The modified Schultz index of the graph G is defined as in which deg (u) denotes vertex degree of and denotes distance of vertices u and v in the graph G

ON THE MODIFIED SCHULTZ INDEX OF C4C8(S)NANOTUBES AND NANOTORUS

ABBAS HEYDARI

Department of Science, Islamic Azad University, Arak branch, Arak, Iran

Let G be a simple undirected connected graph and V (G) denote vertex set of G. The modified Schultz index of the graph G is defined as

in which deg (u) denotes vertex degree of and denotes distance of vertices u and v in the graph G. In this paper we find an exact formula for calculation of modified Schultz index of nanotubes and nanotorus which have square and octagon structure.

∑

⊆

=

} { } , {

*

( ) deg( ) deg( ) ( , )

G V v u

v u d v u G

S

) (G V

v ∈ d ( u , v )

(Received January 26, 2010; accepted February 24, 2010) Keywords: Modified Schultz Index, Nanotubes, Nanotorus

1. Introduction

A topological index is a numeric quantity that is mathematically derived in a direct and unambiguous manner from the structural graph of a molecule. It most be structural invariant, i.e., it most not depends on the labeling or pictorial representation of the graph. Topological indices play an important role in structure property and structure activity studies, particularly when multivariate regression analysis, artificial neural networks, and pattern recognition are used as statistical tools.

Let G be an undirected connected graph without loops or multiple edges. The sets of vertices and edges of G are denoted by V (G) and E (G), respectively. For vertices u and v in V (G), we denote by d (u, v) the topological distance i.e., the number of edges on the shortest path, joining the two vertices of G. Since G is connected, d (u, v) exists for all vertices

u , v ∈ V ( G )

. Suppose deg (u) denotes degree of vertex . Klavzˇar and Gutman defined the modified Schultz index of the graph G as follow [2]

) (G V u ∈

∑

⊆

=

} { } , {

*

( ) deg( ) deg( ) ( , )

G V v u

v u d v u

G

S

(1)

Recently computing topological indices of nanostructures have been the object of many papers [3]-[13]. The modified Schultz index of C4C8(S) nanotubes was introduced by Shubo and Fangli [1]. In this paper we find an exact formula for calculation the modified Schultz index of C4C8(S) nanotubes and nanotorus.

Main results

Suppose a C4C8(S) nanotubes constructed by rolling a lattice of carbon atoms (figure (2)) For following computations we choose a coordinate label for vertices of this lattice as shown in figure (2). Let the graph has q row and 2p column of vertices. In this case we denote the nanotubes by G=T (p, q). If

q ≤ p

nanotubes is called short and if q> p, then nanotubes is called long. We compute by using the Wiener index of C4C8(S) nanotubes where calculated in simple exact formula in [7] as follow:

)

*

( G S

Theorem A. The Wiener index of C4C8(S) nanotubes given by

⎪⎪

⎩

⎪⎪

⎨

⎧

>

− + + +

+

≤ +

+

=

. ,

) 12 16q 2)p 12p ( q 8p -2p 3 (

, ) 8p - 2q - ) 3 ( 8 2 3 ( ) (

3 2

2 3 2

3

p q if p q

p q if q

p pq pq q

G W

Now we compute the summation of distances between vertex

v ∈ V (G )

and vertices of the graph in which placed on kth row below of vertex v. Let x0p and y0p be two vertices of graph placed on the first row of the graph and xkt and ykt be vertices in kth row and tth column of the graph for and . Let dx (k) denotes the summation of distances between vertex x0p and all of the vertices placed in kth row of the graph. Thus

q k <

≤

1 1 ≤ t < 2 p

∑

⁻

=

+

=

¹

0

0

0

) ( , ))

, ( ( ) (

p

i

p kt p

kt

x

k d x x d y x

d

Similarly we define dy (k) as follow:

∑

⁻

=

+

=

¹

0

0

0

) ( , ))

, ( ( ) (

p

i

p kt p

kt

y

k d x y d y y

d

In the following Lemma the value of dx (k) and dy (k) are computed in [7].

Fig. 2. A C4C8(S) Lattice with p=4 and q=6.

Lemma 1. Let

0 ≤ k < q

, then

⎪⎩

⎪⎨

⎧

>

+ +

≤ +

+

= +

. 2 2 4

2 )

( 2 2 )

( ²

2 2

p k if p

p kp

p k if k k kp p

k d_x And

⎪⎩

⎪⎨

⎧

>

− +

≤

− +

= +

. 2 2 4

2 )

( 2 2 )

( ²

2 2

p k if p

p kp

p k if k k kp p

k d_y

Now the modified Schultz index of the graph can be calculated by using Theorem A and Lemma 1.

In the following Theorem to consideration two cases odd and even value for p put

2 ) 1 ( 1 − −

^p

α =

. Theorem 1. The modified Schultz index of C4C8(S) nanotubes is given as

⎩ ⎨

⎧

>

− +

−

− +

+

− +

− +

−

≤

− +

−

− +

− +

− + +

= −

. ,

] ) 2 2 24 24 ( ) 6 12 18 ( ) 4 24 ( 3 [ 2

], 2 4 2 4 3 ) 2 2 12 12 ( ) 4 24 36 [(

) 2

(

_{4 3 2 2 3 2}

2 3 4 2

3 2 2

*

p q if p

q q q p

q q p q p

p

p q if q

q q q p q q q p

q q G p

S α

α

Proof: If denotes the vertex of G where placed on kth row and tth column for and

} , {

_{kt kt}

kt

x y

u ∈

− 2 0 t

1 ≤ k < q ≤ < 2 p

we have

deg( u

_k,t

) = 3

and

deg( x

_1,t

) = deg( x

_q−1,t

) = 2

for

1 ≤ t < 2 p

. Therefore

as

].

4 2 4 3 ) 2 2 12 12 ( ) 4 24 36

[(

2 )

(

^{2 2 3 2 4 3 2}

*

G p q q p q q q p q q q q

S = − + + − + − + − − +

If p is odd, then

].

2 4 2 4 3 ) 2 2 12 12 ( ) 4 24 36

[(

2 )

(

^{2 2 3 2 4 3 2}

*

G = p q − q + p + q − q + q − p + q − q − q + q −

S

Now suppose

p < q

. Similar to previous case by using Theorem A and Lemma 1 if p is even

].

) 2 2 24 24

( ) 6 12 18 ( ) 4 24 ( 3 [ 2 )

(

^{4 3 2 2 3 2}

*

G p p q p q q p q q q p

S = − + − + − + + − − +

And if p is odd, then

].

2 ) 2 2 24 24

( ) 6 12 18 ( ) 4 24 ( 3 [ 2 )

(

^{4 3 2 2 3 2}

*

G = p − p + q − p + q − q + p + q − q − q + p −

S

Therefore the proof is completed.

In continue we compute the modified Schultz index of C4C8(S) nanotorus by using the exact formula for computation Wiener index of this graph which obtained in [11]. We suppose the graph of C4C8(S) nanotorus is constructed by joining the vertices of the graph which placed on first and last row of C4C8(S) nanotubes (figure (3)).

Theorem B. The Wiener index of

G = C

₄

C

₈

( S )

nanotubes is computed as

⎪ ⎪

⎩

⎪⎪ ⎨

⎧

>

− + +

≤

− +

= +

. ,

) 2 3 3 2 3 ( 2

) 4 6

24 3 ( )

(

2 2

2

2 2

p q if q

pq qp p

p q if q

pq pq p

G W

Now we can compute the modified Schultz index of C4C8(S) nanotorus by using the Wiener index of this graph.

Fig. 3. A C4C8(S) nanotprus (a) slide view (b) top view.

Theorem B. The Wiener index of

G = C

₄

C

₈

( S )

nanotubes is computed as

⎪ ⎪

⎩

⎪⎪ ⎨

⎧

>

− + +

≤

− +

= +

. ,

) 2 3 3 2 3 ( 2

) 4 6

24 3 ( )

(

2 2

2

2 2

p q if q

pq qp p

p q if q

pq pq p

G W

Now we can compute the modified Schultz index of C4C8(S) nanotorus.

Theorem 2. The modified Schultz index of

G = C

₄

C

₈

( S )

nanotubes is given by

⎩ ⎨

⎧

>

+

= ≤

. ,

2) - 3qp 2p

+ q(3q 12p

4) - q + 6qp + (24p ) 3pq

(

_{2 2 2}

2 2

2

*

p q if

p q G if

S

Proof: Let

u

_kt

∈ { x

_kt

, y

_kt

}

denote the vertex of G where placed on kth row and tth column for

0 ≤ k < q − 1

and

0 ≤ t < 2 p

we have

deg( u

_k,t

) = 3

. Thus

) ( 9 ) , ( 9 )

, ( ) deg(

) deg(

) (

} { } , { }

{ } , {

* G u v d u v d u v W G

S

G V v u G

V v u

=

=

=

∑ ∑

⊆

⊆

Thus if

q ≤ p

then by using Theorem B we have

4).

- q + 6qp + (24p 3pq ) ( 9 )

(

^{2 2 2}

*

G = W G =

S

If

q > p

then

. 2) - 3qp 2p

+ q(3q 12p ) ( 9 )

(

^{2 2 2}

*

G = W G = +

S

Therefore the proof is done.

References

[1] C. Shubo, X. Fangli, J. Comput. Theor. Nano. Sci. 6(7), 1504 (2007).

[2] S. Klavzˇar, I. Gutman, Disc. Appl. Math., 80, 73 (1997).

[3] A. R. Ashrafi, S. Yousefi, MATCH Commun. Math. Comput. Chem. 57, 403 (2007).

[4] J. Devillers, A. Balaban, Gordon and Breech, Amsterdam 1999.

[5] M. V. Diudea, J. Chem. Inf. Comput. Sci. 36, 833 (1996).

[6] M. V. Diudea, J. Chem. Inf. Comput. Sci. 36, 535 (1996).