Chemistry Reference
In-Depth Information
Chapter 6
Geometry and Topology of Nanotubes
and Nanotori
Fatemeh Koorepazan-Moftakhar, Ali Reza Ashrafi,
Ottorino Ori and Mihai V. Putz
Abstract
A molecular graph is the graph of a molecule in which the vertices are
atoms and edges are chemical bonds. We review the recent results on computing
symmetry of series of armchair polyhex, zig-zag polyhex and C
4
C
8
(R/S) nanotubes
and nanotori. The topological properties of these nanostructures are also investigated.
6.1
Introduction
A
graph G
is a pair (
V, E
) consisting of a set
V
=
V(G)
of vertices and a set
E
=
E(G)
of
unordered pairs
{
x, y
}
xy
of distinct vertices of
G
called edges. Suppose M is a
chemical system like a molecule (having dimensionality
=
=
0), a carbon nanotube
(
3). The
molecular
graph
of M is a simple graph such that
vertices are atoms
and
edges are chemical
bonds
. The
degree δ
of each vertex (the number of chemical bonds) in a molecular
graph is assumed to be at most four. Taking into consideration sp
2
carbon systems,
we mention here that in the Hückel theory only pi electron molecular orbitals are
included because these determine the general properties of these molecules and the
sigma electrons are ignored. Thus, when we are talking about Hückel theory we need
=
1), a graphenic lattice (
=
2) or a diamond crystal (
=
