/mo> ( n 1 ) by attaching a new terminal naphthalene spanned by vertices { s n 1 , t n 1 , g n 1 , h n 1 , x 6 , x 5 , x 4 , x 3 , x 2 , x 1 } (or { r n 1 , s n 1 , t n 1 , g n 1 , x 1 , x 2 , x 3 , x 4 , x 5 , x 6 } ) (see Figure 1(b)).

Lemma 2.1. If D 2 × n is obtained from D 2 × ( n 1 ) by α-type fusion, then

d ( x 1 | D 2 × n ) = 6 n + 13 + d ( s n 1 | D 2 × ( n 1 ) ) , d ( x 2 | D 2 × n ) = 12 n + 1 + d ( s n 1 | D 2 × ( n 1 ) ) ,

d ( x 3 | D 2 × n ) = 6 n + 7 + d ( g n 1 | D 2 × ( n 1 ) ) , d ( x 4 | D 2 × n ) = 12 n + 5 + d ( g n 1 | D 2 × ( n 1 ) ) ,

d ( x 5 | D 2 × n ) = 12 n + 7 + d ( h n 1 | D 2 × ( n 1 ) ) , d ( x 6 | D 2 × n ) = 6 n + 13 + d ( h n 1 | D 2 × ( n 1 ) ) ,

and W ( D 2 × n ) = W ( D 2 × ( n 1 ) ) + 2 [ d ( s n 1 | D 2 × ( n 1 ) ) + d ( g n 1 | D 2 × ( n 1 ) ) + d ( h n 1 | D 2 × ( n 1 ) ) ] + 54 n + 11.

Similarly, if D 2 × n is obtained from D 2 × ( n 1 ) by β-type fusion, then

d ( x 1 | D 2 × n ) = 6 n + 13 + d ( g n 1 | D 2 × ( n 1 ) ) , d ( x 2 | D 2 × n ) = 12 n + 1 + d ( g n 1 | D 2 × ( n 1 ) ) ,

d ( x 3 | D 2 × n ) = 6 n + 7 + d ( s n 1 | D 2 × ( n 1 ) ) , d ( x 4 | D 2 × n ) = 12 n + 5 + d ( s n 1 | D 2 × ( n 1 ) ) ,

d ( x 5 | D 2 × n ) = 12 n + 7 + d ( r n 1 | D 2 × ( n 1 ) ) , d ( x 6 | D 2 × n ) = 6 n + 13 + d ( r n 1 | D 2 × ( n 1 ) ) ,

and W ( D 2 × n ) = W ( D 2 × ( n 1 ) ) + 2 [ d ( s n 1 | D 2 × ( n 1 ) ) + d ( g n 1 | D 2 × ( n 1 ) ) + d ( r n 1 | D 2 × ( n 1 ) ) ] + 54 n + 11 ,

where d ( s 1 | D 2 × 1 ) = d ( g 1 | D 2 × 1 ) = 21 , d ( h 1 | D 2 × 1 ) = d ( r 1 | D 2 × 1 ) = 25 .

Proof. By (1), we have d ( s 1 | D 2 × 1 ) = d ( g 1 | D 2 × 1 ) = 21 , d ( h 1 | D 2 × 1 ) = d ( r 1 | D 2 × 1 ) = 25 . Since a naphthalene is not a vertex rotation symmetry, we must distinguish between two different situations for the Wiener number of the vertex in the terminal naphthalene. In the following, we only consider the case of α-type fusion (the argument for the case of β-type fusion is analogous). Note that D 2 × ( n 1 ) has 6 n 2 vertices. If D 2 × n is obtained from D 2 × ( n 1 ) by α-type fusion then, for n 2 , we have the following relations:

d ( x 1 | D 2 × n ) = 6 n 2 + d ( s n 1 | D 2 × ( n 1 ) ) + 15 ,

d ( x 2 | D 2 × n ) = 2 ( 6 n 2 ) + Δ n 1 6 + 11 ,

d ( x 3 | D 2 × n ) = 6 n 2 + d ( g n 1 | D 2 × ( n 1 ) ) + 9 ,

d ( x 4 | D 2 × n ) = 2 ( 6 n 2 ) + d ( g n 1 | D 2 × ( n 1 ) ) + 9 ,

d ( x 5 | D 2 × n ) = 2 ( 6 n 2 ) + d ( h n 1 | D 2 × ( n 1 ) ) + 11 ,

d ( x 6 | D 2 × n ) = 6 n 2 + d ( h n 1 | D 2 × ( n 1 ) ) + 15.

and W ( D 2 × n ) = W ( D 2 × ( n 1 ) ) + d ( s n 1 | D 2 × ( n 1 ) ) + 2 [ d ( g n 1 | D 2 × ( n 1 ) ) + d ( h n 1 | D 2 × ( n 1 ) ) ] + Δ n 1 + 54 n + 11 ,

where Δ n 1 = min { d ( s n 1 | D 2 × ( n 1 ) ) , d ( g n 1 | D 2 × ( n 1 ) ) } . In this case, s n 1 and g n 1 coincide with the vertices x 2 and x 4 , respectively. Thus, d ( s n 1 | D 2 × ( n 1 ) ) d ( g n 1 | D 2 × ( n 1 ) ) = Δ n 2 4 d ( g n 2 | D 2 × ( n 1 ) ) < 0 . By induction on n, we know that Δ n 1 = d ( s n 1 | D 2 × ( n 1 ) ) . The proof is completed.

For a random double benzenoid chain R D 2 × n , there are two cases to be considered:

Case 1. R D 2 × n R D 2 × ( n + 1 ) by α-type fusion with probability p. In this case, r n , s n , t n , g n , h n (of R D 2 × n ) coincide with the vertices x 1 , x 2 , x 3 , x 4 , x 5 , respectively.

Case 2. R D 2 × n R D 2 × ( n + 1 ) by β-type fusion with probability 1 p . In this case, r n , s n , t n , g n , h n (of R D 2 × n ) coincide with the vertices x 5 , x 4 , x 3 , x 2 , x 1 , respectively.

The distance d ( r n | R D 2 × n ) , d ( s n | R D 2 × n ) , d ( g n | R D 2 × n ) , d ( h n | R D 2 × n ) and W ( R D 2 × ( n 1 ) ) are random variables. We denote their expected values by n = E ( d ( r n | R D 2 × n ) ) , S n = E ( d ( s n | R D 2 × n ) ) , G n = E ( d ( g n | R D 2 × n ) ) , n = E ( d ( h n | R D 2 × n ) ) and W n = E ( W ( R D 2 × n ) ) (or E ( W ( n ) ) in brief), respectively. Then, by Case 1, Case 2 and Lemma 2.1 we have

n = p ( 6 n + 13 + d ( s n 1 | R D 2 × ( n 1 ) ) ) + ( 1 p ) ( 12 n + 7 + d ( r n 1 | R D 2 × ( n 1 ) ) ) ;

S n = p ( 12 n + 1 + d ( s n 1 | R D 2 × ( n 1 ) ) ) + ( 1 p ) ( 12 n + 5 + d ( s n 1 | R D 2 × ( n 1 ) ) ) ;

G n = p ( 12 n + 5 + d ( g n 1 | R D 2 × ( n 1 ) ) ) + ( 1 p ) ( 12 n + 1 + d ( g n 1 | R D 2 × ( n 1 ) ) ) ;

n = p ( 12 n + 7 + d ( h n 1 | R D 2 × ( n 1 ) ) ) + ( 1 p ) ( 6 n + 13 + d ( g n 1 | R D 2 × ( n 1 ) ) )

and

W n = W ( R D 2 × ( n 1 ) ) + 2 ( d ( s n 1 | R D 2 × ( n 1 ) ) + d ( g n 1 | R D 2 × ( n 1 ) ) ) + 2 p d ( h n 1 | R D 2 × ( n 1 ) ) + 2 ( 1 p ) d ( r n 1 | R D 2 × ( n 1 ) ) + 54 n + 11.

Theorem 2.1. For n 2 , then

n = p ( 6 n + 13 + S n 1 ) + ( 1 p ) ( 12 n + 7 + n 1 ) ; S n = 12 n + 5 4 p + S n 1 ; G n = 12 n + 1 + 4 p + G n 1 ; n = p ( 12 n + 7 + n 1 ) + ( 1 p ) ( 6 n + 13 + G n 1 ) ; W n = W n 1 + 2 ( S n 1 + G n 1 ) + 2 p n 1 + 2 ( 1 p ) n 1 + 54 n + 11 (2)

with boundary conditions W 1 = 109 , S 1 = G 1 = 21 and 1 = 1 = 25 .

Proof. Noting that E ( n ) = n , E ( S n ) = S n , E ( G n ) = G n , E ( n ) = n and E ( W n ) = W n . Thus, Theorem 2.1 holds.

3. The Explicit Analytical Expression for W n

From (2), by successive subtraction method we have

S n = 6 n 2 + 11 n 4 p n + 4 p + 4 , G n = 6 n 2 + 7 n + 4 p n 4 p + 8. (3)

Therefore,

n = 12 n + 5 p 7 n p + 6 n 2 p + 8 p 2 4 n p 2 + 7 + ( 1 p ) n 1 , n = 20 + n + 6 n 2 21 p + 15 n p 6 n 2 p + 8 p 2 4 n p 2 + p n 1 , W n = W n 1 + 2 p n 1 + 2 ( 1 p ) n 1 + 23 + 42 n + 24 n 2 . (4)

Thus, we get the following result.

Theorem 3.1. For n 2 , if p = 0 then n = 6 n 2 + 13 n + 6 , S n = 6 n 2 + 11 n + 4 , G n = 6 n 2 + 7 n + 8 , n = 20 + n + 6 n 2 , W n = 12 n 3 + 40 n 2 + 49 n + 8 ; and if p = 1 then n = 6 n 2 + n + 20 , S n = 6 n 2 + 7 n + 8 , G n = 6 n 2 + 11 n + 4 , n = 6 n 2 + 13 n + 6 , W n = 12 n 3 + 40 n 2 + 49 n + 8 .

We now consider the case when 0 < p < 1 by using the method of generating functions. Let

( t ) = n 0 n t n , ( t ) = n 0 n t n , W ( t ) = n 0 W n t n .

Then, by (4) we get

( t ) 0 = ( 1 p ) t ( t ) + ( 19 + 4 p + 4 p 2 ) δ 3 + ( 12 + 5 p 4 p 2 ) δ 2 + 6 p δ 1 , ( t ) 0 = p t ( t ) + ( 27 12 p + 4 p 2 ) δ 3 + ( 13 + 3 p 4 p 2 ) δ 2 + ( 6 6 p ) δ 1 , W ( t ) W 0 = t W ( t ) + 2 p t ( t ) + 2 ( 1 p ) t ( t ) + 89 δ 3 + 90 δ 2 + 24 δ 1 , (5)

where δ 1 = n 0 n 2 t n + 1 , δ 2 = n 0 n t n + 1 and δ 3 = n 0 t n + 1 . Since ( 1 t ) j = n 0 C n j 1 t n j + 1 . Then ( 1 p t ) 1 = n 0 ( p t ) n and ( 1 t ) 1 = n 0 t n . Thus, by (5) we have

( t ) = n 0 ( ( 1 p ) t ) n [ 0 + ( 19 + 4 p + 4 p 2 ) m 1 t m + ( 12 + 5 p 4 p 2 ) m 1 ( m 1 ) t m + 6 p m 1 ( m 1 ) 2 t m ] = n 0 0 ( ( 1 p ) t ) n + k 0 n + m = k , m 1 ( 1 p ) n [ ( 19 + 4 p + 4 p 2 ) + ( 12 + 5 p 4 p 2 ) ( m 1 ) + 6 p ( m 1 ) 2 ] t k = n 0 0 ( ( 1 p ) t ) n + n 0 m = 1 n ( 1 p ) n m [ ( 19 + 4 p + 4 p 2 ) + ( 12 + 5 p 4 p 2 ) ( m 1 ) + 6 p ( m 1 ) 2 ] ] t n = n 0 n t n ,

where,

n = 0 ( 1 p ) n + m = 2 n ( 1 p ) n m [ 7 + 12 m + 5 p 7 m p + 6 m 2 p + 8 p 2 4 m p 2 ] ,

( t ) = n 0 ( p t ) n ( 0 + m 1 [ ( 27 12 p + 4 p 2 ) + ( 13 + 3 p 4 p 2 ) ( m 1 ) + ( 6 6 p ) ( m 1 ) 2 ] t m ) = n 0 0 ( p t ) n + k 0 n + m = k , m 1 p n ( 20 + m + 6 m 2 21 p + 15 m p 6 m 2 p + 8 p 2 4 m p 2 ) t k = n 0 0 ( p t ) n + n 0 m = 1 n p n m [ 20 + m + 6 m 2 21 p + 15 m p 6 m 2 p + 8 p 2 4 m p 2 ] t n = n 0 n t n ,

n = 0 p n + m = 1 n p n m [ 20 + m + 6 m 2 21 p + 15 m p 6 m 2 p + 8 p 2 4 m p 2 ] ,

W ( t ) = n 0 t n [ W 0 + 2 p m 1 m 1 t m + 2 ( 1 p ) m 1 m 1 t m + m 1 ( 23 + 42 m + 24 m 2 ) t m ] = n 0 [ W 0 + m = 1 n ( 2 p m 1 + 2 ( 1 p ) m 1 + 23 + 42 m + 24 m 2 ) t n ] = n 0 W n t n ,

W n = W 0 + m = 1 n ( 2 p m 1 + 2 ( 1 p ) m 1 + 23 + 42 m + 24 m 2 ) .

Thus, we reach the following result.

Theorem 3.2. If n 2 and 0 < p < 1 , then

n = 0 ( 1 p ) n + m = 2 n ( 1 p ) n m [ 7 + 12 m + 5 p 7 m p + 6 m 2 p + 8 p 2 4 m p 2 ] , S n = 6 n 2 + 11 n 4 p n + 4 p + 4 , G n = 6 n 2 + 7 n + 4 p n 4 p + 8 , n = 0 p n + m = 1 n p n m [ 20 + m + 6 m 2 21 p + 15 m p 6 m 2 p + 8 p 2 4 m p 2 ] , W n = W 0 + m = 1 n ( 2 p m 1 + 2 ( 1 p ) m 1 + 23 + 42 m + 24 m 2 ) . (6)

It is easy to verify that m = 0 n p n m = 1 p n + 1 1 p , m = 1 n m p n m = p n ( n + 1 ) p 1 + p n 1 ( p 1 ) 2 , m = 1 n m p m 1 = 1 ( n + 1 ) p n 1 p + p ( 1 p n ) ( 1 p ) 2 and

m = 1 n m 2 p n m = ( n 2 p + 2 n p + 2 n 2 p p 2 2 n p 2 n 2 p 2 + p 1 + n + p 2 + n ) / ( 1 + p ) 3 .

So by (4), we can calculate the values of 0 , 0 and W 0 , and by using Mathematica software(Mathematica 9.0) to (6) we obtain the solution of (6) as follows:

Theorem 3.3. For n 2 and 0 < p < 1 , then

n = 1 ( 1 p ) p ( 8 8 ( 1 p ) n + 5 n p + 6 n 2 p + 6 ( 1 p ) n p 4 p 2 9 n p 2 6 n 2 p 2 4 p 3 + 4 n p 3 ) ,

S n = 6 n 2 + 11 n 4 p n + 4 p + 4 ,

G n = 6 n 2 + 7 n + 4 p n 4 p + 8 ,

n = 1 ( 1 + p ) p ( 20 p n p 6 n 2 p + 16 p 2 3 n p 2 + 6 n 2 p 2 4 p 3 + 4 n p 3 + 2 p n + 6 p 1 + n ) ,

W n = 1 ( 1 + p ) 2 p 2 ( 16 + 16 ( 1 p ) n + 44 p + 16 n p 44 ( 1 p ) n p 32 p 2 + 13 n p 2 + 32 n 2 p 2 + 12 n 3 p 2 + 40 ( 1 p ) n p 2 40 p 3 34 n p 3 72 n 2 p 3 24 n 3 p 3 12 ( 1 p ) n p 3 + 60 p 4 43 n p 4 + 56 n 2 p 4 + 12 n 3 p 4 48 p 5 + 72 n p 5 24 n 2 p 5 + 16 p 6 24 n p 6 + 8 n 2 p 6 + 4 p 2 + n + 12 p 3 + n )

4. Conclusions and Suggestions

From Theorem 3.1 and Theorem 3.3, we know that W n ~ 12 n 3 as n tends to infinity. Note that W n is a polynomial in the variable n, which is different from the expected value of the Wiener number of the random benzenoid chain in the general case. And the limiting behaviors on the annealed entropy l o g W n when the random double hexagonal chain becomes infinite in length are

lim m 1 V log [ W n ] = 0 ,

where V is the number of the vertices in R D 2 × n , i.e., V = 6 n + 4 .

Open problem. For any p 0,1 and n 1 , is it true that the annealed entropy l o g W n of m-tuple random hexagonal chain is zero as m or m , n .

Acknowledgements

Sincere thanks to the members of JAMP for their professional performance, and special thanks to managing editor for a rare attitude of high quality. This research supported by NSFC(11551003) and the Natural Science Foundation of Qinghai(2015-ZJ-911).

Cite this paper
Ren, H. and Su, X. (2017) The Annealed Entropy of Wiener Number on Random Double Hexagonal Chains. Applied Mathematics, 8, 1473-1480. doi: 10.4236/am.2017.810108.
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