Journal of Chemica Acta 1(2013) 1-3
Journal of Chemica Acta
j o u r n a l h o m e p a g e : w w w . j c h e ma c t a . c o m
Fifth Geometric-Arithmetic Index of TURC4C8(S) Nanotubes
Mohammad Reza Farahani a
a
Department of Applied Mathematics, Iran University of Science and Technology (IUST), Narmak, Tehran 16844, Iran
ARTICLE INFO
ABSTRACT
Article history:
Received 14 April 2013
Received in revised form 00 Xxxxxx 2013
Accepted 00 Xxxxxx 2013
Available online 00 Xxxxxx 2013
Among topological descriptors connectivity indices are very important and
they have a prominent role in chemistry.
An important connectivity topological index was introduced by Vukicevic and
Furtula in 2009. The geometric-arithmetic index was defined as

GA G  
uv E (G )
A.
Keywords:
Molecular graph
Topological descriptors,
Connectivity Index,
Geometric-Arithmetic Index,
Nanotube
Graovac
GA5 G  

uv E (G )
et
2 dv d u
dv  d u
al
2 Sv S u
Sv  S u
, where dv denotes degree of vertex v. Recently,
defined
a
where S v 
new

version
uv E (G )
of
GA
index
as
d u . In this paer, we compute
the fifth geometric-arithmetic index of TURC4C8(S) nanotube.
© 2012 Journal of Chemica Acta
1. Introduction
Let G be a simple connected graph in chemical graph
theory. The vertices and edges of a graph also correspond to
the atoms and bonds of the molecular graph, respectively. If
e is an edge/bond of G, connecting the vertices/atoms u and
v, then we write e = uv and say "u and v are adjacent".
A topological index is a numeric quantity from the
structural graph of a molecule and is invariant on the
automorphism of the graph. And computing topological
indices of molecular graphs from chemical graph theory is
an important branch of mathematical chemistry [1-3].
Among topological descriptors, connectivity indices of a
connected graph G=(V,E) (V(G) and E(G) are the vertex
and edge set of G) are very important and they have a
prominent role in chemistry. In chemical graphs, the
vertices of the graph correspond to the atoms of molecules
while the edges represent chemical bonds [1].
One of important connectivity topological indices is
geometric-arithmetic index of G. A class of geometricarithmetic topological indices [4] may be defined as
GA general G  

uv E (G )
2 Qv Qu
Qv  Qu
where Qv is some quantity that in a unique manner can be
associated with the vertex v of the graph G. The first
member of this class for Qv=dv was introduced by Vukicevic
and Furtula [5] in 2009, The first geometric-arithmetic
index was defined as

GA G  
uv E (G )
2 dv d u
dv  d u
where dv denotes degree of vertex v.
A new member of the class of geometric-arithmetic
topological indices was considered by A. Graovac et al [6]
recently, for Qv=Sv is
GA5 G  

uv E (G )
2 Sv S u
Sv  S u
where Sv to be the summation of degrees of all neighbors of
vertex v in G. In other words,
S v  uv E (G ) d u .
In Refs [6-12] some connectivity and geometric-arithmetic
topological indices of some nanotubes and nanotorus are
computed.
* Corresponding author. Tel.:+98- 9192478265; e-mail: [email protected]
© 2013 Journal of Chemica Acta
ISSN: 2314-7083, Jabir Ibn Hayyan Publishing Ltd. All rights reserved
Farahani / Adv. Mat. Corrosion 1 (2013) 1-3
8
The aim of this work is to study this new index and
computing fifth geometric-arithmetic index of famous nano
structure TURC4C8(S) nanotubes. Our notation is standard
and mainly taken from standard books of chemical graph
theory [3].
2.
1 and 2. In addition, for further study and more historical
details, see the paper series [2, 11-12, 14-20].
Main Results and discussion
The goal of this section is is computing a closed formula of
fifth geometric-arithmetic index GA5 of TURC4C8(S)
nanotube. Suppose in the Structure of TURC4C8(S)
nanotube there are rs cycle C8 and C4, so following M.V.
Diudea [13], we denote a molecular graph G=TURC4C8(S)
nanotube by TUC4C8[r,s]. Obviously TUC4C8[r,s]
nanotube has 8rs+2r vertices/atoms and 12rs+r
edges/bonds.
Readers can see the 3-dimensional (cylinder) and 2dimensional lattices of G TUC4C8[r,s] nanotube in Figures
Figure 1. The cylinder lattice of TURC4C8(S) Nanotube.
Figure 2. 2-Dimensional Lattice of TUC4C8[r,s].
Theorem 1. Consider the TUC4C8[r,s] Nanotube (r,s>1),
then its fifth geometric-arithmetic index is equal to


GA5(TUC4C8[r,s]))  12 s  16 10  48 2  7  r


13
17


Proof. Let G be the 2-Dimensional Lattice of TUC4C8[r,s]
nanotube for all integer numbers r,s>1 (Figure 2). The
number of vertices/atoms in this nanotube is equal to
|V(TUC4C8[r,s])|= 8rs+2r and the number of vertices as
degrees 2 and 3 are equal to |V2|=2r+2r and |V3|=8rs-2r,
thus obviously the number of edges/bonds of G is
|E(TUC4C8[r,s])|=
2(4r )  3(8rs  2r )
=12rs+r.
2
Now, by refer to Figure 1, one can see that the summation
of degrees of edge endpoints of this nanotube have five
types e(5,5), e(5,8), e(8,8), e(8,9) and e(9,9) that are shown in
Figure 2 by red, blue, yellow, green and black colors.
From Table 1, there are 2r number of edges e=uv in the
types e(5,5), such that S(v)=S(u)=5 and there are 4r number
of edges f=vw in the types e(5,8), S(v)=5 and S(w)=8 and
other types are similar.
Table 1. The number of al edge types of G=TUC4C8[r,s],
on based the summation of degrees of their endpoints.
Summation of
degrees of edge
e(5,5) e(5,8) e(8,8) e(8,9)
e(9,9)
endpoints
Number of edges
of this type
2r
4r
2r
4r
12rs-11r
Thus, we have following computations, for all r,s≥2.
2 Sv S u
GA5(TUC4C8[r,s])= 
uv E (G ) S v  S u
Farahani / Adv. Mat. Corrosion 1 (2013) 1-3
8
  2r 
2 5 5
2 58
2 88
  4r 
  2r 
55
58
88
2 8 9
2 99
  4r 
 12rs  11r 
89
99
  2r    4r 
  4r 
4 10
  2r 
13
12 2
 12rs  11r 
17
Finally,


GA5(TUC4C8[r,s])  12 s  16 10  48 2  7  r . □


13
17


Corollary 1. Consider the TUC4C8[r,s] Nanotube (r,s>1),
for further applications, we approach its fifth geometricarithmetic index GA5(TUC4C8[r,s]) as
[5]
[6]
[7]
[8]
[9]
[10]
ĜA5 (TUC4C8[r,s])=12rs+0.885r
Corollary 2. By according to fifth geometric-arithmetic
index GA5 (TUC4C8[r,s]) and its approach (ĜA5
(TUC4C8[r,s])) in Theorem 1 and Corollary 1, we see that r
is addition parameter and we can write these connestivity
indices by only one variable s as follow
GA5
r
TUC4C8[r , s] 
GA5 TUC4C8 [r , s]
 12s 
[11]
[12]
[13]
r
16 10 48 2

7
13
17
[14]
[15]
and
ˆ
ˆ  r  TUC C [r , s]  GA5 TUC4C8[r , s] =12s+0.885
GA
5
4 8
r
3.
Conclusion
In chemical graph theory, mathematical chemistry and
mathematical physics, molecular descriptors, topological
and connectivity indices are very important and useful and
have more applications which characterize a molecular
graph topology. In this paper, a new connectivity
topological index called "fifth geometric-arithmetic ty index
(ABC4)" of TURC4C8(S) nanotube was determined.
[16]
[17]
[18]
[19]
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© 2013 Journal of Chemica Acta
ISSN: 2314-7083, Jabir Ibn Hayyan Publishing Ltd. All rights reserved