Geometry and Topology of Nanotubes and Nanotori - Exotic Properties of Carbon Nanomatter - page 138

In the end of this section the concept of eccentric connectivity index of molecular
graphs is presented. Let G be a molecular graph (Sharma et al. 1997 ; Sardana and
Madan 2001 ). The eccentric connectivity index ξ ( G )is defined as:

u ∈ V ( G ) deg G ( u ) ε G ( u )

ξ ( G ) = (6.9)

where deg G ( u ) denotes the degree of vertex u and ε G ( u ) is the largest distance between
u and any other vertex v of the graph G.

6.3 Symmetry Considerations on Nanotubes and Nanotori

A dihedral group is the group of symmetries of a regular polygon, including both
rotations and reflections. These groups play an important role in chemistry. Most of
point group symmetry of molecules can be described by dihedral groups. It is easy
to prove that a group generated by two involutions on a finite domain is a dihedral
group. A dihedral group with 2n symmetry elements is denoted by D n . This group
can be presented as follows:
D n = x, y

1

x n y 2 (xy) 2

| = = =

(6.10)

We now consider the molecular graph of a zig-zag and armchair (achiral) polyhex
nanotorus. Yavari and Ashrafi ( 2009 ) proved in some special cases that the symmetry
group of armchair and zig-zag polyhex nanotorus is constructed from a dihedral
group and a plane symmetry group of order 2.
Arezoomand and Taeri ( 2009 ) presented a generalization of this result. They
proved that:

Theorem 1 (Arezoomand and Taeri) The symmetry group of the molecular graph
of a zig-zag and armchair (achiral) polyhex nanotorus is isomorphic to D 4m × Z 2 ,

where Z 2 denotes the cyclic group of order 2.
Hypergraphs are a generalizations of graphs in which an edge can connect more
than two vertices. In the graph theoretical language, a hypergraph consists of a set
of vertices V and a set of hyper-edges E which is a collection of subsets in V in such
a way than the union of hyper-edges are the whole vertices. A k-hypergraph is a
hypergraph with the property that any edge connects exactly k vertices. The case of
k 2 corresponds to the usual graphs considered so far. The best generalization of

the mentioned result was introduced by Staic and Petrescu-Nita ( 2013 ).They studied
the symmetry group of two special types of carbon nanotori by using the notion of
Cayley hypergraph . To explain this concept, we assume that G is a group and S is a
non-empty subset of G such that S

=

S − 1 =

= and G S . We define a graph Cay(G,

=

S) as follows: (i) V(Cay(G, S)) G, and, (ii) two vertices a and y are adjacent if and

only if xy − 1

∈

S. The graph Cay(G, S) is called the Cayley graph of G constructed
by S . If G is a group and T is a generator set for G containing elements of order 3
such that a T implies that a − 1 ∈

∈ T then 3-hypergraph Cay 3 (G, T) can be defined