Lemma1 and the matrix above, it is easy to check that(10)rxj,v=rvn−1,v+1,forj=1,rvn−1,v+95,forj=2,5,rvn−1,v+115,forj=3,4,for any vertex v of Pn−1α. Also, for a fixed vertex j, we have(11)∑i=15rxj,xi=45+65+65+45=4,where 1≤j≤5. Note that Pn−1α has 5n−1 vertices. Combining the above formulae, we obtain the following formulae:(12)rx1|Pnα=rvn−1|Pn−1α+1×5n−1+4=rvn−1|Pn−1α+5n−1,rx2|Pnα=rx5|Pnα=rvn−1|Pn−1α+95×5n−1+4=rvn−1|Pn−1α+9n−5,(13)rx3|Pnα=rx4|Pnα=rvn−1|Pn−1α+115×5n−1+4=rvn−1|Pn−1α+11n−7.
Hence,(14)KfPnα=KfPn−1α+5rvn−1|Pn−1α+45n−25−12∑i=15∑j=15rxi,xj=KfPn−1α+5rvn−1|Pn−1α+45n−35.
This completes the proof.
Note that bothKfPn;pα and rvn−1|Pn−1;pα are random variables. For convenience, let Rn−1α=Ervn−1|Pn−1;pα for n≥2. We will keep these notations throughout this section. In order to obtain the explicit expression of EKfPn;pα, we need to find the explicit expression of Rn−1α from Lemma 2.Lemma 3.
LetPn;pα be a random alpha-pentachain of length n, where n≥2. Then,(15)Rn−1α=112−pn−12+p−32n−1.Proof.
Whenn=2, P1;pα is isomorphic to a pentagon. So,(16)R1α=Erv1|P1;pα=45+65+65+45=4.
Whenn≥3, there are two cases to be considered as shown in Figure 4. If Pn−1α⟶Pnα1, then vn−1 coincides with the vertex labeled x2 or x5. Consequently, rvn−1|Pn−1;pα is given by equation (12). If Pn−1α⟶Pnα2, then vn−1 coincides with the vertex labeled x3 or x4. Consequently, rvn−1|Pn−1;pα is given by equation (13). By symmetry, we may only consider vn−1 which coincides with the vertex labeled xi+1 for i=1,2.
Recall that the above two cases occur in random alpha-pentachains with probabilitiesp and 1−p. From equations (2⋅i) i=1,2, we obtain that(17)Rn−1α=pErvn−2|Pn−2;pα+9n−1−5+1−pErvn−2|Pn−2;pα+11n−1−7.
Note thatERn−1α=Rn−1α. By applying the expectation operator to the above equation, we get(18)Rn−1α=ERn−1α=EpRn−2α+9n−1−5+1−pRn−2α+11n−1−7=pRn−2α+9n−1−5+1−pRn−2α+11n−1−7.
That is,(19)Rn−1α=Rn−2α+11−2pn−1+2p−7,forn≥3.
Note that equation (19) is a first-order nonhomogeneous linear difference equation with constant coefficients. It is clear that the general solution of the homogeneous part of equation (19) is Rα=c, a constant.
LetRα∗=n−1an−1+b=an−12+bn−1 be a particular solution of equation (19). By substituting Rα∗ into equation (19), we have(20)an−12+bn−1−an−22−bn−2=2an−1+b−a=11−2pn−1+2p−7.
By comparing the coefficients ofn and the constant term, we have(21)a=112−p,b=p−32.
Consequently, the general solution of equation (19) is(22)Rn−1α=Rα∗+Rα=112−pn−12+p−32n−1+c,forn≥2.
By substituting the initial condition, we obtain thatc=0. Hence, we get the lemma.Theorem 2.
LetPn;pα be a random alpha-pentachain of length n, where n≥1. Then,(23)EKfPn;pα=556−53pn3+5p+5n2−103p+256n.Proof.
SinceP1α is unique, E1 is exactly the Kirchhoff index of P1α, namely, E1=KfP1α=10. So, the theorem holds when n=1. When n≥2, by Lemmas 2 and 3, we have(24)En=EKfPn;pα=EKfPn−1;pα+5rvn−1|Pn−1;pα+45n−35=En−1+5Rn−1α+45n−35=En−1+5112−pn−12+p−32n−1+45n−35=En−1+552−5pn2+15p−352n−10p.
Clearly,E=c, a constant, is the general solution for the homogeneous part of equation (24).
LetE∗=a0n3+a1n2+a2n be a particular solution of equation (24). By substituting E∗ into equation (24), we have(25)a0n3+a1n2+a2n−a0n−13+a1n−12+a2n−1=3a0n2+2a1−3a0n+a0−a1+a2=552−5pn2+15p−352n−10p.
By comparing the coefficients ofni term for 0≤i≤2, we have(26)a0=556−53p,a1=5p+5,a2=−103p−256.
Combining the result and the initial condition, we get that the general solution of equation (24) is(27)En=E∗+E=556−53pn3+5p+5n2−103p+256n.
Theorems1 and 2 show that both the function EWPn;pα and EKfPn;pα are asymptotic to a cubic in n and linear in p. In particular, it is easy to obtain the Wiener and Kirchhoff indices of two special alpha-pentachains Pn;1α and Pn;0α when setting p=1 and p=0 from Theorems 1 and 2.Corollary 1.
The Wiener indices of the alpha-pentachainsPn;1α and Pn;0α are(28)WPn;1α=253n3+352n2−656n,WPn;0α=252n3+5n2−52n.Corollary 2.
The Kirchhoff indices of the alpha-pentachainsPn;1α and Pn;0α are(29)KfPn;1α=152n3+10n2−152n,KfPn;0α=556n3+5n2−256n.Klein and Randić [2] proved that KfG≤WG with equality if and only if G is a tree. Combining Theorem 1 with Theorem 2, we can also verify that EKfPn;pα<EWPn;pα.Theorem 3.
LetPn;pα be a random alpha-pentachain. Then,(30)EKfPn;pα<EWPn;pα.Proof.
By Theorems1 and 2, we have(31)6EWPn;pα−6EKfPn;pα=20−20pn3+45pn2+10−30pn=20−20pn3+n45n−30p+10.
Since0≤p≤1, 20−20pn3+n45n−30p+10>0, which implies that(32)EKfPn;pα<EWPn;pα.
Hence, this completes the proof.


## 3. Wiener and Kirchhoff Indices in Spiro-Pentachains

The notation not defined in this section can be found in Section2. A spiro-pentachain of length n, denoted by Pns, can be obtained from an alpha-pentachain Pnα by contracting each bridge between each pair of consecutive pentagons in Pnα. Figure 5 shows a general spiro-pentachain, where vn−1 is a vertex of Cn−1 in Pn−1s. Clearly, Pns may not be unique when n≥3. Corresponding to the structure of alpha-pentachain Pnα, the spiro-pentachain Pns has also two types of local arrangements which are denoted by Pns1 and Pns2 (see Figure 6). We may also assume that for getting Pns from fixed Pn−1s is a random process, namely, the probability of getting Pns1 and Pns2 from fixed Pn−1s is p and 1−p, respectively.Figure 5
A spiro-pentachainPns with n pentagons.Figure 6
Two types of local arrangements in a spiro-pentachain.We assume that the probabilityp is constant and independent of n, i.e., the process described above is a zeroth-order Markov process. After associating probabilities, such a spiro-pentachain is called a random spiro-pentachain and denoted by Pn;ps. In this section, we will consider the Wiener and Kirchhoff indices of random spiro-pentachain.Lemma 4.
LetPns be a spiro-pentachain of length n, where n≥2. Then, the Wiener index WPns of Pns is(33)WPn−1s+4dvn−1|Pn−1s+24n−9.Proof.
Keeping the notation defined in Figure5, it is easy to check that(34)dxj,v=dvn−1,v,forj=1,dvn−1,v+1,forj=2,5,dvn−1,v+2,forj=3,4,for any vertex v of Pn−1s. Also, for a fixed vertex j, we have(35)∑i=15dxj,xi=1+1+2+2=6,where 1≤j≤5. Note that Pn−1s has 5n−1−n−2=4n−3 vertices and x1 coincides with vn−1 in Pns. Combining the above formulae, we obtain the following formulae:(36)dx1|Pns=dvn−1|Pns=dvn−1|Pn−1s+6,dx2|Pns=dx5|Pns=dvn−1|Pn−1s+4n−1+6=dvn−1|Pn−1s+4n+2,(37)dx3|Pns=dx4|Pns=dvn−1|Pn−1s+2×4n−1+6=dvn−1|Pn−1s+8n−2.
Hence,(38)WPns=WPn−1s+6+4dvn−1|Pn−1s+24n−12∑i=15∑j=15dxi,xj=WPn−1s+4dvn−1|Pn−1s+24n−9.
This completes the proof.
Note that bothWPn;ps and rvn−1|Pn−1;ps are also random variables. In order to obtain the explicit expression of EWPn;ps, we need to find the explicit expression of Dn−1s≜Edvn−1|Pn−1s for n≥2 from Lemma 4.Lemma 5.
LetPn;ps be a random spiro-pentachain of length n, where n≥2. Then,(39)Dn−1s=4−2pn−12+2p+2n−1.Proof.
Whenn=2, P1;ps is isomorphic to a pentagon. So,(40)D1s=Edv1|P1;ps=6.
Whenn≥3, there are two cases to be considered as shown in Figure 6. If Pn−1s⟶Pns1, then vn−1 coincides with the vertex labeled x2 or x5. Consequently, dvn−1|Pn−1;ps is given by equation (36). If Pn−1s⟶Pns2, then vn−1 coincides with the vertex labeled x3 or x4. Consequently, dvn−1|Pn−1;ps is given by equation (37). By symmetry, we may only consider vn−1 which coincides with the vertex labeled xi+1 for i=1,2.
Recall that the above two cases occur in random spiro-pentachains with probabilitiesp and 1−p. From equations (3·i) i=1,2, we obtain that(41)Dn−1s=pEdvn−2|Pn−2;ps+4n−1+2+1−pEdvn−2|Pn−2;ps+8n−1−2.
Note thatEDn−1s=Dn−1s. By applying the expectation operator to the above equation, we get(42)Dn−1s=EDn−1s=EpDn−2s+4n−1+2+1−pDn−2s+8n−1−2=pDn−2s+4n−1+2+1−pDn−2s+8n−1−2.
That is,(43)Dn−1s=Dn−2s+8−4pn−1+4p−2,forn≥2.
Using recurrence relation (43) and the initial condition, we have(44)Dn−1s=4−2pn−12+2p+2n−1.
Hence, we get the lemma.Theorem 4.
LetPn;ps be a random spiro-pentachain of length n, where n≥1. Then,(45)EWPn;ps=163−83pn3+8p+8n2+53−163pn.Proof.
For convenience, letEnw=EWPn;ps. Since P1s is unique, E1w is exactly the Wiener index of P1s, namely, E1w=WP1s=15. So, the theorem holds when n=1. When n≥2, by Lemmas 4 and 5, we have(46)Enw=EWPn;ps=EWPn−1;ps+4dvn−1|Pn−1;ps+24n−9=En−1w+4Dn−1s+24n−9=En−1w+44−2pn−12+2p+2n−1+24n−9=En−1w+16−8pn2+24pn−1+16p.
Using recurrence relation (46) and the initial condition, we have(47)EWPn;ps=163−83pn3+8p+8n2+53−163pn.
We now give the formula for the expected value of the Kirchhoff index of a random spiro-pentachain.Lemma 6.
LetPns be a spiro-pentachain of length n, where n≥2. Then,(48)KfPns=KfPn−1s+4rvn−1|Pn−1s+16n−6.Proof.
Keeping the notation defined in Figure5, note that the resistance distance matrix of the graph induced by x1,x2,x3,x4,x5 is(49)045656545450656545654504565656545045456565450,by (1). By Lemma 1 and the matrix above, it is easy to check that(50)rxj,v=rvn−1,v,forj=1,rvn−1,v+45,forj=2,5,rvn−1,v+65,forj=3,4,for any vertex v in Pn−1s. From the proof of Lemma 2, we have(51)∑i=15rxj,xi=4,for a fixed j, 1≤j≤5. Note that Pn−1s has 5n−1−n−2=4n−3 vertices and x1 coincides with vn−1 in Pns. Combining the formulae above, we obtain the following formulae:(52)rx1|Pns=rvn−1|Pns=rvn−1|Pn−1s+4,rx2|Pns=rx5|Pns=rvn−1|Pn−1s+45×4n−1+4=rvn−1|Pn−1s+454n+1,(53)rx3|Pns=rx4|Pns=rvn−1|Pn−1s+65×4n−1+4=rvn−1|Pn−1s+456n−1.
Hence,(54)KfPns=KfPn−1s+4+rvn−1|Pn−1s+454n+1×2+rvn−1|Pn−1s+456n−1×2−12∑i=15∑j=15rxi,xj=KfPn−1s+4rvn−1|Pn−1s+16n−6.
This completes the proof.Lemma 7.
LetPn;ps be a random spiro-pentachain of length n, where n≥2. Then,(55)Rn−1s=453−pn−12+452+pn−1,where Rn−1s=Ervn−1|Pn−1;ps.Proof.
Whenn=2,(56)R1s=Erv1|P1;ps=45+65+65+45=4.
Whenn≥3, there are two cases to be considered as shown in Figure 6. If Pn−1s⟶Pns1, then vn−1 coincides with the vertex labeled x2 or x5. Consequently, rvn−1|Pn−1;ps is given by equation (52). If Pn−1s⟶Pns2, then vn−1 coincides with the vertex labeled x3 or x4. Consequently, rvn−1|Pn−1;ps is given by equation (53). By symmetry, we may only consider vn−1 which coincides with the vertex labeled xi+1 for i=1,2.
Recall that the above two cases occur in random spiro-pentachains with probabilitiesp and 1−p. From equations (3·i) i=5,6, we obtain that(57)Rn−1s=Eprvn−2|Pn−2;ps+454n−1+1+1−prvn−2|Pn−2;ps+456n−1−1.
By applying the expectation operator to the above equation, we get(58)Rn−1s=Rn−2s+853−pn−1+452p−1,forn≥2.
Using recurrence relation (58) and the initial condition, we have(59)Rn−1s=453−pn−12+452+pn−1.
Hence, we get the lemma.
Similar to the proof in Theorem2, now we have the following result by Lemmas 6 and 7.Theorem 5.
LetPn;ps be a random spiro-pentachain of length n, where n≥1. Then,(60)EKfPn;ps=16153−pn3+1652+pn2+2153−16pn.
Similarly, Theorems4 and 5 show that both the functions EWPn;ps and EKfPn;ps are asymptotic to a cubic in n and linear in p. Then, we easily get the Wiener and Kirchhoff indices of two special spiro-pentachains Pn;1s and Pn;0s when setting p=1 and p=0 from Theorems 4 and 5.Corollary 3.
The Wiener indices of the spiro-pentachainsPn;1s and Pn;0s are(61)WPn;1s=83n3+16n2−113n,WPn;0s=163n3+8n2+53n.Corollary 4.
The Kirchhoff indices of the spiro-pentachainsPn;1s and Pn;0s are(62)KfPn;1s=3215n3+485n2−2615n,KfPn;0s=4815n3+325n2+25n.By Theorems4 and 5, similar to Theorem 3, we also have the following result.Theorem 6.
LetPn;ps be a random spiro-pentachain. Then,(63)EKfPn;ps<EWPn;ps.


## 4. The Average Values of the Kirchhoff and Wiener Indices in Random Pentachains

In this section, we present the average values of the Kirchhoff and Wiener indices in random pentachains with respect to the set of all alpha-pentachains and all spiro-pentachains withn pentagons. Let Pnα and Pns be the sets of all alpha-pentachains and all spiro-pentachains with n pentagons. The average values of Wiener indices of Pnα and Pns are defined by(64)WavrPnα=1Pnα∑G∈PnsWG,WavrPns=1Pns∑G∈PnsWG,respectively. Similarly, the average values of Kirchhoff indices of Pnα and Pns are defined by(65)KfavrPnα=1Pnα∑G∈PnsKfG,KfavrPns=1Pns∑G∈PnsKfG,respectively.Actually, these are the population means of the Wiener and Kirchhoff indices of all elements inPnα or Pns. Since every element occurring in Pnα or Pns has the same probability, we have p=1−p. Thus, we may apply Theorems 1, 2, 4, and 5 by putting p=1−p=1/2 and obtain the following results.Theorem 7.
The average values of the Wiener indices with respect toPns and Pnα are(66)WavrPnα=12512n3+454n2−203n,WavrPns=4n3+12n2−n.Theorem 8.
The average values of the Kirchhoff indices with respect toPns and Pnα are(67)KfavrPnα=253n3+152n2−356n,KfavrPns=4n3+12n2−n.It is not difficult to note that the average values of the Wiener indices with respect toPn;1α,Pn;0α and Pn;1s,Pn;0s are(68)WPn;1α+WPn;0α2=12512n3+454n2−203n,WPn;1s+WPn;0s2=83n3+8n2−23n,from Corollaries 1 and 3 and Theorem 8. We also can find that the average values of the Kirchhoff indices with respect to Pn;1α,Pn;0α and Pn;1s,Pn;0s are(69)KfPn;1α+KfPn;0α2=253n3+152n2−356n,KfPn;1s+KfPn;0s2=83n3+8n2−23n,from Corollaries 2 and 4 and Theorem 8. The interesting results show that both the average values of the Wiener indices and the average values of the Kirchhoff indices with respect to Pnα and Pns are exactly equal to the average value of the Wiener and Kirchhoff indices with respect to Pn;1α,Pn;0α and Pn;1s,Pn;0s, respectively.



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