chemia - Studia

THE WIENER INDEX OF CARBON NANOJUNCTIONS
distances between vertices of the core and the set D 1 is 3480. In what
follows, we obtain a recursive formula for computing S 2 .
• The summation of distances between vertices of the core and the
set D 1 is 3480,
• The summation of distances between vertices of the core and the
set D 1 ∪ D 2 is 3480 + 12 × 384,
• The summation of distances between vertices of the core and the
set D 1 ∪ D 2 ∪ D 3 is 3480 + 12 × 384 + 12 × (384 + 96),
• The summation of distances between vertices of the core and the
set D 1 ∪ D 2 ∪ D 3 ∪ D 4 is 3480 + 12 × 384 + 12 × (384 + 96) + 12 ×
(384 + 2 × 96),
• The summation of distances between vertices of the core and the
set D 1 ∪ ... ∪ D n is 3480 + 12 × 384 (n – 1) + 12 × 96 ×
⎡ 1 ⎤ 1 1
⎢
( −1)
2 − + .
2⎥
n n
⎣ ⎦ 2 2
Therefore, S 2 = −1128 + 4608n + 576(n−1)2 – 576n + 576. Notice
that for computing the Wiener index, we should compute 4S 2 .
We now calculate the quantity S 3 . To do this, we assume that R i R j
denote the summation of distances between vertices of D i and D j in
subgraph L, Figure 4. For computing S 3 it is enough to compute R i R j , for 1 ≤
i, j ≤ n. In Table 1, the occurrence of R i R j in S 3 is computed.
Table 1. The Number of R i R j in Computing S 3 .
# Rows The Number of R i R j
1 R 1 R 1
2 2R 1 R 1 + R 1 R 2
3 3R 1 R 1 + 2R 1 R 2 + R 1 R 3
N n−
216n +528(n − 1) + [ 792 + 288( i −1)
][ n − ( i 1 ]
∑ +
i= 1 2
)
From Table 1, one can compute S 3 as follows:
n−
S 3 = 216n +528(n − 1) + ∑ i= 1 2 [ 792 + 288( i −1)
][ n − ( i + 1)
]
2
2
= 360n
( n −1)
+ 84n
+ 132 − 252( n −1)
+ 144n(
n −1)
− 96( n −1)
Notice that in computing the Wiener index of G[n], we should
consider 4S 3 , Figure 1.
To compute S 4 , we assume that D i and E i , 1 ≤ i ≤ n, denote the set
of vertices in the i th row of two different arm tubes in G[n]. Using a similar
argument as above, we assume that R i S j denote the summation of distances
between vertices of D i and E j , 1 ≤ i, j ≤ n. For computing S 4 it is enough to
compute R i S j , for 1 ≤ i, j ≤ n. In Table 2, the occurrence of R i S j in S 4 is computed.
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