Chemistry Reference
In-Depth Information
as follows: take the set of vertices to be G. A cyclically ordered 3-subset {g
1
,g
2
,g
3
}
is a hyper-edge if there exists a
∈
T such that g
2
=
g
1
a and g
3
=
g
1
a. The main result
of Staica and A. Petrescu-Nita is as follows:
Theorem 2 (Staic and Petrescu-Nita)
Suppose T
n
is a group presented as follows:
1;(ab
2
)
n
a
3
1;b
3
1;(ab)
3
T
n
={
a; b
|
=
=
=
=
1
}
(6.11)
Then T
n
is a semi-direct product of Z
n
×
Z
n
by Z
3
. Moreover, the Cayley hypergraph
associated with the group T
n
can be placed on a torus.
If SL(2,3) denotes the set of all 2
2 matrices over a field of order 3 then a
Cayley hypergraph associated with this group can be placed on a torus.
We end this section by a result on the symmetry group of nanotubes. Suppose
A[p, q], B[p, q], C(R)[p, q] and C(S)[p, q] are zig-zag polyhex, armchair polyhex,
C
4
C
8
(R) and C
4
C
8
(S) nanotubes with parameters p and q, respectively.
Theorem 3
The symmetry groups of A[p, q], B[p, q], C(R)[p, q] and C(S)[p, q] are
computed as follows:
×
⎧
⎨
D
4
p
p
=
q
1. Aut(A[p, q])
=
,
⎩
C
2
×
=
D
2
p
p
q
⎧
⎨
D
2
q
piseven
q
2
isodd
2. If q is even and p
=
3 then Aut(B[p, q]) =
D
2
q
⎩
pisodd
q
2
iseven
C
2
×
D
q
2 then Aut(C(R)[p, q])
=
Aut(C(S)[p, q])
=
3. If p
=
Z
2
×
D
2p
.
Moreover, if p
=
2
then Aut(C(S)[2,1])
=
Z
2
×
D
8
and Aut(C(S)[2, q])
=
C
2
×
C
2
×
C
2
, when q
=
1.
Previous theorems clearly illustrate the power of topological methods in predicting
geometrical properties of carbon nanostructures.
6.4
Topology of Nanotubes and Nanotori
The history of computing topological indices of nanotubes and nanotori started by
publishing two paper by Diudea and his co-workers (Diudea et al.
2004
; John and
Diudea
2004
) about Wiener index armchair and zig-zag polyhex nanotubes. After
publishing these seminal papers, several scientists focus on computing such numbers
for the molecular graphs of nanostructures. For the sake of completeness we record
the mentioned results of Diudea and his co-workers in Theorem 4.
=
=
Theorem 4
B[p, q] are zig-zag and armchair polyhex
nanotubes, respectively. Then the Wiener index of these nanotubes can be computed
by the following formulas:
Suppose A
A[p, q] and B
