Chemistry Reference
In-Depth Information
the molecule to have a pi network and therefore all molecular graphs should have
max degree 3 or less—a vertex of degree 4 represents a saturated carbon atom that
cannot be part of a pi system and which is typical of diamond-like bulk structures.
The max degree 4 also holds for alkanes, which are not included in the present
research.
Being this chapter devoted to the investigation of the structural and topological
properties of some families of nanostructures, a few formal tools have to be shortly
introduced here starting with a simple recap of the concept of
group
.
A
group
is a mathematical structure that is usually used to describe the symmetries
characterizing a given set of mathematical objects. It is a set of elements
G
equipped
with a binary operation of multiplication *:
G
×
G
→
G
such that:
i) it is associative, e.g. for all
x
,
y
,
z
∈
G
,
x
∗
(
y
∗
z
)
=
(
x
∗
y
)
∗
z
;
ii) there exists an element
e
∈
G
such that for an arbitrary element
g
in
G
,
g
∗
e
=
g
;
iii) and, for each
x
e
∗
g
=
∈
G
there exists
y
∈
G
such that
x
∗
y
=
y
∗
x
=
e.
A group is then called
finite
if the underlying set
G
is
finite
. The
symmetry
structure
of
G
can be formalized by the notion of
finite group action
. To describe it, we assume
G
is a group and
X
is a set. We also assume that there is a map
φ
:
G
×
→
X
X
with
the following two properties:
a. for each
x
x
,
b. and, for all elements
x
∈
X
and
g
,
h
∈
G
,
φ
(
g
,
φ
(
h
,
x
))
∈
X
,
φ
(
e
,
x
)
=
=
φ
(
gh
,
x
)
.
In this case,
G
and
X
are called a
transformation group
and a
G-set
, respectively.
The mapping
φ
is called a
group action
. For simplicity it is convenient to define
gx
φ
(
g
,
x
)
.
Suppose now
G
is a group and
H
is a non-empty subset of
G
.
H
is said to be
a
subgroup
of
G
,if
H
is closed under group multiplication. A subgroup
H
of
G
is
called to be
normal
in
G
, if for all
g
=
G
we have
g
−
1
Hg
∈
=
H
. When both
H
and
K
are subgroups of
G
such that
H
is normal in
G
,
H
HK
, then
G
is
called a semi-direct product of
H
by
K
. Here,
HK
is the set of all elements of
G
in
the form of
xy
such that
x
∩
K
=
1, and
G
=
K
.
An
automorphism
of
G
is a permutation
g
of
V(G)
such that
g(u)
and
g(v)
are
adjacent if and only if
u
and
v
are adjacent, where
u
,
v
∈
H
and
y
∈
V
(
G
). It is well-known that
the set of all automorphisms of
G
, with the operation of composition of permutations,
is a permutation group on
V(G)
, denoted
Aut(G)
. The name
topological symmetry
is also used for this algebraic structure. Randi´c(
1974
,
1976
) showed that a graph
can be depicted in different ways such that its point group symmetry or even the 3D
representation may differ, but its automorphism group symmetry remains the same.
The
topological symmetry
of a molecular graph is not usually the same as its point
group symmetry but it corresponds to the maximal symmetry the geometrical real-
ization of a given topological structure may possess. This very important topological
feature will play an instrumental role in the present investigations.
Computationally, several software packages are may be sourced on line for solving
computational problems related to finite groups. GAP is an abbreviation for Groups,
∈
