Chemistry Reference
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3. (Ashrafi and Rezaei
2007
)
⎧
⎨
9p
2
q
2
pq
2
12p
2
q
−
−
+
4pq
q
≥
2p
PI(C) =
⎩
9p
2
q
2
7pq
2
−
+
4pq
q
<
2p
Theorem 8
The PI index of TUC
4
C
8
(R/S) nanotubes and TC
4
C
8
(R/S) nanotori can
be computed as follows:
1. (Ashrafi and Loghman
2008
)
⎧
⎨
⎩
⎨
⎩
36p
2
q
2
−
26pq
2
−
2p
2
q
+
8pq
q
<
p
if
p & q are even
36p
2
q
2
−
26p
2
q
−
2pq
2
+
8pq
q
≥
p
⎧
⎨
⎩
36p
2
q
2
10pq
2
2p
2
q
−
−
q
<
p
if
p & q are odd
36p
2
q
2
10p
2
q
2pq
2
−
−
q
≥
p
PI(E[p,q])
=
⎧
⎨
⎩
36p
2
q
2
−
18pq
2
−
2p
2
q
q
<
p
if
piseven & qisodd
36p
2
q
2
−
18p
2
q
−
2pq
2
+
8pq
q
>
p
⎨
⎩
36p
2
q
2
−
18p
2
q
−
2pq
2
q
>
p
if
p is odd & q is even
36p
2
q
2
−
18pq
2
−
2p
2
q
+
8pq
q
<
p
2. (Ashrafi and Loghman
2006c
)
⎧
⎨
Xq
≤
p
PI(F[4p, q])
=
,
⎩
Yq
≥
p
36p
2
q
2
- 28p
2
q
8p
2
8pq
2
and Y
36p
2
q
2
- 36p
2
q - 4pq
2
4p
3
where X
=
+
−
=
+
4pq
+
4p
2
.
+
3. (Ashrafi et al.
2009
)
⎨
36p
2
q
2
8p
2
q
10pq
2
−
−
+
4pq
q
≤
2p
PI(G[p, q])
=
.
⎩
36p
2
q
2
20p
2
q
4pq
2
−
−
+
4pq
q
>
2p
We are now ready to investigate the Szeged index of nanotubes and nanotori.
Dobrynin and Gutman (
1994
) proved that if G is a connected bipartite graph with
n vertices and m edges, then
Sz
(
G
)
4
n
2
m
d
(
v
))
2
. Using this
−
e
1
=
−
(
d
(
u
)
∈
E
(
G
)
result Yousefi et al. (
2008d
) proved that the Szeged index of a polyhex nanotorus
is computed by Sz(C)
3
=
8
p
3
q
3
.
Another application of the mentioned result
