Comparison of the Wiener and Kirchhoff Indices of Random Pentachains

. Let G be a connected (molecule) graph. The Wiener index W ( G ) and Kirchhoﬀ index Kf ( G ) 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.


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 V(G) � v 1 , v 2 , . . ., v n   and edge set E(G). e traditional distance d(v r , v s ) between vertices v r and v s 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 r(v i , v j ) the resistance distance between the vertices v i and v j 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/n)J) − 1 .en, the resistance distance between vertices i and j of G is r(i, j) � X i,i + X j,j − 2X i,j . ( e Wiener index W(G) [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. is 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., where d(v i |G) denotes the sum of distances between v i and all other vertices of G, defined by For a survey of results and bibliography on the chemical applications and the mathematical literature of the Wiener index, see [7][8][9][10][11][12][13][14][15] and the references cited therein.
Similar to the Wiener index, the Kirchhoff index Kf(G) of a graph G [2,16] is defined as where r(v i |G) denotes the sum of resistance distances between v i and all other vertices of G, defined by Klein and Randić [2] proved that r(v i , v j ) ≤ d(v i , v j ) and Kf(G) ≤ W(G) 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 distanceregular graphs and Möbius ladders [17][18][19][20][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 R top � (Kf/n 2 ) of macromolecules containing cyclic fragments.Many mathematical properties have also been established (see [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41] and the references cited therein).
Motivated by the works in [42][43][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 spiropentachain.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 E[Ξ] the expected value of a random variable.

Wiener and Kirchhoff Indices in Alpha-Pentachains
An alpha-pentachain of length n is obtained from a sequence of 5 cycles (pentagons) C 1 , C 2 , . . ., C n by adding a bridge to each pair of consecutive 5 cycles, which is denoted by P α n .e cycle C i will be called the i-th pentagon of P α n , 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 v n− 1 is a vertex of C n− 1 in P α n− 1 .Note that there are two ways to add a bridge between two consecutive cycles.So, P α n may not be unique when n ≥ 3. Consider P α n− 1 for n ≥ 3. Let C n− 1 � x 1 x 2 x 3 x 4 x 5 x 1 , and there is a bridge connecting x 1 and v n− 2 which is a vertex in C n− 2 .ere are two possible constructions of getting P a n from P α n− 1 by the symmetry of the structure of an alpha-pentachain (see Figure 4).Precisely, let P n from a fixed P α n− 1 is p and 1 − p, respectively.We assume that the probability p is constant and independent of n, i.e., the process described above is a zerothorder Markov process.After associating probabilities, such an alpha-pentachain is called a random alpha-pentachain and denoted by P α n;p .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]).Let P α n;p be a random alpha-pentachain.en, 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][53][54][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]).Let y 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. en, Lemma 2. Let P α n be an alpha-pentachain of length n, where n ≥ 2. en, 2

Journal of Mathematics
Proof.Keeping the notation defined in Figure 3, note that by (1), the resistance distance matrix of the graph induced by By Lemma 1 and the matrix above, it is easy to check that for any vertex v of P α n− 1 .Also, for a fixed vertex j, we have where 1 ≤ j ≤ 5. Note that P α n− 1 has 5(n − 1) vertices.Combining the above formulae, we obtain the following formulae:  Hence, is completes the proof.Note that both Kf(P α n;p ) and r( for n ≥ 2. We will keep these notations throughout this section.In order to obtain the explicit expression of E[Kf(P α n;p )], we need to find the explicit expression of R α n− 1 from Lemma 2. □ Lemma 3. Let P α n;p be a random alpha-pentachain of length n, where n ≥ 2. en, Proof.When n � 2, P α 1;p is isomorphic to a pentagon.So, When n ≥ 3, there are two cases to be considered as shown in Figure 4.
n , then v n− 1 coincides with the vertex labeled x 3 or x 4 .Consequently, r(v n− 1 |P α n− 1;p ) is given by equation (13).By symmetry, we may only consider v n− 1 which coincides with the vertex labeled x i+1 for i � 1, 2.
Recall that the above two cases occur in random alphapentachains with probabilities p and 1 − p. From equations By applying the expectation operator to the above equation, we get at is, 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 be a particular solution of equation (19).By substituting R α * into equation (19), we have By comparing the coefficients of n and the constant term, we have Consequently, the general solution of equation ( 19) is By substituting the initial condition, we obtain that c � 0. Hence, we get the lemma.(

□
Proof.Since P α 1 is unique, E 1 is exactly the Kirchhoff index of P α 1 , namely, E 1 � Kf(P α 1 ) � 10.So, the theorem holds when n � 1.When n ≥ 2, by Lemmas 2 and 3, we have Clearly, E � c, a constant, is the general solution for the homogeneous part of equation (24).
Let E * � a 0 n 3 + a 1 n 2 + a 2 n be a particular solution of equation (24).By substituting E * into equation ( 24), we have By comparing the coefficients of n i term for 0 ≤ i ≤ 2, we have Combining the result and the initial condition, we get that the general solution of equation ( 24) is Hence, this completes the proof.

Wiener and Kirchhoff Indices in Spiro-Pentachains
e notation not defined in this section can be found in Section 2. A spiro-pentachain of length n, denoted by P s n , can be obtained from an alpha-pentachain P α n by contracting each bridge between each pair of consecutive pentagons in P α n .Figure 5 shows a general spiro-pentachain, where v n− 1 is a vertex of C n− 1 in P s n− 1 .Clearly, P s n may not be unique when n ≥ 3. Corresponding to the structure of alpha-pentachain P α n , the spiro-pentachain P s n has also two types of local arrangements which are denoted by P s 1 n and P s 2 n (see Figure 6).We may also assume that for getting P s n from fixed P s n− 1 is a random process, namely, the probability of getting P s 1 n and P s 2 n from fixed P s n− 1 is p and 1 − p, respectively.We assume that the probability p is constant and independent of n, i.e., the process described above is a zerothorder Markov process.After associating probabilities, such a spiro-pentachain is called a random spiro-pentachain and denoted by P s n;p .In this section, we will consider the Wiener and Kirchhoff indices of random spiro-pentachain.

Lemma 4. Let P s
n be a spiro-pentachain of length n, where n ≥ 2. en, the Wiener index W(P s n ) of P s n is Proof.Keeping the notation defined in Figure 5, it is easy to check that for any vertex v of P s n− 1 .Also, for a fixed vertex j, we have where 1 ≤ j ≤ 5. Note that P s n− 1 has 5(n − 1) − (n − 2) � 4n − 3 vertices and x 1 coincides with v n− 1 in P s n .Combining the above formulae, we obtain the following formulae: Hence, is completes the proof.Note that both W(P s n;p ) and r(v n− 1 |P s n− 1;p ) are also random variables.In order to obtain the explicit expression of E[W(P s n;p )], we need to find the explicit expression of

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Lemma 5. Let P s n;p be a random spiro-pentachain of length n, where n ≥ 2. en, Proof.When n � 2, P s 1;p is isomorphic to a pentagon.So, When n ≥ 3, there are two cases to be considered as shown in Figure 6 ) is given by equation (37).By symmetry, we may only consider v n− 1 which coincides with the vertex labeled x i+1 for i � 1, 2.
Recall that the above two cases occur in random spiropentachains with probabilities p and 1 − p. From equations (3•i) (i � 1, 2), we obtain that 6 Journal of Mathematics Note By applying the expectation operator to the above equation, we get at is, Using recurrence relation (43) and the initial condition, we have Hence, we get the lemma.

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Theorem 4. Let P s n;p be a random spiro-pentachain of length n, where n ≥ 1. en, Proof.For convenience, let 1 is exactly the Wiener index of P s 1 , namely, E w 1 � W(P s 1 ) � 15.So, the theorem holds when n � 1.When n ≥ 2, by Lemmas 4 and 5, we have Using recurrence relation (46) and the initial condition, we have We now give the formula for the expected value of the Kirchhoff index of a random spiro-pentachain.□ Lemma 6.Let P s n be a spiro-pentachain of length n, where n ≥ 2. en, by (1).By Lemma 1 and the matrix above, it is easy to check that , for j � 2, 5, for any vertex v in P s n− 1 .From the proof of Lemma 2, we have for a fixed j, 1 ≤ j ≤ 5. Note that P s n− 1 has 5(n − 1) − (n − 2) � 4n − 3 vertices and x 1 coincides with v n− 1 in P s n .Combining the formulae above, we obtain the following formulae: Hence, is completes the proof.

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Lemma 7. Let P s n;p be a random spiro-pentachain of length n, where n ≥ 2. en, where n , then v n− 1 coincides with the vertex labeled x 3 or x 4 .Consequently, r(v n− 1 |P s n− 1;p ) is given by equation (53).By symmetry, we may only consider v n− 1 which coincides with the vertex labeled x i+1 for i � 1, 2.
Recall that the above two cases occur in random spiropentachains with probabilities p and 1 − p. From equations (3•i) (i � 5, 6), we obtain that

Figure 4 :
Figure 4: Two types of local arrangements in an alpha-pentachain.

Figure 5 :
Figure 5: A spiro-pentachain P s n with n pentagons.

Figure 6 :
Figure 6: Two types of local arrangements in a spiro-pentachain.

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 with n pentagons.Let P α n and P s n be the sets of all alpha-pentachains and all spiro-pentachains with n pentagons.eaveragevaluesofWienerindices of P α n and P s n are defined by are the population means of the Wiener and Kirchhoff indices of all elements in P α n or P s n .Since every element occurring in P α n or P s n has the same probability, we have p � 1 − p. us, we may apply eorems 1, 2, 4, and 5 by putting p � 1 − p � (1/2) and obtain the following results.It is not difficult to note that the average values of the Wiener indices with respect to P α n;1 , P α from Corollaries 2 and 4 and eorem 8. e interesting results show that both the average values of the Wiener indices and the average values of the Kirchhoff indices with respect to P α n and P s n are exactly equal to the average value of the Wiener and Kirchhoff indices with respect to