Chemistry Reference
In-Depth Information
⎧
⎨
qp
2
24
q
2
p
12
p
2
+
(4q
2
(q
2
+
3qp
−
4)
+
−
1) q
≤
1
1) W(A)
=
,
⎩
qp
2
24
p
3
192
(p
2
4) q
>
p
(8q
2
p
2
+
−
6)
−
−
2
+
1
⎧
⎨
p
12
[p
2
(12q
2
1)
p
)]
2p
2
8pq(p
2
q
2
−
+
8)
+
+
−
2)
+
3(
−
1
+
(
−
q
≥
p
2) W(B)
=
.
p
12
[24p
2
q
2
⎩
2q
4
8q
2
1)
p
(1
1)
q
)]
+
−
+
3(
−
−
(
−
q
≤
p
In a series of papers (Yousefi and Ashrafi
2006
,
2007
,
2008a
,
2011
; Yousefi et al.
2008c
,
d
; Ashrafi and Yousefi
2007a
,
b
) introduced a matrix method to recalculate
the Wiener number of armchair, zig-zag polyhex nanotubes, C
4
C
8
(R/S) nanotube,
polyhex nanotorus and C
4
C
8
(R/S) nanotorus. Choose a zig-zag polyhex nanotube
T. The main idea of this matrix method is choosing a base vertex b from the 2-
dimensional lattice of a T and then label vertices of hexagon by starting from the
hexagon containing b. By computing distances between b and other vertices, we
can obtain a distance matrix of nanotubes related to the vertex b. It is clear that by
choosing different base vertices, one can find different distance matrices for T, but
the summation of all entries will be the same. To explain, we assume that b is a base
vertex from the 2-dimensional lattice of T and x
ij
is the (i, j)
th
vertex of T, Fig.
6.6
.
Define D
(1,1)
m
×
n
[d
(1,1)
i,j
], where d
(1,1)
i,j
=
is distance between (1,1) and (i, j), i
=
1, 2,
...
,
1, 2,
...
, n. There are two separates cases for the (1,1)
th
m and j
=
vertex, where in
the first case d
(1,1)
1,1
0,d
(1,1)
1,2
d
(1,1)
2,1
1 and in the second case d
(1,1)
1,1
0,d
(1,1)
1,2
=
=
=
=
=
1,d
(1,1)
3
.
Suppose D
(p,q)
2,1
=
m
×
n
is distance matrix of T related to the vertex (p, q) and
s
(p,q)
i
rowofD
(p,q)
is the sum of i
th
m
×
n
. Then there are two distance matrix related to (p,
q) such that s
(p,2k
−
1)
i
s
(p,1)
i
;s
(p,2k)
i
s
(p,2)
i
m.
If b varies on a column of T then the sum of entries in the row containing base vertex
is equal to the sum of entries in the first row of D
(1,1)
=
=
;1
≤
k
≤
n/2, 1
≤
i
≤
m, 1
≤
p
≤
m
×
n
. On the other hand, one can
compute the summation of all entries in other rows by distance from the position of
base vertex. Therefore, if2|(i
+
j) then
⎧
⎨
s
(1,1)
i
−
k
+
1
1
≤
k
≤
i
≤
m,
1
≤
j
≤
n
s
(i,j)
k
=
⎩
s
(1,2)
k
1
≤
i
≤
k
≤
m,
1
≤
j
≤
n
−
i
+
1
⎧
⎨
s
(1,2)
i
1
≤
k
≤
i
≤
m,
1
≤
j
≤
n
j) then we have s
(i,j)
k
−
k
+
1
and if 2
|
(i
+
=
.
⎩
s
(1,1)
k
1
≤
i
≤
k
≤
m,
1
≤
j
≤
n
−
i
+
1
We now describe our algorithm to compute distance matrix of a zig-zag polyhex
nanotube. To do this, we define matrices A
(a
m
×
(n/2
+
1)
=
[a
ij
], B
m
×
(n/2
+
1)
=
[b
ij
] and
A
(b)
m
=
[c
ij
] as follows: (Fig.
6.13
).
×
(n/2
+
1)
