Chemistry Reference
In-Depth Information
Fig. 6.11
The 2D lattice of
C
4
C
8
(S) nanotorus
Fig. 6.12
The molecular
graph of a polyhex nanotorus
subgroup, their intersection is trivial and the components generate the whole group.
In the
semi-direct product
instead we admit one of the subgroup to be non-normal.
In mathematical exact phrasing:
in semi-direct product a group can be constructed
from two subgroups, one of which is a normal subgroup, the intersection of two
subgroups is trivial and they generate the group.
It is easy to see that if N and H are groups and
ϕ
is a homomorphism of H into
the automorphism group of N then the set N
×
H by operation (n
1
,h
1
)
·
(n
2
,h
2
)
=
(n
1
ϕ
h1
(n
2
), h
1
h
2
), n
1
,n
2
∈
N and h
1
,h
2
∈
H, has a group structure. This group is
denoted by N
×
ϕ
H and called the
semi-direct product
ofNbyH.
A
graph invariant
is a quantity that is
invariant under all graph automorphisms
.
The
topological indices
are numerical
graph invariants
used in theoretical chemistry
to encode molecules for the classification an design of chemical compounds with
given physico-chemical properties or given pharmacological and biological activities
(Trinajstic
1992
) (MIHAI). Notice that the bond relations between atoms do not fully
determine the molecular geometry and so, in general, topological indices cannot
uniquely determine a chemical compound, but they are usually useful to obtain
information on some physico-chemical properties of compounds.
