(1)COMPUTER CALCULATION OF THE EDGE WIENER INDEX OF AN INFINITE FAMILY OF FULLERENES M

COMPUTER CALCULATION OF THE EDGE WIENER INDEX OF AN INFINITE FAMILY OF FULLERENES

M. GHORBANI, M. B. AHMADI, M. HEMMASI*

Institute of Nanoscience and Nanotechnology, University of Kashan, Kashan 87317-51167, I. R. Iran

Department of Mathematics, University of Shiraz, Shiraz 71454, I. R. Iran

The edge-Wiener index of G is defined as the sum of the distances between all pairs of edges of G. In this paper, the first and the second edge-Wiener index of an infinite family of fullerenes is computed.

(Received June 2, 2009; accepted July 19, 2009) Keywords: Fullerene, Edge-Wiener index.

1. Introduction

Let G be a simple molecular graph without directed and multiple edges and without loops, the vertex and edge-sets of which are represented by V(G) and E(G), respectively. If x and y are two vertices of G then d(x,y) denotes the length of a minimal path connecting x and y. A topological index for G is a numeric quantity that is invariant under automorphisms of G. The oldest topological index is the Wiener index which introduced by Harold Wiener.¹ This index is defined as the sum of all distances between vertices of G, i.e.

W G ( )

= ∑x y V G_{, ∈ ( )}

d x y ( , )

. The most important works on computing topological indices of nanostructures were done by Diudea and his co-authors.^{2- 7}.

Also, the edge-Wiener index of G is defined as the sum of the distances (in the line graph) between all pairs of edges of G, i.e.,

{ , } ( )

( ) ( , )

e e f E G

W G d e f

= ∑⊆ , where the distance between two edges is the distance between the corresponding vertices in the line graph of G⁸. The first edge- Wiener index is:

{ }

0 0

, ( )

( ) ( , ),

e e f E G

W G d e f

= ∑⊆

where ₀ ¹

( , ) 1

and

( , )

0

d e f e f d e f

e f

+ ≠

= ⎨⎧⎩ =

^d

₁

^{( e , f )}

⁼

^min { ^{d ( x , u ), d ( x , v ), d ( y , u ), d ( y , v )} }

such that e=xy and f =uv. This version satisfy in

0( ) ( ( )))

W Ge =W L G . The second edge-

_________________________

•Corresponding author: [email protected]

488

Wiener index is:

{ }

4 4

, ( )

( ) ( , )

e e f E G

W G d e f

⊆

= ∑ ^,

where ₄ ²

( , )

and

( , )

0

d e f e f d e f

e f

⎧ ≠

= ⎨⎩ =

d

2

⁽ e ^, f ⁾

=

^max { d ⁽ x ^, u ^), d ⁽ x ^, v ^), d ⁽ y ^, u ^), d ⁽ y ^, v ⁾ }

such that e= xy and f =uv.

Example1. Suppose Kn denotes the complete graph on n vertices and Cn be a cycle of length n.

Then, we have W K W K

0( 3) 6,

e = _e ( ₄) 36

0 = , W K

0( 5) 120,

e =

0( 4) 16, W Ce =

W K W K

0( 5) 30

W Ce = , W Ce =

1( 3) 6,

e =

1( 4) 30,

0( 6) 54, e = W K

1( 5) 90

e = , W C

1( 4) 24

e = , W C and

By continue this process one can see that

1( ) 405

e =

1( 6) 78.

W Ce = ₀

3

3

4 2 | ( )

4 2 |

e n

n n

n n

n

⎧⎪

= ⎨⎪

⎪ −

⎪⎩ /

W C ,

1

2

2

( 2)

3 2 |

( ) 4

( 2) 17

4 2 |

e n

n n

W C

n n

⎧ + −

⎪⎪

= ⎨⎪ + −

⎪⎩ /

, and

0

( ) ( 1) (2 2) / 2

e n

W K =n n− n−

0

2( 1)2 ( 1)

( )

4 2

e n

n n n n

W K − −

= − .

)

Example2. Suppose Sn denotes the Star graph on n+1 vertices. Then for every ,

and so

, (

_n

e f

∈

E S

0

( , )

4

( , )

d e f

=

d e f W

_e0

( ) S

_n =

W S

_e1

( ) (

_n =

n

−

1)( n

−

2)

.

We encourage the reader to consult^9-11 and references therein for background material as well as basic computational techniques. Our notation is standard and mainly taken from standard books of graph theory and the books of Trinajestic^12-17.

2. Main results and discussion

The fullerene era was started in 1985 with the discovery of a stable C60 cluster and its interpretation as a cage structure with the familiar shape of a soccer ball, by Kroto and his co- authors.¹⁸ The well-known fullerene, the C60 molecule(figure 1), is a closed-cage carbon molecule with three-coordinate carbon atoms tiling the spherical or nearly spherical surface with a truncated icosahedral structure formed by 20 hexagonal and 12 pentagonal rings.¹⁹ Let p, h, n and m be the number of pentagons, hexagons, carbon atoms and bonds between them, in a given fullerene F.

Since each atom lies in exactly 3 faces and each edge lies in 2 faces, the number of atoms is n = (5p+6h)/3, the number of edges is m = (5p+6h)/2 = 3/2n and the number of faces is f = p + h. By the Euler’s formula n − m + f = 2, one can deduce that (5p+6h)/3 – (5p+6h)/2 + p + h = 2, and therefore p = 12, v = 2h + 20 and e = 3h + 30. This implies that such molecules made up entirely of n carbon atoms and having 12 pentagonal and (n/2 − 10) hexagonal faces, where n ≠ 22 is a natural number equal or greater than 20.^20-23

Fig. 1. Fullerene graph C60

The adjacency matrix of a molecular graph G with n vertices is an n × n matrix A = [aij] defined by: aij = 1, if vertices i and j are connected by an edge and, aij = 0, otherwise. The distance matrix D = [dij] of G is another n × n matrix defined by dij is the length of a minimum path connecting vertices i and j, i ≠ j, and zero otherwise.

In this section, a computer program is presented which is useful for computing the edge- Wiener index of a connected graph. To do this, we first draw the molecule by HyperChem²⁴ and then compute the distance matrix of the molecular graph by TopoCluj.²⁵ Finally, we prepare a GAP²⁶ program for computing the first and the second edge- Wiener indices of any connected graph G. We apply this program to compute the first and the second edge-Wiener index of the molecular graph of fullerene C12n+4, Figure 2. In Table 1, we calculate the first and the second edge- Wiener indices of C_12n+4, for 2 ≤ n ≤ 14. Then by curve fitting method, we will find a polynomial of degree ≤ 12, through the values of Table 1. This polynomial will be the edge- Wiener index of fullerene C_12n+4.

By the calculation, the first edge-Wiener index of fullerene C_12n+4 is computed as , where

0

12 11 10

12 4 12 11 10 1 0

( ) +

e n

W C ₊ = a n + a n +a n L+a n a+

+

a₁₂=-7.732750788306343861899417455E-6, a₁₁ =7.4249438832772166105499438833 E-4, a₁₀= -0.03199037330981775426219870664, a₉=0.8167052469135802469135802469, a₈=-13.740003858024691358024691358025, a₇= 160.223576388888888888888888889, a₆=-1325.580640064667842445620223398, a₅= 7823.88132716049382716049382710, a₄=-32613.334900058788947677836566725, a₃=93511.05845679012345679012345679, a2=-172645.81245791245791245791245791, a1=187523.5191919191919191919191919 and a0=-88892. The second edge-Wiener index of fullerene C12n+4 is computed as

4

12 11 10

12 4 12 11 10 1 0

( )

e n

W C ₊ = a n + a n + a n + +L a n a ^{, where}

a12=-2.1486358291913847469403024959E-5, a11=0.002119458473625140291806958473625, a₁₀=-0.0939505254262198706643151087595, a₉=2.4710289902998236331569664903, a₈=-42.87634562389770723104056437389, a₇=516.08070601851851851851851851852, a₆=-4408.673514568636096413874191652, a₅=26863.394656635802469135802469136, a₄=-115532.73604754556143445032333921, a₃=341024.9050242504409171075837743, a₂=-651211.62012025012025012025012025, a₁=724107.1464646464646464646464646 and a₀=-351288.

490

Fig. 2. The molecular graph of fullerene C12n+4.

Table1. Values of the first and the second edge-Wiener of C12n+4 (2 ≤ n ≤ 14).

0

(

12 2

)

n

e n

W C

_{+ n}

4

(

12 2

)

e n

W C

₊

4

(

12 2

)

e n

W C

₊

0

(

12 2

)

n

e n

W C

₊ n

217574 9

5484 2

195849 9

4752 2

287840 10

14342 3

260544 10

12414 3

371930 11

28460 4

338418 11

24812 4

471140 12

48914 5

430764 12

42926 5

586766 13

76760 6

538878 13

67848 6

720104 14

113330 7

664056 14

100794 7

159836 8

143028 8

Fig. 3. The curve of for 2≤ n ≤ 14

e0

W

Fig. 4. The curve of for 2≤ n ≤ 14.

e4

W

492

A GAP Program For Computing The Edge Wiener Index Of Graphs f:= function(M)

local l, s, ss, e, i, j, a, b;

l:=Length(M);s:=0;ss:=0;e:=[];

for i in [1..l]do for j in[i+1..l] do if M[i][j]=1 then Add(e,[i,j]);

fi;

od;

od;

for a in e do for b in e do if a<> b then

s:=s+Minimum(M[a[1]][b[1]],M[a[1]][b[2]],M[a[2]][b[1]],M[a[2]][b[2]])+1;

ss:=ss+Maximum(M[a[1]][b[1]],M[a[1]][b[2]],M[a[2]][b[1]],M[a[2]][b[2]]);

fi;

od;

od;

Print("**************************************","\n");Print("\n");

Print("The first edge - Wiener number is: ", s);Print("\n");Print("\n");

Print("The second edge - Wiener number is: ", ss);Print("\n");

Print("**************************************","\n");

return;

end;

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