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Int. J. Mol. Sci. 2008, 9(10), 2016-2026; doi:10.3390/ijms9102016

Article
Schultz Index of Armchair Polyhex Nanotubes
Mehdi Eliasi * and Nafiseh Salehi
Islamic Azad University Najafabad Branch, Isfahan, Iran; E-Mail: nsalehi@iaun.ac.ir
Author to whom correspondence should be addressed. E-mail: eliasi@math.iut.ac.ir
Received: 3 July 2008; in revised form: 23 August 2008 / Accepted: 14 October 2008 /
Published: 29 October 2008

Abstract

: The study of topological indices – graph invariants that can be used for describing and predicting physicochemical or pharmacological properties of organic compounds – is currently one of the most active research fields in chemical graph theory. In this paper we study the Schultz index and find a relation with the Wiener index of the armchair polyhex nanotubes TUV C6[2p; q]. An exact expression for Schultz index of this molecule is also found.
Keywords:
Topological index; Wiener index; Schultz index; Armchair nanotube; Molecular graph; Distance; Carbon Nanotube

1. Introduction

Topological indices are a convenient method of translating chemical constitution into numerical values that can be used for correlations with physical, chemical or biological properties. This method has been introduced by Harold Wiener as a descriptor for explaining the boiling points of paraffins [13]. If d(u, v) is the distance of the vertices uand νof the undirected connected graph G (i.e., the number of edges in the shortest path that connects u and v) and V (G) is the vertex set of G, then the Wiener index of G is the half sum of distances over all its vertex pairs (u, v):

W ( G ) = 1 2 u V ( G ) ν V ( G ) d ( u , ν ) .

A unified approach to the Wiener topological index and its various recent modifications is presented. Among these modifications particular attention is paid to the Hyper-Wiener, Harary, Szeged, Cluj and Schultz indices as well as their numerous variants and generalizations [410]. The Schultz index of the graph was introduced by Schultz [14] in 1989 and is defined as follows:

S ( G ) = 1 2 u V ( G ) ν V ( G ) ( deg ( u ) + deg ( ν ) ) d ( u , ν ) ,
where deg(u) is the degree of the vertex u.

The main chemical applications and mathematical properties of this index were established in a series of studies [1215]. Also a comparative study of molecular descriptors showed that the Schultz index and Wiener index are mutually related [1618].

Carbon nanotubes, the one-dimensional carbon allotropes, are intensively studied with respect to their promise to exhibit unique physical properties: mechanical, optical electronic etc. [1921]. In [19], Diudea et al. obtained the Wiener index of TUV C6[2p; q], the armchair polyhex nanotube (see Figure 1). Here we find a relation between the Schultz index and Wiener index of this molecule. By using this relation we find an exact expression for the Schultz index of the same. The Appendix includes a Maple program [22] to produce the graph of TUV C6[2p; q], and to compute the Schultz index of the graph.

2. Schultz index of armchair polyhex nanotubes

Throughout this paper G := TUV C6[2p; q]denotes an arbitrary armchair polyhex nanotube in terms of its circumference 2p and their length q, see Figure 2. At first we consider an armchair lattice and choose a coordinate label for it, as illustrated in Figure 2. The distance of a vertex u of G is defined as

d ( u ) = x V ( G ) d ( u , x ) ,
the summation of distances between v and all vertices of G. By considering this notation the following lemma gives us a relation between the Schultz and Wiener index of G.

Lemma 1. For the graph G = TUV C6[2p; q]we have

S ( G ) = 6 W ( G ) - 2 u l e v e l 1 d ( u ) .

Proof: For each k such that 1 ≤ kq put Ak := {u ε V (G)u; level k}( see Figure 2). Then

S ( G ) = 1 2 u V ( G ) ν V ( G ) ( deg ( u ) + deg ( ν ) ) d ( u , ν ) = 1 2 u V ( G ) ν V ( G ) deg ( u ) d ( u , ν ) + 1 2 u V ( G ) ν V ( G ) deg ( ν ) d ( u , ν ) = 1 2 u V ( G ) ν V ( G ) deg ( u ) d ( u , ν ) + 1 2 ν V ( G ) u V ( G ) deg ( ν ) d ( u , ν ) = 1 2 u V ( G ) deg ( u ) ν V ( G ) d ( u , ν ) + 1 2 ν V ( G ) deg ( ν ) u V ( G ) d ( u , ν ) = 1 2 u V ( G ) deg ( u ) d ( u ) + 1 2 ν V ( G ) deg ( ν ) d ( ν ) = u V ( G ) deg ( u ) d ( u )
But
deg ( u ) = { 2 if u A 1 A q 3 if o t h e r w i s e .

Also in the graph G it is clear that u A 1 d ( u ) = u A q d ( u ). Therefore

S ( G ) = u V ( G ) deg ( u ) d ( u ) = u A 1 A q deg ( u ) d ( u ) + u V ( G ) \ ( A 1 A q ) d e g ( u ) d ( u ) = u A 1 A q 2 d ( u ) + u V ( G ) \ ( A 1 A q ) 3 d ( u ) = 3 u V ( G ) d ( u ) - 2 u A 1 d ( u ) = 6 W ( G ) - 2 u A 1 d ( u ) .
This completes the proof.

To compute the d(u) in the graph G, when u is a vertex in level 1, we first prove the following lemma.

Lemma 2.The sum of distances of one vertex of level 1 to all vertices of level k is given by

w k : = x l e v e l k d ( x 10 , x ) = x l e v e l k d ( x 11 , x ) = { 2 p 2 + k 2 - 2 k - 2 p + 1 + H ( p , k ) if 1 k < p p ( p + 2 k - 2 ) if k p ,
where
H ( p , k ) = { 2 p - 1 if k + p is    even 2 p if k + p is  odd .

Proof:We calculate the value of wk. We consider that the tube can be built up from two halves collapsing at the polygon line joining x10 to xq,0 (see Figure 2). The right part is the graph G1 which consists of vertical polygon lines 0, 1,. . . . . p and x10 is one of the vertices in the first row of the graph G1. The left part is the graph G2 which consists of vertical polygon lines (p + 1); (p + 2),. . . . , 2p –1. We change the indices of the vertices of G2 in the following way:

V ( G 2 ) = { x ^ j i | x ^ j , i = x j , 2 p - i V ( G ) }

(See Figure 3)

We must consider two cases:Case 1: If k ≥ p. In the graphs G1 and for 0 i < k we have

d ( x 10 , x k , i ) = k + i - 1.

Also in the graphs G2 and for 1 ≤.i < k we have

d ( x 10 , x ^ k , i ) = k + i - 1.
So
x l e v e l k d ( x 10 , x ) = 2 i = 1 p - 1 ( k + i - 1 ) + ( 0 + k - 1 ) + ( p + k - 1 ) = p ( p + 2 k - 2 ) .

Case 2: If k < p. First suppose that 1 ≤ i < k. In the graphs G1 and G2 we have

d ( x 10 , x k , i ) = k + i - 1 = d ( x 10 , x ^ k , i ) = k + i - 1.

Now suppose that kip. Then in the graph G1 we can see that if k is odd, then

d ( x 10 , x k , i ) = { 2 i if i is even 2 i - 1 if i is odd 
and if k is even, then
d ( x 10 , x k , i ) = { 2 i - 1 if i is even 2 i if i is odd
Also in G2 we have
d ( x 10 , x ^ k , i ) = { 2 i if i is even 2 i + 1 if i is odd 
if k is odd
d ( x 10 , x ^ k , i ) = { 2 i + 1 if i is even 2 i if i is odd 
and if k is even.

All of this distances give us

x l e v e l k d ( x 10 , x ) = 2 p 2 + k 2 - 2 k - 2 p + 1 + H ( p , k ) .

For other vertices we can convert those to x10 by changing transfer vertices and apply a similar argument by choosing suitable G1 and G2 and compute wk.

By a straightforward computation (if irem means the positive integer remainder) we can see:

H ( p , k ) = 2 p - 1 + irem(k + p , 2 ) = 2 p - 1 + 1 2 + 1 2 ( - 1 ) k - irem ( p , 2 ) + 1 ,
where
irem ( p , 2 ) = { 0 if p is even 1 if p is odd .

So, by Lemma 1, when 1 ≤ kp, we have

w k = 2 p 2 + k 2 - 2 k + 1 2 + 1 2 ( - 1 ) k - irem ( p , 2 ) + 1 .

Also in the graph G,

d ( x 10 ) = x l e v e l 0 d ( x 10 , x ) + x l e v e l 1 d ( x 10 , x ) + + x l e v e l q d ( x 10 , x ) = w 1 + w 2 + + w q .
So
d ( x 10 ) = d ( x 11 ) = = d ( x 2 p - 1 , 1 ) = w 1 + w 2 + + w q .

This leads us to the following corollary:

Corollary 1. For each vertex u on level 1 we have

d ( u ) = w 1 + w 2 + + w q .

Now suppose that p > q. Then by lemma 2 and equation (1) we have

d ( u ) = k = 1 q ( 2 p 2 + k 2 - 2 k + 1 2 + 1 2 ( - 1 ) k - irem ( p , 2 )+ 1 ) = 2 p 2 q + q 3 3 - q 2 2 - q 3 + 1 4 ( - 1 ) - i r e m ( p , 2 ) + 1 + q + 1 4 ( - 1 ) - i r e m ( p , 2 ) .

Also if p ≤ q, then by Lemma 1 and equation (1) we have

d ( u ) = w 1 + w 2 + + w p - 1 + w p + w p + 1 + + w q = k = 1 p - 1 ( 2 p 2 + k 2 - 2 k + 1 2 + 1 2 ( - 1 ) k - irem ( p , 2 ) + 1 ) + k = p q p ( p + 2 k - 2 ) = p 3 3 + p 2 2 - p 3 - 1 4 ( - 1 ) - i r e m ( p , 2 ) + 1 + p - 1 2 - 1 4 ( - 1 ) - i r e m ( p , 2 ) + 1 + p 2 q - p q + p q 2

We summarize the above results in the following proposition

Corollary 2. For each vertex u on level 1, d(u) is given by

Case 1: p is even

d ( u ) = { 2 p 2 q + q 3 3 - q 2 2 - q 3 + 1 4 + 1 4 ( - 1 ) q + 1 if p > q p 6 [ 2 p 2 + 3 p - 2 + 6 p q - 6 q + 6 q 2 ] if p q

Case 2: p is odd

d ( u ) = { 2 p 2 q + q 3 3 - q 2 2 - q 3 + 1 4 + 1 4 ( - 1 ) q if p > q p 3 3 + p 2 2 - p 3 - 1 2 + p 2 q - p q + p q 2 if p q

Theorem 1. The Wiener index of G := TUV C6[2p; q] nanotubes is given by

Case 1: p is even

W ( G ) = { p 12 [ 3 ( - 1 ) q + 1 + 3 + 24 q 2 p 2 - 8 q 2 + 2 q 4 ] if p > q - p 2 6 [ 8 q - 4 p + p 3 - 4 q p 2 - 4 q 3 - 6 q 2 p ] if p q

Case 2: p is odd

W ( G ) = { p 12 [ 3 ( - 1 ) q - 3 + 24 q 2 p 2 - 8 q 2 + 2 q 4 ] if p > q - p 6 [ - 4 p 3 q - 4 p q 3 - 6 q 2 p 2 + 3 + 8 q p - 4 p 2 + p 4 ] if p q

Proof: See [19].

Now we are in the position to prove the main result of this section.

Theorem 2. The Schultz index of G:= TUV C6[2p; q] nanotubes is given by

Case 1: p is even

S ( G ) = { p 6 [ - 48 q 2 p + 72 p 2 q 2 + 3 ( - 1 ) q + 1 + 3 - 8 q 3 - 12 q 2 + 6 q 4 + 8 q ] if p > q - p 2 3 [ - 18 q 2 p + 3 p 3 - 6 p - 12 p 2 q - 12 q 3 + 12 q + 4 p 2 - 4 + 12 p q + 12 q 2 ] if p q

Case 2: p is odd

S ( G ) = { p 6 [ 72 q 2 p 2 + 6 q 4 - 12 q 2 - 3 + 3 ( - ) q - 48 p 2 q - 8 q 3 + 8 q ] if p > q - p 3 [ - 12 p 3 q - 12 p q 3 - 18 p 2 q 2 + 3 + 12 p q - 6 p 2 + 3 p 4 + 4 p 3 - 4 p + 12 p 2 q + 12 p q 2 ] if p q

Proof: According to Lemma 1 we must calculate 6W(G) –∑ulevel 1 d(u). But by corollary 1 we have

d ( u ) = w 1 + w 2 + + w q .

Since there are 2p vertices on level 1 therefore

S ( G ) = 6 W ( G ) - 4 p d ( u )

Finally by replacing d(u) from corollary 1 in the equation (2) the result obtains.

3. Experimental Section

Tables 1 and 2 show the numerical data for the Schultz index in tubes TUV C6[2p; q] of various dimensions.

4. Appendix

The following code is the MAPLE program [22] used to produce the graph of TUHC6[2p; q] and to compute the Schultz index of the graph.

> restart;with(networks):

> l:=proc(p,q) (*generating the graph *)

local G,i,j,k,ff,cc;G:=new();

  for i from 0 to (2*p–1) do

      for j from 1 to q do

      addvertex(a[i,j],G);

      end do;

  end do;

  for i from 0 to (2*p–1) do

        for j from 1 to (q–1) do

          addedge ({a[i,j],a[i,j+1]},G);

       end do;

  end do;

     for i from 0 to (2*p–2)/2 do

           for k from 1 to iquo(q,2) do

                addedge({a[2*i,2*k–1],a[2*i+1,2*k–1]},G);

           end do;

       end do;

  for i from 0 to (2*p–4)/2 do

    for k from 1 to iquo(q,2) do

      addedge({a[2&ast;i+1,2&ast;k],a[2&ast;i+2,2&ast;k]},G);

    end do;

  end do;

for ff from 1 to iquo(q,2) do

   addedge({a[2*p–1,2*ff],a[0,2*ff]},G);

  end do;

  if irem(q,2)=1 then

  for cc from 0 to 2*p/2–1 do

   addedge({a[2*cc,q],a[2*cc+1,q]},G); end do;

   end if ;return(G);

   end proc:

> m:=l(3,8):(#Graph G:=TUVC_6[2*3,8]#)

> t :=edges(m):

> ii:=vertices(m):

> T := allpairs(m,p):

> Sch:=proc(u)

   local b,o,pp;

   b:=0;

   for o in ii do

     for pp in ii do

         b:=b+ T[(pp,o)]*(vdegree(o,m)+vdegree(pp,m));

      end do;

   end do;

return(b/2);

   end proc:

> Sch(u); 27648(#The Schultz index of the graph #)

This work was supported by a grant from the Center of Research of Islamic Azad University, Najafabad Branch, Isfahan, Iran.

References and Notes

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Figure 1. A TUV C6[2p; q] Lattice with p = 5 and q = 7.

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Figure 1. A TUV C6[2p; q] Lattice with p = 5 and q = 7.
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Figure 2. An armchair polyhex nanotube [19].

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Figure 2. An armchair polyhex nanotube [19].
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Figure 3. Distances from x01 to all vertices of TUV C6[2p; q] with p = 5 and q = 7.

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Figure 3. Distances from x01 to all vertices of TUV C6[2p; q] with p = 5 and q = 7.
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Table Table 1. Schultz index of short tubes, p > q.

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Table 1. Schultz index of short tubes, p > q.
pqS(G)pqS(G)
626912524000
63183665310650
64354245420720
655865695193266
1023200096288432
10526416097404514
10639344098542880
1075505601582425440
1087369601571823310
1099544001561310160
Table Table 2. Schultz index of long tubes, pq.

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Table 2. Schultz index of long tubes, pq.
pqS(G)pqS(G)
4410816344752
4518304358262
46283523613104
47413443719494
48576643827648
1021681040011111954502
1022764160011122371952
1023853680011132839524
1024949840011143359312
10251052880011153935030
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