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Balaban and Randic Indices of IPR C

80

Fullerene Isomers,

Zigzag Nanotubes and Graphene

A. Iranmanesh

*

and Y. Alizadeh

Department of Mathematics, Tarbiat Modares University, Tehran, I.R. Iran

(*) Corresponding author: iranmanesh@modares.ac.ir

(Received: 25 Oct. 2010 and Accepted: 25 Feb. 2011)

Abstract:

In this paper an algorithm for computing the Balaban and Randic indices of any simple connected graph was introduced. Also these indices were computed for IPR C80 fullerene isomers, Zigzag nanotubes and graphene by GAP program.

Keywords: Balaban index, Randic index, IRP C80 Fullerene isomers, Zigzag nanotubes, Graphene, GAP program.

1. INTRODUCTION

Topological descriptors are derived from hydrogen-suppressed molecular graphs, in which the atoms are represented by vertices and the bonds by edges. The connections between the atoms can be described by various types of topological matrices (e.g., distance or adjacency matrices), which can be mathematically manipulated so as to derive a single number, usually known as graph invariant, graph-theoretical index or topological index[1,2].

Topological indices based on distances, are used to describe the molecular structure of a series of alkylcupferrons used as mineral collectors in the

beneficiation of a Canadian uranium ore. There is

a linear relation between any of these topological

indices and the separation efficiency of the alkylcupferrons considered. The data fit into two

separate curves, differentiating the alkylcupferrons into two subclasses, one with rigid rod-like

methyl substituents and the other with flexible

alkyl substituents. As in the case of Wiener index, connectivity index has also been correlated with physical properties such as density and heat of vaporization [3-7].

Let G be a connected graph. The vertex-set and edge-set of G are denoted by V(G) and E(G) respectively. The distance between the vertices u and v, d(u,v), in a graph is the number of edges in a shortest path connecting them. Two graph vertices are adjacent if they are joined by a graph edge. The degree of a vertex i V G∈ ( )is the number of vertices joining to i and denoted by

δ

i. The Balaban

index of a molecular graph G was introduced by

A.T. Balaban [8,9]. It is denoted by J(G) and defined

as

( )

1

( )

1

ij E G

( ) ( )

m

J G

d i d j

µ

=

+

where m is the

number of edges of G and

µ

( )

G

is the cyclomatic number of G. Noting that the cyclomatic number is the minimum number of edges that must be removed from G in order to transform it to an acyclic graph; it can be calculated using ( )µ G = − +m n 1where n is the number of vertices and d(i) is the sum of distances between vertex i and all other vertices of G, and the summation goes over all edges from the edge set E(G).

The Balaban index appears to be a very useful molecular descriptor with attractive properties

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29

[10,11]. The connectivity indices are extensively

used as molecular descriptors in predicting the retention indices in chromatographic analysis of various isomeric aliphatic, aromatic and polycyclic hydrocarbons [12-14]. Randic or connectivity index

was introduced by Milan Randic in 1975 [15] defined

as

( )

1

,

ij i j

R G

δ δ

=

where

ij

runs over all edges

in G. In a series of papers, Balaban and Randic indices of some nanotubes are computed [16-18]. In this paper we give an algorithm for computing the Balaban and Randic indices of any graph. Also, by using GAP program [19], we compute these indices for IPR C80 fullerene isomers, zigzag nanotubes and graphene. IPR fullerenes are described in [20-22].

2. An algorithm for computing the Balaban

and Randic indices

In this section an algorithm for computing the Balaban and Randic indices of any graph was introduced.

I. At first, one number was assigned to each

vertex.

II. The set of vertices that are adjacent to vertex i is denoted by N(i).

The set of vertices that their distance to vertex

u

is equal to

t

(

t

0

)

is denoted by

D u

t

( )

and

consider D u0( ) { }.= u So, the following relations

are obtained:

0

( ) t( ), ( )

t

V G D u u V G

=

∀ ∈

( ) 1

( ) ( , ) t( ) , ( )

v V G t

d u d u v t D u u V G

∈ ≥

=

=

× ∀ ∈

δ

u

=

N u

( )

According to the above relations, we can obtain the Balaban index by determining the

D ( )

t

u

for every vertex u.

III. The distance between vertex i and its adjacent vertices is equal to 1, therefore Di,1=N i( ).

For eachjDi,t,t≥1, the distance

be-tween each vertex of N(j)\(Di,tDi,t1) and the vertex i is equal to

t

+

1

. Thus we have:

.

1

,

)

(

\

)

(

(

, , 1

1

,+

=

∈ ,

N

j

D

D

t

D

it

j Dit it

it

According to the above equation we can obtain

,, 2,

i t

D t≥ for each i V G∈ ( ). By determining the sets Di,t we can compute the Balaban and Randic indices of G.

3. Computing the Balaban and Randic

indices for IPR C

80

fullerene isomers

Fullerenes consist of the networks of pentagons and hexagons. To be a closed shape, a fullerene should exactly have 12 pentagon faces, but the number of hexagons faces can be extremely variable. Fuller-enes were discovered in 1985 by Robert Curl, Har-old Kroto and Richard Smalley at the University of Sussex and Rice University, and are named after Richard Buckminster Fuller. The fullerene gallery at the address http://www.cochem2.tutkie.tut.ac.jp/ Fuller/fsl/fsl.html presents many of them including non-IPR ones. Seven possible IPR isomers C80 are shown in Figure 1.

Ac determ 3 Fu fuller be ex and R after http:/ non-I For c to any progr to any ccording to mining the s

. Computi ullerenes con rene should xtremely var Richard Sma Richard //www.coche IPR ones. Se

computing th y vertex of t ram to determ

y vertex of t

the above e sets Di,twe c

ng the Bala nsist of the n

exactly hav riable. Fuller alley at the

Buckminst em2.tutkie.tu even possible

Figure1. Sev ac

hese indices the graph and

mineN(i) an the graph on

equation we an compute

aban and Ra networks of p

e 12 pentag renes were d University ter Fuller. ut.ac.jp/Full e IPR isome

ven possible ccording to an

for IPR C80 d then accor nd Di,t . Tak e number as

3

can obtain the Balaban

andic indice pentagons an gon faces, bu

discovered in of Sussex a The ful er/fsl/fsl.htm ers C80 are sh

isomers of C8

n Atlas of fulle

fullerene iso rding to the a ke the IPR is

s in Fig. 2.

, , 2,

i t

D t

n and Randic

es for IPR C nd hexagons ut the numbe n 1985 by R and Rice Un llerene gal ml presents

hown in Fig.

80 with their s

erene [23].

omers, one n above algori somer D5h of

for each c indices of G

C80 fullerene

s. To be a cl er of hexago Robert Curl,

niversity, an llery at

many of th 1.

ymmetry

number is fir

ithm, we can f C80 fulleren

( ) i V G . B G.

e isomers losed shape,

ons faces ca Harold Krot nd are name the addres em includin

rstly assigne

n write a GA ne and assig By a an to ed ss ng ed AP gn

Figure 1: Seven possible isomers of C80 with their symmetry

according to an Atlas of fullerene [23].

For computing these indices for IPR C80 fullerene

isomers, one number is firstly assigned to any ver -tex of the graph and then according to the above algorithm, we can write a GAP program to deter-mine N(i) and Di,t . Take the IPR isomer D5h of C80

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fullerene and assign to any vertex of the graph one number as in Figure 2.

Figure 2: The IPR isomer D5h of C80 Fullerene which is

numbered.

Using the following GAP program, the Balaban and Randic indices for the above IPR isomer of C80 fullerene are computed as follows:

n:=80; k:=[]; N:=[];

k[1]:=[1..5]; k[2]:=[6..20]; k[3]:=[21..40]; k[4]:=[41..60]; k[5]:=[61..75]; k[6]:=[76..80]; for i in [1..6] do

y:=Size(k[i]); for j in [1..y] do x:=k[i][j]; N[x]:=[x-1,x+1]; od;

od;

D1:=[9,12,15,18]; for i in [1..4] do

x:=D1[i]; N[i][3]:=x; N[x][3]:=i; od;

D2:=Difference(k[2],Filtered(k[2],i->(i mod 3)=0));

D3:=Filtered(k[3],i->(i mod 2)=1); for i in [1..9] do

x:=D3[i+1]; N[D2[i]][3]:=x; N[x][3]:=D2[i]; od;

D4:=Difference(k[3],D3);

D5:= Filtered(k[4],i->i mod 2=1); for i in [1..9] do

x:=D5[i+1]; N[D4[i]][3]:=x; N[x][3]:=D4[i]; od;

D6:=Difference(k[4],D5);

D7:= Filtered(k[5],i->(i mod 3)<>2); for i in [1..9] do

x:=D7[i+1]; N[D6[i]][3]:=x; N[x][3]:=D6[i]; od;

D8:=Difference(k[5],D7); for i in [1..4] do

x:=k[6][i+1]; N[D8[i]][3]:=x; N[x][3]:=D8[i]; od;

N[1]:=[2,5,9]; N[5]:=[1,4,6]; N[6]:=[5,7, 20];N[20]:=[6,19,21];N[21]:=[20,22,40]; N[40]:=[21,39,41]; N[41]:=[40,42,60];N[60]:=[ 61,59,41];N[61]:=[60,62,75]; N[74]:=[73,75,76];

N[75]:=[58,61,74]; N[76]:=[74,77,80]; N[80]:=[71,76,79];

v:=[]; D:=[]; for i in [1..n] do

D[i]:=[]; u:=[i]; D[i][1]:=N[i]; v[i]:=Size(N[i]); u:=Union(u,D[i][1]);

r:=1; t:=1; while r<>0 do D[i][t+1]:=[]; for j in D[i][t] do

for m in Difference (N[j],u) do AddSet(D[i][t+1],m);

od; od;

u:=Union(u,D[i][t+1]);

if D[i][t+1]=[] then r:=0;fi;

t:=t+1; od;od;

m:=(1/2)*Sum(v)-n+1; d:=[];deg:=[];

for i in [1..n] do d[i]:=0;

deg[i]:=Size(N[i]); for t in [1..Size(D[i])] do d[i]:=d[i]+t*Size(D[i][t]); od;od;

B:=0;R:=0; for i in [1..n] do

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31

for j in N[i] do

B:=B+ER(1/((d[i]*d[j]))); R:=R+ ER(1/((deg[i]*deg[j]))); od;

od;

B:=m*B/2;#(this value is equal to Balaban index of the graph)

R:=R/2; #(this value is equal to Randic index of the graph)

Similar programs were applied for computing the Balaban and Randic indices of other IPR isomers C80 fullerene and results are shown in Table 1. For all C80 isomers, the Randic index is constant, and it’s equal to 13.333.

Table 1: The Balaban index of IPR C80 Fullerene isomers.

Simil IPR Rand

4

In thi

comp

To do rows nanot

p:=3 n:=2 K1:= for i i if i m else od; N[1]:

lar programs

isomers C80

dic index is c

T

. Computi graphene

is section, th

putedsimilar

o this we de

byq. In ea

tubes is equa

; q:=8;#(for *p*q; N:=[] =[1..2*p]; V1

in V1 do mod 2=0 the e N[i]:=[i-1,

:=[2,2*p]; N

s were appli

0 fullerene a

constant, and

IPR is fu

Table1. The

ng the Bal e.

he Balaban a r to those giv

enote the nu ach row, ther

al to2pq. Th

r example) ];

1:=[2..2*p-1

en N[i]:=[i-1 ,i+1]; fi;

N[2*p]:=[1,

ied for com

and results d it’s equal to

somers of C8

fullerene

D5d

D2

C2v

D3

C`2v

D5h

lh

Balaban ind

laban and

and Randic ven for IPR C

Figure3. Zi

umber of he

re are 2pve

he following

1];

1,i+1,i+2*p]

2*p-1,4*p];

6

mputing the B are shown o 13.333.

80 Ba

dex of IPR C

Randic in

indices of z

C80 fullerene

igzag[8,8] nan

exagons in th ertices and h g program is

];

;

Balaban and in Table1.

alaban index 0.81074 0.80994 0.80701 0.80915 0.80499 0.80281 0.79822

C80 Fullerene

ndices for Z

zigzag nanot e.

notube.

the first row hence the nu the same as

d Randic ind

For all C80

x

e isomers.

Zigzag nan

tubes and gr

w bypand th

number of ve s the last pro

dices of othe isomers, th

notubes an

rapheme wer

he number o ertices in th ogram.

er he

d

re

of is

4. Computing the Balaban and Randic

indices for Zigzag nanotubes and graphene

In this section, the Balaban and Randic indices of zigzag nanotubes and grapheme were computed similar to those given for IPR C80 fullerene.

Figure 3: Zigzag[8,8] nanotube.

To do this we denote the number of hexagons in the

first row by and the number of rows by . In each

row, there are vertices and hence the number of vertices in this nanotubes is equal to . The following program is the same as the last program.

p:=3; q:=8;#(for example) n:=2*p*q; N:=[];

K1:=[1..2*p]; V1:=[2..2*p-1]; for i in V1 do

if i mod 2=0 then N[i]:=[i-1,i+1,i+2*p];

else N[i]:=[i-1,i+1]; fi;

od;

N[1]:=[2,2*p]; N[2*p]:=[1,2*p-1,4*p]; K:=[2*p+1..n];

K2:=Filtered(K,i->i mod (4*p) in [1..2*p]); for i in K2 do

if i mod 2 =0 then N[i]:=[i-1,i+1,i+2*p];

else N[i]:=[i-1,i+1,i-2*p]; fi;

if i mod (4*p)=1 then N[i]:=[i+1,i-2*p,i+2*p-1];

fi;

if i mod (4*p)=2*p then

N[i]:=[i-1,i-2*p+1,i+2*p];fi;

od;

K3:=Filtered(K,i->i mod (4*p) in Union([2*p+1..4*p-1],[0]));

for i in K3 do

if i mod 2=0 then N[i]:=[i-1,i+1,i-2*p];

else N[i]:=[i-1,i+1,i+2*p]; fi;

if i mod (4*p)=2*p+1 then

N[i]:=[i+1,i+2*p-1,i+2*p]; fi;

if i mod (4*p)=0 then N[i]:=[i-1,i-2*p,i-2*p+1];

fi;

od;

for i in [n-2*p+1..n] do if q mod 2=1 then

if i mod 2 =0 then N[i]:=[i-1,i+1];

else N[i]:=[i-1,i+1,i-2*p]; fi;

if i mod (4*p)=1 then N[i]:=[i+1,i-2*p,i+2*p-1];

fi;

if i mod (4*p)=2*p then N[i]:=[i-1,i-2*p+1];fi;

else

if i mod 2=0 then N[i]:=[i-1,i+1,i-2*p];

else N[i]:=[i-1,i+1]; fi;

if i mod (4*p)=2*p+1 then N[i]:=[i+1,i+2*p-1];

fi;

if i mod (4*p)=0 then N[i]:=[i-1,i-2*p,i-2*p+1];

fi;

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32

fi;

od;

v:=[]; D:=[]; for i in [1..n] do

D[i]:=[]; u:=[i]; D[i][1]:=N[i]; v[i]:=Size(N[i]); u:=Union(u,D[i][1]);

r:=1; t:=1; while r<>0 do D[i][t+1]:=[]; for j in D[i][t] do

for m in Difference (N[j],u) do AddSet(D[i][t+1],m);

od; od;

u:=Union(u,D[i][t+1]);

if D[i][t+1]=[] then r:=0;fi;

t:=t+1; od;od;

m:=(1/2)*Sum(v)-n+1; d:=[];deg:=[];

for i in [1..n] do d[i]:=0;

deg[i]:=Size(N[i]); for t in [1..Size(D[i])] do d[i]:=d[i]+t*Size(D[i][t]); od;od;

B:=0;R:=0;

for i in [1..n] do for j in N[i] do

B:=B+ER(1/((d[i]*d[j]))); R:=R+ ER(1/((deg[i]*deg[j]))); od;od;

B:=m*B/2;#(this value is equal to Balaban index of the graph)

R:=R/2; #(this value is equal to Randic index of the graph)

By a similar way, the Balaban and Randic indices for graphene can be obtained. The number of

hexagons in the first row is denoted by and the

number of rows by .

Figure 4: Graphene[4,8] nanotube.

9

P q Balaban index of

zigzag[p,q]

Randic index of

zigzag[p,q]

Balaban index of

grapheme[p,q]

Randic index of

grapheme[p,q]

2 2

1.95970

3.93265

1.71379

7.93265

2 3

1.67044

5.93265

1.42196

11.93265299

4 5

1.096412

19.86531

0.82065

39.86530

3 5

1.16766

14.89898

0.92423

29.89897

6 4

1.03259

23.79796

0.70617

47.79795

5 3

1.27712

14.83163

0.88425

29.83163

7 7

0.73099

48.76429

0.51655

97.76428

Table2. Balaban and Randic indices of zigzag[p,q] nanotubes and graphene[p,q].

Conclusion

An algorithm has been presented for computing the Balaban and Randic indices of any

connected simple graph. According to this algorithm and using the GAP program, a

program

can be written

to compute these indices quickly. This method has been used

here for the first time. We tested the algorithm to calculate the Balaban and Randic

indices of IPR C

80

fullerene isomers, zigzag nanotubes and graphene.

Acknowledgment. This research is partially supported by Iran National Science Foundation (INSF) (Grant No. 87040351). The authors would like to thank the referees for the valuable comments.

References

1. Gonzalez-Diaz, H., Vilar, S., Santana, L., Uriarte, E., 2007,

"

Medicinal chemistry

and bioinformatics--current trends in drugs discovery with networks topological

indices,

"

Current Topics in Medicinal Chemistry., 7, pp. 1015–1029.

2. Gonzalez-Diaz, H., Gonzalez-Diaz, Y., Santana, L., Ubeira, FM., Uriarte, E.,

2008,

"

Proteomics, networks and connectivity indices,

"

Proteomics., 8, pp. 750–

78.

3. Wiener, H., 1947,

"

Structural determination of paraffin boiling points

"

, J. Am.

Chem. Soc., 69, pp. 17-20.

4. Wiener, H., 1947,

"

Correlation of heats of isomerization and differences in heats

of vaporization of isomers among the paraffin hydrocarbons, J. Am. Chem. Soc.,

69, pp. 2636-2638.

5. Wiener, H., 1947, "Influence of interatomic forces on paraffin properties," J.

Chem. Phys., 15, pp. 766-766.

6. Wiener, H., 1948, "Vapor pressure-temperature relationships among the branched

paraffin hydrocarbons," J. Phys. Chem., 52, pp. 425-430.

Table 2: Balaban and Randic indices of zigzag[p,q] nanotubes and graphene[p,q].

(6)

33

The Randic index of zigzag[p,q] nanotubes and

graphene are equal to

d[i]:= deg[i] for t i d[i]:= od;od B:=0 for i i for j B:=B R:=R od;od B:=m R:=R By a numb The R ( R zi exam nanot =0; i]:=Size(N[i] in [1..Size(D =d[i]+t*Size d; 0;R:=0; in [1..n] do

in N[i] do B+ER(1/((d[ R+ ER(1/((de

d;

m*B/2;#(this R/2; #(this va

similar way ber of hexag

Randic index ) igzag  p

mple, we ob tubes for som

]); D[i])] do

e(D[i][t]);

[i]*d[j]))); eg[i]*deg[j])

s value is equ alue is equal

y, the Balab ons in the fir

x of zigzag[p 2 6 3 5

3 q

 

tained the B me p and q b

])));

ual to Balaba l to Randic in

ban and Ran rst row is de

Figure4. Gra

p,q] nanotub 5and

R g(

Balaban and by GAP prog

8 an index of t ndex of the g

ndic indices enoted bypa

aphene[4,8] n

bes and graph ) graphene

d Randic in gram. In Tab

the graph) graph)

s for graphe and the numb

anotube.

hene are equ 5 2

3 p q

  

ndices of zi ble 2, these n

ne can be o

ber of rows

ual to 4

6

 

, resp

igzag[p,q] a numbers are

obtained. Th

byq.

ectively. Fo and graphen presented. he or ne d[i]:= deg[i] for t i d[i]:= od;od B:=0 for i i for j B:=B R:=R od;od B:=m R:=R By a numb The R ( R zi exam nanot =0; i]:=Size(N[i] in [1..Size(D =d[i]+t*Size d; 0;R:=0; in [1..n] do

in N[i] do B+ER(1/((d[ R+ ER(1/((de

d;

m*B/2;#(this R/2; #(this va

similar way ber of hexag

Randic index ) igzag  p

mple, we ob tubes for som

]); D[i])] do

e(D[i][t]);

[i]*d[j]))); eg[i]*deg[j])

s value is equ alue is equal

y, the Balab ons in the fir

x of zigzag[p 2 6 3 5

3 q

 

tained the B me p and q b

])));

ual to Balaba l to Randic in

ban and Ran rst row is de

Figure4. Gra

p,q] nanotub 5and

R g(

Balaban and by GAP prog

8 an index of t ndex of the g

ndic indices enoted bypa

aphene[4,8] n

bes and graph ) graphene

d Randic in gram. In Tab

the graph) graph)

s for graphe and the numb

anotube.

hene are equ 5 2

3 p q

  

ndices of zi ble 2, these n

ne can be o

ber of rows

ual to 4

6

 

, resp

igzag[p,q] a numbers are

obtained. Th

byq.

ectively. Fo and graphen presented. he or ne

respectively. For example, we obtained the Balaban and Randic indices of zigzag[p,q] and graphene nanotubes for some p and q by GAP program. In Table 2, these numbers are presented.

5. CONCLUSION

An algorithm has been presented for computing the Balaban and Randic indices of any connected simple graph. According to this algorithm and using the GAP program, a program can be written to compute these indices quickly. This method

has been used here for the first time. We tested

the algorithm to calculate the Balaban and Randic indices of IPR C80 fullerene isomers, zigzag nanotubes and graphene.

ACKNOWLEDGMENT

This research is partially supported by Iran National Science Foundation (INSF) (Grant No. 87040351). The authors would like to thank the referees for the valuable comments.

REFERENCES

1. Gonzalez-Diaz, H., Vilar, S., Santana, L.,

Uriarte, E., 2007, “Medicinal chemistry and bioinformatics--current trends in drugs discovery with networks topological indices,” Current Topics in Medicinal Chemistry., 7, pp. 1015– 1029.

2. Gonzalez-Diaz, H., Gonzalez-Diaz, Y., Santana, L., Ubeira, FM., Uriarte, E., 2008, “Proteomics, networks and connectivity indices,” Proteomics., 8, pp. 750–78.

3. Wiener, H., 1947, “Structural determination of

paraffin boiling points”, J. Am. Chem. Soc., 69,

pp. 17-20.

4. Wiener, H., 1947, “Correlation of heats of isomerization and differences in heats of

vaporization of isomers among the paraffin

hydrocarbons, J. Am. Chem. Soc., 69, pp. 2636-2638.

5. Wiener, H., 1947, “Influence of interatomic forces on paraffin properties,” J. Chem. Phys., 15, pp.

766-766.

6. Wiener, H., 1948, “Vapor pressure-temperature

relationships among the branched paraffin

hydrocarbons,” J. Phys. Chem., 52, pp. 425-430.

7. Wiener, H., 1948, “Relation of the physical

properties of the isomeric alkanes to molecular structure”, J. Phys. Chem., 52, pp. 1082-1089. 8. Balaban, A. T., 1982, “Highly discriminating

distance-based topological index”, Chem. Phys. Letters., 82, pp. 399-404.

9. Balaban, A. T., 1983, “Topological indices based on topological distances in molecular graphs,” Pure Appl. Chem., 55, pp. 199-206.

10. Trinajstic, N., 1992, “Chemical Graph Theory,” CRC Press, Boca Raton, FL. 246.

11. Diudea, M. V., Florescu, M. S., Khadikar, P. V., 2006, “Molecular Topology and its Applications,”

Eficon Press, Bucarest.

12. Ehrman, L., Parsons, P.A., 1981, “in Behaviour Genetics and Evolutions,” McGraw-Hill, New York.

13. Ringo, J., 1996, “Sexual Receptivity in Insects,” Annu. Rev. Entomol., 41, pp. 473-494.

14. Spieth, H.T., Ringo, 1983, J.M., “In the Genetics and Biology of Drosophila,” Academic Press, London., 3, pp. 223-284.

15. Randic, M., 1975, “On the characterization of molecular branching”, J. Amer. Chem. Soc., 97, pp. 6609-6615.

16. Iranmanesh, A., Ashrafi, A. R., 2007, “Balaban

Index of an Armchair Polyhex,” Journal of Computational and Theoretical Nanoscience, 4, pp. 514-517.

(7)

17. Yousefi-Azari, H., Ashrafi, A. R., KHalifeh, M. H.,

2008, “Topological indices of nanotubes, nanotori and nanostars,” Digest Journal of Nanomaterials and Biostructures., 3, pp. 251-255.

18. Iranmanesh, A., Alizadeh, Y., 2009, “Computing Higher Randic index of HAC5C7[p,q] and TUZC6[p,q] Nanotubes by GAP program”, MATCH Commun. Math. Compute. Chem., 62, pp. 285-294.

19. 19. Schonert, M., et al., 1992, “GAP, Groups, Algorithms and Programming,” Lehrstuhl D fuer Mathematik, RWTH, Achen.

20. Kroto, H. W., 1987, “The Stability of the

Fullerenes,” CnNature., 329, pp. 529–531. 21. Schmalz, T. G., Seitz, W.A., Klein, D.J., Hite, G.

E., “Elemental carbon cages,” J. Am.Chem. Soc., 110, pp. 1113–1127.

22. Hirsch, A., 1994, “The Chemistry of the

Fullerenes”, Georg Thieme, Stuttgart.

23. Fowler, P.W., Manolopoulos, D.E., 1995, “An atlas of fullerenes ”. Oxford University Press, Oxford.

Figure

Figure 2: The IPR isomer D5h of C80 Fullerene which is numbered.
Table 2: Balaban and Randic indices of zigzag[p,q] nanotubes and graphene[p,q].

References

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