RESEARCH ARTICLE
Copyright © 2009 American Scientific Publishers
All rights reserved
Printed in the United States of America
Journal of
Computational and Theoretical Nanoscience
Vol. 6, 1670–1679, 2009
Szeged Index of HAC
5
C
7
rp Nanotubes
A. Iranmanesh
*
and O. Khormali
Department of Mathematics, Tarbiat Modares University, P.O. Box. 14115-137, Tehran, Iran
A C
5
C
7
net is a trivalent decoration made by alternating C
5
and C
7
. It can cover either a cylinder
or a torus. In this paper we compute the Szeged index of HAC
5
C
7
rp nanotube.
Keywords: Nanotube, Molecular Graph, Szeged Index.
1. INTRODUCTION
A graph G consists of a set of vertices V G and a set of
edges EG. In chemical graphs, each vertex represented
an atom of the molecule and covalent bonds between
atoms are represented by edges between the corresponding
vertices. This shape derived form a chemical compound is
often called its molecular graph, and can be a path, a tree
or in general a graph.
A topological index is a single number, derived
following a certain rule, which can be used to character-
ize the molecule.
1
Usage of topological indices in biol-
ogy and chemistry began in 1947 when chemist Harold
Wiener
2
introduced Wiener index to demonstrate correla-
tions between physico-chemical properties of organic com-
pounds and the index of their molecular graphs. Wiener
originally defined his index (W ) on trees and studied
its use for correlation of physico chemical properties of
alkanes, alcohols, amins and their analogous compounds.
3
A number of successful QSAR studies have been made
based in the Wiener index and its decomposition forms.
4
In series of papers, the Wiener index of some nanotubes
is computed.
5–11
Another topological index was introduced by Gutman
and called the Szeged index, abbreviated as Sz.
3
The
Szeged index was conceived by Gutman at the Attila Jozef
University in Szeged.
4
This index received considerable
attention. It has attractive mathematical characteristics
8
.
Let e be an edge of a graph G connecting the
vertices u and v. Define two sets N
1
e G and
N
2
e G as N
1
e G = x ∈ V G dx u < dx v
and N
2
e G = x ∈ V G dx v < dx u. The num-
ber of elements of N
1
e G and N
2
e G are denoted
by n
1
e G and n
2
e G respectively. The Szeged
index of the graph G is defined as SzG = Sz =
∑
e∈EG
n
1
e Gn
2
e G. The Szeged index, is a modi-
fication of Wiener index to cyclic molecules.
*
Author to whom correspondence should be addressed.
In Refs. [12–16], another topological index of some
nanotubes is computed. In this paper, we compute the
Szeged index of HAC
5
C
7
rp nanotube.
We denote the number of heptagons in one row by p,
the number of the periods by k and each period consist of
three rows as in Figure 2, which shows the m-th period,
1 ≤ m ≤ k.
2. SZEGED INDEX OF HAC
5
C
7
rp
NANOTUBE
Let e be an edge in Figure 1. Denote:
E
1
= e ∈ EG e is an oblique edge between two
heptagons
E
2
= e ∈ EG e is a horizontal edge
E
3
= e ∈ EG e is a vertical edge
E
4
= e ∈ EG e is an oblique edge between
heptagon and pentagon
E
5
= e ∈ EG e is an oblique edge between two
pentagons
Also, we can define some subsets of E
i
’s as follows:
E
2
′ = e ∈ E
2
e is an edge in 3m − 1-th row
E
2
′′ = e ∈ E
2
e is an edge in 3m-th row
so that E
2
= E
2
′ ∪ E
2
′′
E
3
′ = e ∈ E
3
e is an edge between 3m − 1-th and
3m − 2-th rows
E
3
′′ = e ∈ E
3
e is an edge between 3m-th and
3m + 1 − 2-th rows
so that E
3
= E
3
′ ∪ E
3
′′ .