• No results found

Hosoya polynomials of random benzenoid chains

N/A
N/A
Protected

Academic year: 2020

Share "Hosoya polynomials of random benzenoid chains"

Copied!
10
0
0

Loading.... (view fulltext now)

Full text

(1)

Hosoya Polynomials of Random Benzenoid Chains

SHOU-JUN XUa, QING-HUA HEa, SHAN ZHOUb AND WAI HONG CHANc

a

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

b School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, China

c

Department of Mathematics and Information Technology, The Hong Kong Institute of Education, Tai Po, Hong Kong, R. R. China

Correspondence should be addressed to shjxu@lzu.edu.cn (S.J. Xu).

Received 27 October 2014; Accepted 30 November 2014

ACADEMIC EDITOR:ALI REZA ASHRAFI

ABSTRACT Let G be a molecular graph with vertex set V(G) and dG(u,v) be the topological distance between vertices u and v in G. The Hosoya polynomial H(G,x) of G is a polynomial

{, } ( ) ) , (

G V v u

v u G d

x in variable x. In this paper, we obtain an explicit analytical expression

for the expected value of the Hosoya polynomial of a random benzenoid chain with n

hexagons. Furthermore, as corollaries, the expected values of the well-known topological indices: Wiener index, hyper-Wiener index and TratchStankevitchZefirov index of a random benzenoid chain with n hexagons can be obtained by simple mathematical calculations, which generates the results given by I. Gutman et al. [Wiener numbers of random benzenoid chains, Chem. Phys. Lett.173 (1990) 403408].

KEYWORDS Wiener index • random benzenoid chain • Hosoya polynomial • expected value • generating function.

1.

I

NTRODUCTION

A molecular graph ( or chemical graph) is a representation of the structural formula of a

chemical compound in terms of graph theory. Specifically, a molecular graph is a simple

graph whose vertices correspond to the atoms of the compound and edges correspond to

chemical bonds. Note that hydrogen atoms are often omitted. For example, benzenoid

chains are molecular graphs of unbranched catacondensed benzenoid hydrocarbons.

(2)

invariants and are used for Quantitative Structure-Activity Relationship (QSAR) and Quantitative Structure-Property Relationship (QSPR) studies, which mainly focus on structure-dependent chemical behaviours of molecules [4, 18].

Let G be a molecular graph with vertex set V(G), dG(u,v) be the topological distance

(or distance for short) between vertices u and v inG, i.e., the length of a shortest path

connecting uand v in G. The subscript is omitted when there is no risk of confusion. The

Hosoya polynomial} in variable x of G , introduced by Hosoya [12], is defined as ,

) ,

(G x {u,v}V(G)xdG(u,v)

H where the sum is taken over all unordered pairs of (not

necessarily distinct) vertices in G. Hence the polynomial contains the number of vertices

as the constant term.

The Hosoya polynomial not only contains more information concerning distance in the molecular graph than any of the hither to proposed distance-based molecular structure descriptors, which were extensively studied in chemical graph theory, see for instance the

surveys [16, 17], but also deduces some of them. For example, Wiener index W G( ) of a

molecular graph G [20], the oldest and most well-studied molecular structure descriptor so

far, is equal to the first derivative of the Hosoya polynomial in x1, i.e.,

. ) , ( )

(  x1

x

x G H d

d G

W (1)

The chemical applications and mathematical properties of W G( ) are well

documented [5, 6, 9, 10]. Moreover, hyper-Wiener index WW G( ) [14],

TratchStankevitchZefirov index TSZ G( ) [19] can be deduced from H G x( , ) as follows:

, ) , ( 2

1 )

( 2 1

2

xH G x x

dx d G

WW (2)

. ) , ( !

3 1 )

( 3 2 1

3

x H G x x

dx d G

TSZ (3)

Two classes of general molecular structure descriptors

1 1

) , ( !

1

 

x k

k k

x G H x dx

d

k and ! ( , ) 1

1

x k

k

x G H dx

d k

for positive integers k were also studied in Refs. [2, 15]. On the other hand, recently

(3)

quantities derived from it will play a significant role in QSAR and QSPR researches, and abundant literature appeared on this topic [3, 8,21, 22, 23].

Let Bn+1 denote a benzenoid chain with n+1 hexagons (n0). There are obviously

unique benzenoid chains Bn+1 for n0,1. More generally, a benzenoid chain Bn1 can be

regarded as a benzenoid chain Bn to which a new terminal hexagon un, y1, y2, y3, y4, vn has

been adjoined. However, when n2, the terminal hexagon can be attached in three ways,

resulting in the local arrangements we describe as B1n1, 2

1

n

B , 3

1

n

B , according to the related

position of the terminal hexagon shown in Figure 1.

1

y

2

y

3

y

4

y

1

y y2 3

y

4

y

1

y

2

y

3

y

4

y

n

B Bn

n

B

n

u un

n

u

n

v vn

n

v

1 1

? n

B 2

1

? n

B Bn3?1

1

y

2

y

3

y

4

y

1

y y2 3

y

4

y

1

y

2

y

3

y

4

y

n

B Bn

n

B

n

u un

n

u

n

v vn

n

v

1 1

? n

B 2

1

? n

B Bn3?1

Figure 1. The three types of local arrangements in benzenoid chains Bn1

A random benzenoid chain, Rn1 with n1 hexagons, is a benzenoid chain obtained

by stepwise additions of terminal hexagons. As the initial steps, R1B1, R2 B2, and for

each step k ( 2kn ) a random selection is made from one of the three possible

constructions:

1 1 

k

k B

B , with probability p1,

2 1 

k k B

B , with probability p2 or

3 1 

k k B

B , with probability q=1- p1- p2.

We assume the probabilities p1 and p2 are constants, invariant to the step parameter

k. That is, the process described is a Markov chain of order zero with a state space

consisting of three states [7].

In the present paper, we calculate the expected value of the Hosoya polynomial of a

random benzenoid chain Rn and give an explicit analytical expression by using the

(4)

2.

R

ECURSION

R

ELATIONS OF

H

OSOYA

P

OLYNOMIALS OF

R

ANDOM

B

ENZENOID

C

HAINS

Let G be a connected graph with vertex set ( )V G . For the simplicity, we define one

notation as follows: for a vertex uV(G),

, )

; (

) (

) , (

G V v

v u d G u x x

H

i.e., the contribution of the vertexu to the Hosoya polynomial H(G,x) of G. As described

above in the previous section, a benzenoid chain Bn+1 is obtained by attaching to a

benzenoid chain Bn a terminal hexagon consisting of vertices un, y1, y2, y3, y4, vn (see

Figure 1). For this construction the following relations are easily obtained [10]:

1 )

; ( )

;

( 1 3 2

1     

y x xH u x x x x

HBn Bn n (4a)

, 1 2 )

; ( )

;

( 2 2 2

1    

y x x H u x x x

H n

n B n

B (4b)

, 1 2 )

; ( )

;

( 3 2 2

1    

y x x H v x x x

HBn Bn n (4c)

, 1 )

; ( )

;

( 4 3 2

1     

y x xH v x x x x

HBn Bn n (4d)

and

   

 

 

4

1

2 3 1

1, ) ( , ) ( ; ) ( 2 3 ).

(

i

i n B n

n x H B x H y x x x x

B

H (5)

Note that the last term on the right-hand side of Eq. (5) appears because the contribution of pairs of vertices yi and yj (1 i j4) to H(Bn+1,x) are calculated twice

in the second term on the right-hand side of Eq. (5). Substituting Eq. (4) for Eq.(5), we get

. 4 3 2 ))

; ( ) ; ( ( ) 1 ( ) , ( ) ,

(B1 xH B xx xH u xH v xx3 x2  x

H n n Bn n Bn n (6)

In fact, the equations discussed above associated with a concrete benzenoid chain are valid for a random benzenoid chain, i.e., Eqs. (4)-(6) still hold when we simultaneously replace Bn+1 for Rn+1and Bn for Rn.

In the following we consider contributions of un1 and vn1 to H(Bn+1,x) according to

the positions of un+1 and vn+1. There are three cases to consider:

Case 1. Bn+1 B1n2 . In this case, un+1 = y1 and vn+1=y2=. Consequently,

) ; ( )

;

( 1 1 1

1 u x H y x

H

n

n n B

B    and HBn1(vn1;x)HBn1(y2;x), which are given by Eqs. (4a)

(5)

Case 2. Bn+1 Bn2 . In this case, un1 y3 and vn1y4 . Consequently,

) ; ( )

;

( 1 3

1 1 u x H y x

H

n

n n B

B    and HBn1(vn1;x)HBn1(y4;x),, which are given by Eqs. (4c)

and (4d), respectively.

Case 3. Bn+1 Bn32 . In this case, un+1=y2 and vn1 y3 . Consequently,

, ) ; ( )

;

( 1 1 2

1 u x H y x

H

n

n n B

B    and HBn1(vn1;x)HBn1(y3;x),, which are given by Eqs. (4b)

and (4c), respectively.

For a random benzenoid chain Rn+1, H(Rn+1,x), HRn1(un1;x) and HRn1(vn1;x)are

random variables and we denote their expected values by Hn1(x),Un1(x) and Vn1(x),

respectively, i.e.,

)). ; ( ( ) (

, )) ; ( ( ) ( , )) , ( ( ) (

1 1

1 1

1 1

1

1

x v H E x V

x u H E x U x R H E x H

n R n

n R n

n n

n

n

 

 

 

 

Since the above three cases occur in random benzenoid chains with probabilities p1, p2

and 1p1p2, respectively, by the definition of the expected value we immediately obtain

), ; ( )

; ( )

; ( )

( 1 1 2 3 2

1 x pH 1 y x p H 1 y x qH 1 y x

Un  Rn  Rn  Rn (7a)

;

(

;

)

(

;

),

)

(

1 2 2 4 3

1

x

p

H

1

y

x

p

H

1

y

x

qH

1

y

x

V

n n

n R R

R

n

(7b)

Substituting the corresponding analogues associated with random benzenoid chains Rn

and Rn+1 to Eq. (4) for Eq. (7), we get

, ) 1 ( ) ( ) ; ( )

; ( ) (

)

( 2

1 3 2

2 2

1

1       

x px qx H u x p x H v x x x p x

Un R n R n

n

n (8a)

, ) 1 ( ) ( ) ; ( )

; ( ) (

)

( 2

2 3 2

1 2

2

1       

x p x qx H v x p x H u x x x p x

Vn R n R n

n

n (8b)

By applying the expectation operator to Eq. (8), and noting that E(Un1(x))Un1(x) and E(Vn1(x))Vn1(x), we obtain

, ) 1 ( ) ( ) ( )

( ) (

)

( 2

1 3 2

2 2

1

1       

x px qx U x p x V x x x p x

Un n n (9a)

2 2

3 2

1 2

2

1( )(  ) ( ) ( )(  ) ( 1)

x p x qx V x px U x x x P x

Vn n n (9b)

A recursion relation for the expected value of the Hosoya polynomial of a random

benzenoid chain can be obtained from Eq. (6) by using Rk in place of Bk (k=n, n+1) and by

using the expectation operator:

. 4 3 2 ))

( ) ( ( ) ( ) ( )

( 2 3 2

1        

x H x x x U x V x x x x

(6)

The system of recursion equations (9) and (10) holds for n ≥ 0, and has boundary conditions: . 1 ) ( , 1 ) ( , 2 )

( 0 0

0 xxU xxV xx

H (11)

3.

S

OLUTION FOR THE

S

YSTEM OF

R

ECURSION

E

QUATIONS

To solve the recursion equations (9) and (10), we use the method of the generating function

[1]. First define the following generating functions in variable t. Let

        0 0 0 . 1 0 , ) ( ) ( , ) ( ) ( , ) ( ) ( n n n n n n n n

n x t V t V x t H t H x t t

U t

U

From Eqs. (9)(11), we get relations of their generating functions as follows:

1 1 ) 1 ( ) ( ) ( ) ( ) ( ) ( 2 1 3 2 2 2

1  

        x t x t p x x t t V x t p t U qx x p t t

U (12a)

, 1 1 ) 1 ( ) ( ) ( ) ( ) ( ) ( 2 2 3 2 1 2

2  

        x t x t p x x t t U x t p t V qx x p t t

V (12b)

2 1 ) 4 3 2 ( )) ( ) ( ( ) ( ) ( ) ( 2 3 2            x t x x x t t V t U t x x t H t t

H (12c)

As Eqs. (12a) and (12b) comprise a system of two linear equations in two variables

U(t) and V(t), a straight forward calculation results in

, 1 1 1 1 ) 1 )( 1 ( ) 1 ( ) ( ) 1 )( 1 ( ) 1 )( 1 ( ) 1 )( 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) ( 2 2 3 2 2 1 2 2 1 2 1 2 1                             xt t x qt t x x t p p p t x x x p t x x x x p xt x x x p t

U (13a)

. 1 1 1 1 ) 1 )( 1 ( ) 1 ( ) ( ) 1 )( 1 ( ) 1 )( 1 ( ) 1 )( 1 ( ) 1 ( ) 1 ( ) 1 )( 1 ( ) 1 ( ) ( 2 2 3 2 1 2 1 2 2 2 2 2 2                             xt t x qt t x x t p p p t x x x p t x x x x p xt x x x p t

V (13b)

Substituting Eq. (13) for Eq. (12) and then rearranging, we can easily get:

). 1 1 1 1 ( ) 1 )( 1 ( ) 1 ( ) ( ) 1 )( 1 ( ) 2 ) 1 (( ) 1 ( ) 1 )( 1 )( 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 4 3 2 ( 1 2 ) ( 2 2 3 3 4 2 2 1 2 3 2 2 2 2 3 2 2 2 3 xt t x t at t x x p p t x t x x q x t x t x t x x q xt t x t x x q t t x x x t x t H                                   (14)

(7)

   0 1 1 n n y

y and  2 

0( 1)

) 1 ( 1 n n y n y

to Eq. (14) and then rearranging it, we get

                                                3 0 3 0 2 3 4 2 2 1 2 2 2 2 3 2 3 2 3 2 3 2 2 3 4 5 2 3 . ))] )( 2 ( ( ) 1 ( ) ( ) 1 ( ) 1 )( 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( 1 ) 2 ) 1 (( ) 1 ( ) 4 3 2 ( 2 [ ) 10 11 14 12 6 2 ( ) 2 2 2 ( 3 2 ) ( n l l n k n k k l n n n t x x k l n q x x p p x x x x q x x x x q x x x q x n x x x n x t x x x x x t x x x x t H

.. (15)

4.

R

ESULTS AND

D

ISCUSSION

From Eq. (15), we have the following main theorem.

Theorem 4.1. Let Hn(x) be the expected value of the Hosoya polynomial of a random

benzenoid chain with

n

hexagons. Then

; 6 6 6 3 )

( 3 2

1 xxxx

H ; 10 11 14 12 6 2 )

( 5 4 3 2

2 xxxxxx

H

and when and n ≥3,

                            3 0 3 0 2 3 4 2 2 1 2 2 2 3 2 2 3 )). )( 2 ( ( ) 1 ( ) ( ) 1 ( ) 1 )( 1 ( ) 1 ( 1 ) 2 ) 1 (( ) 1 ( ) 4 3 2 ( 2 ) ( n l l n k k k l n n x x k l n q x x p p x x x x q x x x q x n x x x n x x H

We can obtain some corollaries by taking parameters as special values or Eqs. (1)(3).

When q =1 (in this case p1 = p2 = 0), a random benezoid chain is definitely a linear benzenoid chain, i.e., a benzoid chain without no turns. So from Theorem 4.1 we have

Corollary 4.2. [21] Let G be a benzenoid chain with n hexagons. If G has no turns, then

the Hosoya polynomial of G is

2 2 2 2

2

( 4)( 1) 2 ( 1)( 1)

( , ) 2 .

1 ( 1)

n

n x x x x x x

H G x x

x x

    

   

 

If p1=1 or p2=2, a random benzenoid chain with n hexagons is definitely a helicene with n

(8)

Corollary 4.3. [21] Let G be a helicene with n hexagons. Then the Hosoya polynomial of

G is

x x x x x x n x x x x n x x n x x x x x x G H n                      1 ) 4 3 2 ( ) 1 ( ) 2 2 ) 3 ( ) 1 ( ) 1 )(( 1 ( 2 ) , ( 2 3 4 5 2 2 3 4 5 3 2

In addition, from Eqs. (1)(3), we can obtain the expected values of some molecular

structure descriptors from Theorem 4.1.

Corollary 4.4. [13] The expected value Wn of the Wiener index of a random benzenoid

chain with n hexagons is

               3 0 3 2 2 1 2 3 2 3 ) 2 )( 1 ( ) ( 3 4 ) 2 3 ( 3 4 1 6 16 4 n l l n

n n n n q n n n p p l l l q

W

Corollary 4.5. The expected value WWn of the hyper-Wiener index of a random benzenoid

chain with n hexagons is

                  3 0 3 2 2 1 4 3 2 ) 9 )( 2 )( 1 ( ) ( ] ) 3 5 ( ) 4 28 ( ) 33 79 ( ) 26 11 ( 3 [ 3 1 n l l n n q l l l l p p n q n q n q n q WW

Corollary 4.6. The expected value TSZn of the Tratch-Stankevitch-Zefirov index of a

random benzenoid chain with n hexagons is

                        3 0 3 2 2 2 1 5 4 3 2 ) 510 ) 3 ( 139 ) 3 ( 14 )( 2 )( 1 ( ) ( ) 14 18 ( ) 65 135 ( ) 10 490 ( ) 635 1185 ( ) 566 58 ( 30 [ 30 1 n l l n n q l l l l l p p n q n q n q n q n q TSZ

A

CKNOWLEDGEMENTS

This work is partially supported by National Natural Science Foundation of China (Grants Nos. 11001113, 61202315). Part of this work was completed while S.-J. Xu was visiting Department of Mathematics and Information Technology at Hong Kong Institute of Education, Tai Po, Hong Kong SAR, China.

R

EFERENCES

(9)

2. F. M. Brückler, T. Došlić, A. Graovac, I. Gutman, On a class of distance based

molecular structure descriptors. Chem. Phys. Lett.503 (2001), 336–338.

3. J. Chen, S.-J. Xu, H. Zhang, Hosoya polynomials of TUC4C8(R) nanotubes. Int. J.

Quantum Chem.109 (4) (2009), 641–649.

4. M. V. Diudea, I. Gutman, L. Jantschi, Molecular Topology. Nova Science: New

York, 2001.

5. A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: theory and

applications. Acta Appl. Math.66 (2001), 211–249.

6. A. Dobrynin, I. Gutman, S. Klavžar, P. Žigert, Wiener index of hexagonal systems.

Acta Appl. Math.72 (2002), 247–294.

7. D. Freedman, Markov Chains. SpringerVerlag: New York, 1983.

8. I. Gutman, E. Estrada, O. Ivanciuc, Some properties of the Hosoya polynomial of

trees. Graph Theory Notes New York36 (1999), 7–13.

9. I. Gutman, S. Klavžar, B. Mohar, (eds.) Fifty years of the Wiener index. MATCH

Commun. Math. Comput. Chem.35 (1997), 1–259.

10.I. Gutman, S. Klavžar, B. Mohar, (eds.) Fiftieth anniversary of the Wiener index.

Discrete Appl. Math.80(1) (1997), 1–113.

11.I. Gutman, S. Klavžar, M. Petkovšek, P. On some counting polynomials in

chemistry. MATCH Commun. Math. Comput. Chem. 43 (2001), 49–66.

12. H. Hosoya, On some counting polynomials in chemistry. Discrete Appl. Math., 19

(1988), 239–257.

13. I. Gutman, J. W. Kennedy, L. V. Quintas, Wiener numbers of random benzenoid

chains. Chem. Phys. Lett.173 (1990), 403–408.

14.D. J. Klein, I. Lukovits, I. Gutman, On the definition of the hyperWiener index

for cyclecontaining structures. J. Chem. Inf. Comput. Sci.35 (1995), 50–52.

15.E. V. Konstantinova, M. V. Diudea, The Wiener polynomial derivatives and other

topological indices in chemical research. Croat. Chem. Acta.73 (2000), 383–403.

16.B. Lučić, I. Lukovits, S. Nikolić, N. Trinajstić, Distancerelated indexes in the

quantitative structureproperty relationship modeling. J. Chem. Inf. Comput. Sci.

41 (2001), 527–535.

17.Z. Mihalić, N. Trinajstić, A graphtheoretical approach to structureproperty

relationships. J. Chem. Educ.69 (1992), 701–712.

18.R. Todeschini, V. Consonni, Handbook of Molecular Descriptors; Wiley-vch:

Weinheim, 2000.

19.S. S. Tratch, M. I. Stankevitch, N. S. Zefirov, Combinatorial models and

algorithms in chemistry. The expanded Wiener number  A novel topological

(10)

20.H. Wiener, Structural determination of paraffin boiling points. J. Amer. Chem. Soc.

69 (1947), 17–20.

21.S.J. Xu, H. Zhang, Hosoya polynomials under gated amalgamations. Discrete

Appl. Math.156 (2008), 2407–2419.

22.S.J. Xu, H. Zhang, The Hosoya polynomial decomposition for catacondensed

benzenoid graphs. Discrete Appl. Math.156 (2008), 2930–2938.

23.B. Y. Yang, Y.N. Yeh, Wiener polynomials of some chemically interesting

Figure

Figure 1. The three types of local arrangements in benzenoid chains Bn 1

References

Related documents

and 11 placebo) patients had known SSE before study entry/before first study treatment and therefore were excluded from the pre-SSE analysis population. These 27 patients, however,

Sequel to expansion of the Nigeria 330kV from 9 power stations, 32 transmission lines, 28-bus system to 16 generating power stations, 45 transmission lines and

The researcher made attempt to whether micro finance programmers properly support the financial inclusion process in India by analyzing and interpreting the role

@Mayas Publication UGC Approved Journal Page 1 A STUDY ON INVESTOR BEHAVIOR WITH THE SPECIAL REFERENCE ON..

Wideband spectral broadening or supercontinuum formation can occur inside a small infrared-resonant microcavity, under high intracavity wave intensity, provided that

@Mayas Publication UGC Approved Journal Page 89 The converted Christians of Igbo people. who killed the python demonstrate the

Most of the studies have attempted to study the Indian capital market in weak form and semi strong form of efficiency with various corporate announcements like

Simulations with 5DOF were performed for oblique wave cases for a tanker ship using RaNS solver and the results were compared with head wave simulation results based on