ARS MATHEMATICA CONTEMPORANEA
Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 9 (2015) 197-207
The expected values of Kirchhoff indices in the random polyphenyl and spiro chains*
Guihua Huang
College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, P. R. China
Meijun Kuang
College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, P. R. China
Hanyuan Deng t
College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, P. R. China
Received 14 March 2013, accepted 22 April 2014, published online 28 November 2014
The Kirchhoff index Kf (G) of a graph G is the sum of resistance distances between all pairs of vertices in G. In this paper, we obtain exact formulas for the expected values of the Kirchhoff indices of the random polyphenyl and spiro chains, which are graphs of a class of unbranched multispiro molecules and polycyclic aromatic hydrocarbons. Moreover, we obtain a relation between the expected values of the Kirchhoff indices of a random polyphenyl and its random hexagonal squeeze, and the average values for the Kirchhoff indices of all polyphenyl chains and all spiro chains with n hexagons, respectively.
Keywords: Expected value, average value, Kirchhoff index, resistance distance, polyphenyl chain, spiro chain.
Math. Subj. Class.: 05C12, 05C80, 05C90, 05D40
* Project supported by Hunan Provincial Natural Science Foundation of China(13JJ3053). t Corresponding Author.
E-mail addresses: 380026412@qq.com (Guihua Huang), 1075998525@qq.com (Meijun Kuang), hydeng@hunnu.edu.cn (Hanyuan Deng)
©® This work is licensed under http://creativecommons.org/licenses/by/3.0/
Abstract
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Ars Math. Contemp. 9 (2015) 165-186
1 Introduction
Based on the electrical network theory, Klein and Randic [13] introduced the concept of resistance distance. A connected graph G with vertex set {v1,v2, • • • ,vn} is viewed as an electrical network N by replacing each edge of G with a unit resistor, the resistance distance between vi and vj, denoted by rG(vi,vj) or r(vi,vj), is the elective resistance between them as computed by the methods of the theory of resistive electrical networks based on Ohm's and Kirchhoff's laws in N.
The Kirchhoff index of G, denoted by Kf (G), is the sum of resistance distances between all pairs of vertices in G, namely,
Kf (G) = ^ rG(vi,vj) i<j
Like many topological indices, Kirchhoff index is a structure descriptor. The resistance distance is also intrinsic to the graph with some nice purely mathematical and physical interpretations [14] [15]. Also, the Kirchhoff index has been found very useful in chemistry, such as in assessing cyclicity of polycyclic structures including fullerenes, linear hexagonal chains and some special molecular graphs such as circulant graphs, distance-regular graphs and Mobius ladders [1] [18] [22] [24]. Bonchev et al. [4] used it in polymer science and found that the Kirchhoff index in their approach is especially useful for defining the topological radius Rtop = of macromolecules containing cyclic fragments. Some closed-form formulae for Kirchhoff index have been given for circulant graphs, linear hexagonal chains and so on [1] [16] [19] [22]. The resistance distance is also well studied in mathematical literatures. Much work has been done to compute Kirchhoff index of some classes of graphs, or give some bounds for Kirchhoff index of graphs and characterize extremal graphs. For instance, unicyclic and bicyclic graphs with extremal Kirchhoff index are characterized and sharp bounds for Kirchhoff index of such graphs are obtained [6] [12] [21] [25] [26].
Polyphenyls and their derivatives, which can be used in organic synthesis, drug synthesis, heat exchangers, etc., attracted the attention of chemists for many years [11] [17] [20]. Spiro compounds are an important class of cycloalkanes in organic chemistry. A spiro union in spiro compounds is a linkage between two rings that consists of a single atom common to both rings and a free spiro union is a linkage that consists of the only direct union between the rings. Some results on energy, Merrifield-Simmons index, Hosoya index and Wiener index of the spiro and polyphenyl chains were reported in [2] [5] [9] [27]. Recently, Deng [7] [8] [10] gave the recurrences or explicit formulae for computing the Wiener index and Kirchhoff index of spiro and polyphenyl chains. Yang and Zhang [23] obtained a simple exact formula for the expected value of the Wiener index of a random polyphenyl chain. In this paper, we will consider the expected values of the Kirchhoff index of random polyphenyl and spiro chains.
A polyphenyl chain PPCn with n hexagons can be regarded as a polyphenyl chain PPCn-1 with n - 1 hexagons to which a new terminal hexagon has been adjoined by a cut edge, see Figure 1.
Let PPCn = H1H2 • • • Hn be a polyphenyl chain with n(n > 2) hexagons, where Hk is the k-th hexagon of PPCn attached to Hk-1 by a cut edge uk-1ck, k = 2, 3, • • • , n. A vertex v of Hk is said to be ortho-, meta- and para-vertex of Hk if the distance between v and ck is 1,2 and 3, denoted by ok, mk and pk, respectively. Examples of ortho-, meta-, and
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199
para-vertices are shown in Figure 1. Except the first hexagon, any hexagon in a polyphenyl chain has two ortho-vertices, two meta-vertices and one para-vertex.
X2 X3
Figure 1: A polyphenyl chain PPCn with n hexagons, cn = x\ and ortho-vertices on = x2,x6, meta-vertices mn = x3,x5, and para-vertex pn = x4 in Hn.
A polyphenyl chain PPCn is a polyphenyl ortho-chain if uk = ok for 2 < k < n — 1. A polyphenyl chain PPCn is a polyphenyl meta-chain if uk = mk for 2 < k < n — 1. A polyphenyl chain PPCn is a polyphenyl para-chain if uk = pk for 2 < k < n — 1. The polyphenyl ortho-, meta- and para-chain with n hexagons are denoted by On, Mn and Pn, respectively.
For n > 3, the terminal hexagon can be attached to meta-, ortho-, or para-vertex in three ways, which results in the local arrangements we describe as PPC*n+1, PPC*n+1, PPCn+1, see Figure 2.
Figure 2: The three types of local arrangements in polyphenyl chains.
A random polyphenyl chain PPC(n,p1,p2) with n hexagons is a polyphenyl chain obtained by stepwise addition of terminal hexagons. At each step k(= 3,4, • • • , n), a random selection is made from one of the three possible constructions:
(i)PPCk-1	^ PPC1 with probability pu
(ii)PPCfc_1	^ PPC2 with probability p2,
(iii)PPCk-1	^ PPCf with probability 1 — p1 — p2
where the probabilities p1 and p2 are constants, irrespective to the step parameter k.
Specially, the random polyphenyl chain PPC(n, 1,0) is the polyphenyl meta-chain Mn, PPC(n, 0,1) is the polyphenyl orth-chain On, and PPC(n, 0,0) is the polyphenyl para-chain Pn, respectively.
Also, a spiro chain SPCn with n hexagons can be regarded as a spiro chain SPCn-1 with n — 1 hexagons to which a new terminal hexagon has been adjoined, see Figure 3.
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Ars Math. Contemp. 9 (2015) 165-186
_ x6 x 5
SPCn-1 Un-^Xi Hn^ X4
X2 X3
Figure 3: A spiro chain SPCn with n hexagons.
For n > 3, the terminal hexagon can also be attached in three ways, which results in the local arrangements we describe as SPCn+1, SPCn+1, SPC*n+1, see Figure 4.
SPCn
o r>r~t 1	cpri2	cprt
SPCn+1	SPCn+1	SPCn+1
Figure 4: The three types of local arrangements in spiro chains.
A random spiro chain SPC(n,p1,p2) with n hexagons is a spiro chain obtained by stepwise addition of terminal hexagons. At each step k(= 3,4, • • • , n), a random selection is made from one of the three possible constructions:
(i)SPCfc_1	^ SPC1 with probability p1,
(ii)SPCfc_1	^ SPC2 with probability p,
(iii)SPCfc_1	^ SPC| with probability 1 - p - p2
where the probabilities p1 and p2 are constants, irrelative to the step parameter k.
Similarly, the random spiro chain SPC(n, 1,0), PPC(n, 0, l) and PPC(n, 0,0) are the spiro meta-chain Mn, the spiro orth-chain On and the spiro para-chain Pn, respectively.
For a random polyphenyl chain PPC(n,p1,p2) and a random spiro chain SPC(n, p1,p2), their Kirchhoff indices are random variables. In this paper, we will obtain exact formulas for the expected values E(Kf (PPC(n,p1,p2))) and E(Kf (SPC(n,p1,p2))) of the Kirchhoff indices of random polyphenyl and spiro chains, respectively.
2 Main results
2.1 The Kirchhoff index of the random polyphenyl chain
In this section, we will consider the Kirchhoff index of the random polyphenyl chain.
Theorem 2.1. For n > 1, the expected value of the Kirchhoff index of the random polyphenyl chain PPC(n,p1,p2) is
E(Kf (PPC(n,p1,p2))) = (15 -P1 -4p2)n3 + (3p + 12p + 8)n2 - (2p + 8p + y)n
Proof. Note that the polyphenyl chain PPCn is obtained by attaching PPCn-1 a new terminal hexagon by an edge, we suppose that the terminal hexagon is spanned by vertices x1, x2, x3, x4, x5, x6, and the new edge is un_1x1 (see Fig.1). Then (i) For any v e PPCn-1,
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r(xi,v) = r(w„_i,v) + 1, r(x2,v) = r(w„_i,v) + 1 + |, r(x3, v) = r(u„_i,v) + 1 + §, r(x4, v) = r(u„_i, v) + 1 + §, r(xi, v) = r(u„_i,v) + 1 + §, r(x6, v) = r(u„_i, v) + 1 + 1 ; (ii) PPCn_i has 6(n - 1) vertices;
(iii)For k G {1, 2, 3,4, 5, 6}, £ r(xk,x,) = 35
i=i
So, we have
r(xi r(x2 r(x3
r(w„_i|PPC„_i) + 1 x 6(n - 1) + f r(u„_i|PPC„_i) + (1 + 6) x 6(n - 1) + 3|
PPC„) PPC„)
PPC„) = r(u„_i|PPC„_i) + (1 + 4) x 6(n - 1) + f r(u„_i|PPC„_i) + (1 + 3) x 6(n - 1) + f
r(x4|PPC„) r(xi|PPC„) = r(x3|PPC„_i) r(x6|PPC„) = r(x2|PPC„_i) where r(x|G) = J2 r(x, y), and
yev(G)
66
Kf (PPG„) = Kf (PPG„_i) + 6r(u„_i|PPG„_i) + 71n - 36 - - ]T ]Tr(v4, v;)
i=i j=i
Then
35
= Kf (PPC„_i) + 6r(u„_i|PPG„_i) + 71n - 36 - 35
35
Kf (PPGn+i) = Kf (PPG„) + 6r(u„|PPG„)+71n + 35	(2.1)
For a random polyphenyl chain PPC(n,pi,p2), the resistance number r(un|PPC(n, pi, p2)) is a random variable, and its expected value is denoted by
U„ = E(r(u„|PPC (n,pi,p2))).
By the expectation operator and (1), we can obtain a recursive relation for the expected value of the Kirchhoff number of a random polyphenyl chain PPC(n, pi, p2 )
35
E (Kf (PPC (n + 1,pi,p2)) = E (Kf (PPC (n,pi,p2))) + 6U„ + 71n + 35 (2.2)
Now, we consider computing Un.
(i)	If PPCn ^ PPCn+1 with probability pi, then un coincides with the vertex x3 or x5. Consequently, r(w„|PPC„) is given by r(x3|PPCn) with probability p1.
(ii)	If PPCn ^ PPGn+1 with probability p2, then un coincides with the vertex x2 or x6. Consequently, r(un|PPCn) is given r(x2 |PPCn) with probability p2.
(iii)	If PPCn ^ PPGn+1 with probability 1 - p1 - p2, then un coincides with the vertex x4. Consequently, r(w„|PPC„) is given by r(x4|PPCn) with probability 1 -p1 -p2.
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Ars Math. Contemp. 9 (2015) 165-186
From (i)-(iii) above, we immediately obtain
Un =r(x3\PPCn)pi + r(x2\PPCn)p2 + r(x4\PPCn )(1 - pi - P2 )
35
=Pi[r(un_i\PPC(n - 1,pi,p2)) + 14(n - 1) + —]
6
35
+ P2[r(un-i\PPC(n - 1,pi,p2)) + 11(n - 1) + — ]
6
35
+ (1 - pi - p2)[r(un-i\PPC(n - 1,pi,p2)) + 15(n - 1) + — ]
6
By applying the expectation operator to the above equation, we obtain
55
Un = Un-i + (15 - pi - 4p2)n + pi +4p2 - —
6
And Ui = E(r(ui\PPC(1,pi,p2))) = 3f, using the above recurrence relation, we have „ (15 - pi - 4p2) 2 , fpi , o 5\
Un = -^-n + (y + 2p2 - 3)n
From (2),
E(Kf (PPC (n +1,pi,p2))
= E(Kf (PPC(n,pi,p2))) + 6[(i5-p2-4p2)n2 + ( f + 2p2 - f )n] + 71n + 35 = E(Kf (PPC(n,pi,p2))) + (45 - 3pi - 12p2)n2 + (3pi + 12p2 + 61)n + ^
and E(Kf (PPC (1,pi,p2))) = ^l5.
Using the above recurrence relation, we have
E(Kf (PPC (n,pi,p2))) = (15 - pi - 4p2)n3 + (3pi + 12p2 +8)n2 - (2pi + 8p2 +
□
Specially, by taking (pi,p2) = (1,0), (0,1) or (0,0), respectively, and Theorem 2.1, we have
Corollary 2.2. ([8]) The Kirchhoff indices of the polyphenyl meta-chain Mn, the poly-phenyl ortho-chain On and the polyphenyl para-chain Pn are
__15
Kf (Mn) = 14n3 + 11n2 - y n
__27
Kf (On) = 11n3 + 20n2 - — n
Kf (Pn) = 15n3 + 8n2 - y n
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2.2 The Kirchhoff index of the random spiro chain
In this section, we will consider the Kirchhoff index of the random spiro chain.
Theorem 2.3. For n > 1, the expected value of the Kirchhoff index of the random spiro chain SPC(n,pi,p2) is
E(Kf (SPC(n,pi,p2))) = (f - §Pi - f p2)n3 + (§p + f p + 1f )n2
.25 50 5.
— ( — p +--p2--)n.
V18^ 9 2 6'
Proof. Note that the spiro chain SPCn is obtained by attaching SPCn-1 a new terminal hexagon, we suppose that the terminal hexagon is spanned by vertices x1, x2, x3, x4, x5,
x6, and the vertex x1 is wn-1 (see Fig.3). Then
(i)	For any v e SPC„_i,
r(xi, v) = r(un_i, v), r(x2, v) = r(u„_i,v) + 5, r(x3, v) = r(u„_i,v) + 3, r(x4, v) = r(u„_i,v) + 3, r(x5, v) = r(un_i, v) + 3, r(x6, v) = r(u„_i,v) + g;
(ii)	SPCn_i has 5(n - 1) + 1 vertices;
6
(iii)	For k e {1, 2, 3,4, 5, 6}, ^ r(xfc,x,) = 35.
i=i
So, we have
r(xi|SPC„) = r(u„_i|SPC„_i) + 35
r(x2|SPC„) = rK_i|SPC„_i) + 5 x(5n-4)+g + 3 + f+4 = r(u„_1 |SPC„_i) +
25 x (n -1)-+ 35
r(x3|SPC„) = r(u„_i|SPC„_i) + f x (n - 1) + f r(x4|SPC„) = r(u„_i|SPC„_i) + f x (n - 1) + 35 r(x5|SPC„ ) = r(x3|SPC„_i) r(xg|SPC„ ) = r(x2|SPC„_i) where r(x|G) = J2 r(x, y),and
yev(G)
Kf (SPC„) = Kf (SPC„_i) + 5rK_i|SPC„_i)+
(2.3)
175(n - 1) or 1 AA .
—4-) +35 - 9 EEr(Vi'Vj)
i=i j=i
= Kf (SPCn-i) + 5r(u„_i|SPC„_i) +
175n 35
3

6
Then
175n 35
Kf (SPCn+i) = Kf (SPC„) + 5r(u„|SPC„) + -— + —
6 ■ 2 (24) For a random spiro chain SPC(n, pi, p2 ), the resistance number r (un |SPC(n, pi, p2 )) is a random variable, and its expected value is denoted by
U„ = E (r(u„|SPC (n,pi ,^2))).
By the expectation operator and (3), we can obtain a recursive relation for the expected value of the Kirchhoff number of a random spiro chain SPC (n, pi, p2 )
175n 35
E (Kf (SPC (n + 1,pi,p2 )) = E(Kf (SPC (n,pi,p2))) + 5U„ + — + 35 (2.5)
62
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Ars Math. Contemp. 9 (2015) 165-186
Now, we consider computing Un.
(i)	If SPCn ^ SPCn+1 with probability pi, then un is the vertex x3 or x5. Consequently, r(un|SPCn) is given by r(x3|SPCn) with probability p1.
(ii)	If SPCn ^ SPCn+1 with probability p2, then un is the vertex x2 or x6. Consequently, r(un|SPCn) is given r(x2|SPCn) with probability p2.
(iii)	If SPCn ^ SPCn+1 with probability 1 - p1 - p2, then un is the vertex x4. Consequently, r(un |SPCn) is given by r(x4 |SPCn) with probability 1 - p1 - p2.
From (i)-(iii) above, we immediately obtain
Un =r(x3|SPCn )p1 + r(x2|SPCn)p2 + rfo |SPCn)(1 - p1 - p2 )
20	35
=p1[r(un-1|SPC(n - 1,p1,p2)) + 2°(n - 1) +
3	6
25	35
+ p2[r(un-1|SPC(n - 1,p1,p2)) + -7r(n - 1) + —]
6 6
15	35
+ (1 - p1 - p2)[r(un-1|SPC(n - 1,p1,p2)) + ^-(n - 1) + —]
26
By applying the expectation operator to the above equation, we obtain
15 5 10	5 10 5
Un = Un-1 + (y - ^p1 - yp2)n + ^p1 + yp2 - 3
And U1 = E(r(u1|SPC(1,p1,p2))) = 35, using the above recurrence relation, we have
15 5 5 2 25 5 5 Un = - ^p1 - 3p2)n + + ^p1 + 3p2)n
From (4),
E (Kf (SPC (n +1,p1,p2)) =
= E(Kf (SPC(n,p1,p2))) + 5[(f - 152p1 - |p2)n2 + (1 + 152 P1 + fp2)n] + n + f and E(Kf (SPC (1,p1,p2))) = f.
Using the above recurrence relation, we have
E(Kf (SPC(n,p1,p2))) = (25 - 36p1 - 25p2)n3 + (ffp1 + f p2 + ^n2
25 50 5
— (— p +--p2--)n.
V18^ 91 2 6'
□
Specially, by taking (p1?p2) = (1,0), (0,1) or (0,0), respectively, and Theorem 2.3, we have
Corollary 2.4. ([8]) The Kirchhoff indices of the spiro meta-chain Mn, the spiro orthochain On and the spiro para-chain Pn are
50 3 25 2 5 Kf (Mn) = -n3 + 25n2 - -n
Kf (On) = 125n3 + — n2 - 85n J\ n 36 1 4	18
Kf (Pn) = 24" n3 + 125 n2 + 5 n.
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2.3	A relation between E(Kf (PPC)) and E(Kf (SPC))
Since a spiro chain can be obtained from a polyphenyl chain by squeezing off its cut edges, it is straightforward by Rayleigh short-cut principle in the classical theory of electricity that the Kirchhoff index of the spiro chain is less than the polyphenyl chain. In fact, a relation between the Kirchhoff indices of a polyphenyl chain and its corresponding spiro chain obtained by squeezing off its cut edges was given in [8]. Here, we can also obtain a relation between the expected values of their Kirchhoff indices of the random polyphenyl chain PPC(n,pi,p2) and the random spiro chain SPC(n,pi,p2) with the same probabilities pi andp2 from Theorems 2.1 and 2.3.
Theorem 2.5. For a random polyphenyl chain PPC(n,pi,p2) and a random spiro chain SPC (n,pi,p2) with n hexagons, the expected values of their Kirchhoff indices are related as
50E(Kf (PPC(n,pi,p2))) = 72E(Kf (SPC(n,pi,p2))) + 300n3 - 350n2 - 335n.
Theorem 2.5 also shows that the expected value of Kirchhoff index of the random spiro chain is less than the random polyphenyl chain. In fact, for n > 2, E(Kf (SPC(n, pi,p2))) < 35E(Kf (PPC(n,pi,p2))). The reason is quite obvious. Dividing both sides of the equation in Theorem 2.5 yields
36	67
E(Kf (PPC(n,pi,p2))) = -E(Kf (SPC(n,pi,p2))) + 6n3 - 7n2 - — n
25	10
and it is easily seen that for n > 2, 6n3 - 7n2 - 67n > 0.
2.4	The average value of the Kirchhoff index
Let Gn is the set of all polyphenyl chains with n hexagons. The average value of the Kirchhoff indices with respect to G n is
KfaVr (Gn) = £ Kf (G) . |Gn| G6G„
In order to obtain the average value of the Kirchhoff indices with respect to Gn, we only need to take pi = p2 = i in the random polyphenyl chain PPC(n,pi,p2), i.e., the average value of the Kirchhoff indices with respect to Gn is just the expected value of the Kirchhoff index of the random polyphenyl chain PPC(n,pi,p2) for pi = p2 = i. From Theorem 2.1, we have
Theorem 2.6. The average value of the Kirchhoff indices with respect to Gn is
Kfavr(Gn) = n3 + 13n2 - 53n.
36
Similarly, let Gn is the set of all spiro chains with n hexagons. The average value of the Kirchhoff indices with respect to Gn is
Kfavr (Gn) = £ Kf (G) . |Gn| oeGn
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Ars Math. Contemp. 9 (2015) 165-186
And the average value of the Kirchhoff indices with respect to Gn is just the the expected value of the Kirchhoff index of the random spiro chain SPC(n,pl,p2) for pi = p2 = 1. From Theorem 2.3, we have
Theorem 2.7. The average value of the Kirchhoff indices with respect to Gn is
. 275 3 125 2 40
Kfavr [Qn) = -54n + -9-n - n.
Acknowledgments. The authors would like to thank the anonymous referees for their valuable comments and suggestions on the original manuscript.
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