PI POLYNOMIAL OF TUC4C8(S) NANOTUBES AND NANOTORUS
AMIR LOGHMAN•, LEILA BADAKHSHIANa
Department of Mathematics, University of Isfahan, Isfahan 81746-73441, Iran
aIslamic Azad University Najafabad Branch, Isfahan, Iran;
A C4C8 net is a trivalent decoration made by alternating squares C4 and octagons C8. Such a covering can be derived from square net by the leapfrog operation. The PI polynomial of a molecular graph G is defined as A +Σx|E(G)|−N(e), where N(e) is the number of edges parallel to e, A=1/2|V(G)|(|V(G)|+1)-|E(G)| and summation goes over all edges of G. In this paper, the PI polynomial of TUC4C8(S) Nanotubes and Nanotorus are computed.
(Received September 8, 2009; accepted October 2, 2009)
Keywords: PI index, PI polynomial, TUC4C8(S) nanotube, nanotorus
1. Introduction
A graph G consists of a set of vertices V(G) and a set of edges E(G). The vertices in G are connected by an edge if there exists an edge UiUj ∈ E(G) connecting the vertices Ui and Uj in G such that Ui, Uj ∈ V(G). In chemical graphs, the vertices of the graph correspond to the atoms of the molecule, and the edges represent the chemical bonds. The number of vertices and edges in a graph will be denoted by |V(G)| and |E(G)|, respectively. The distance between a pair of vertices u and w of G is denoted by d(u,v).
A topological index is a real number related to a graph. It must be a structural invariant, i.e., it is fixed by any automorphism of the graph. There are several topological indices have been defined and many of them have found applications as means to model chemical, pharmaceutical and other properties of molecules. The Wiener index W is the first topological index to be used in chemistry. It was introduced in 1947 by Harold Wiener, as the path number for characterization of alkanes, [16]. In a graph theoretical language, the Wiener index is equal to the count of all shortest distances in a graph, [9,16].
Let G be a graph and e = uv an edge of G. neu(e|G) denotes the number of edges lying closer to the vertex u than the vertex v, and nev(e|G) is the number of edges lying closer to the vertex v than the vertex u. The Padmakar–Ivan (PI) index of a graph G is defined as PI(G)
= where PI(f) = nfu(f|G) + nfv(f|G) see for details [8,10-12]. In this definition, edges equidistant from the two ends of the edge e = uv are not counted. We call such edges parallel to e.
The number of edges parallel to e is denoted by N(e).
∈∑( ) ( )
G E f
f PI
The PI polynomial, introduced by Ashrafi, Manoochehrian and Yousefi-Azari [5], of a connected graph G is defined as PI(G,x)= where N(u,v)=N(f) if f=uv∈E(G) and N(u,v)=0 if uv∉E(G). We can see that (1)
∑
⊆− ) ( } , {
) , ( )|
(
| G V v u
v u N G
x
EPI(G,x)=
| ( ) |
(1)2 1
| ) (
|
) (
) (
|) (
|
V G E G
x
G E f
f N G
E
⎟⎟ −
⎠
⎜⎜ ⎞
⎝
⎛ +
∑ +
∈
−
• Corresponding author: [email protected]
In a series of papers [1-4,6], Ashrafi and Loghman computed PI index of some nanotubes and nanotori. In [7,13] we computed polynomial of zig-zag and armchair polyhex nanotubes and nanotori. Here we continue this progress to compute the PI polynomial of the TUC4C8(S) polyhex nanotubes and nanotorus. Our notation is standard and mainly taken from [14,15]. Throughout this paper T = TUC4C8(S)[4p,q] an arbitrary C4C8 nanotubes and T'= T[2p,2q] denotes a C4C8 nanotorus, see Figure 1, 2.
2. PI Polynomial of TUC4C8[4p,q]
In this section, the PI polynomial of the graph T = TUC4C8[4p,q] were computed. From Figures 1(a) and 1(b), it is easy to see that |E(T)| = 2p(3q-1). In the following theorem we compute the PI polynomial of the molecular graph T in Figure 1.
(a). The TUC4C8(S) Nanotubes (b). A TUC4C8(S) Lattice with p = 4 and q = 8 Fig. 1
(a). The C4C8(S) nanotorus (b). A C4C8(S) nanotorus Lattice Fig. 2
Theorem 1. The PI polynomial of TUC4C8(S)[4p,q] nanotube is as follows:
, 2 )
1 2 ) 1 (
( 2 2
2 )
2 1 2
) 1 (
( 2 2
2 f(x) x) PI(T,
2 2 2 ) 1 ( 6
2 ) 1 ( 2 ) ( 2 6
) ( 2 6
⎪ ⎪
⎪
⎩
⎪⎪
⎪
⎨
⎧
≥
−
− +
−
<
<
−
−
− +
−
≤ +
=
+
−
+ + −
− +
−
p q p
x q px x
p q p q
x p px x
p q pqx
p q
p
p q q
p pq
q p pq
where
| ( ) |
.2 1
| ) ( 2 |
) 1 ( 2
f(x)
2 (3 2) 6 2( )V T E T
pqx x
q
p
p q pq p q⎟⎟ ⎠ −
⎜⎜ ⎞
⎝
⎛ +
+ +
−
=
− − +Proof. To compute the PI polynomial of T, it is enough to calculate N(e). To do this, we consider three cases that e is vertical, horizontal or oblique. If e is horizontal a similar proof as Lemma 1,2 in [2] shows that N(e)=2q and N(e)=2p for vertical edge e. Also, by Lemma 3 in [2], if e is an oblique edge in the (2k-1)th row, 1≤ k ≤ p, of the TUC4C8(S)[4p,q] lattice of T, then N(e)
= .
⎩ ⎨
⎧
− +
≤
− +
≥
− +
1 2
1 2
2k 2p
k p q q
k p q
Therefore we consider Eij denote the oblique edge of T in the ith row and jth column. We first notice that for every i, 1≤ i ≤ q, N(E(2i-1)1) = N(E(2i-1)2) = ⋅⋅⋅ = N(E(2i-1)(2p)), Figure 1(b). Suppose A, B and C are the set of all horizontal, vertical and oblique edges of T. It is easy to see that
|A|=|C|=2pq and |B|=2p(q-1). Then Since T is symmetric, we have:
+ ∑ +
−
=
∑ +
∑ +
∑ +
=
∑ +
∑ +
∑ +
=
⎟⎟ −
⎠
⎜⎜ ⎞
⎝
⎛ +
∑ +
=
∈
− +
−
−
∈
−
∈
−
∈
−
∈
−
∈
−
∈
−
∈
−
C f
f N T E q
p pq q
p
C f
f N T E B
f
p T E A
f
q T E
C f
f N T E B
f
f N T E A
f
f N T E
K T
E f
f N T E
x pqx
x q p
K x
x x
K x
x x
T T E
x V
) (
|) (
| )
( 2 6 )
2 3 ( 2
) (
|) (
| 2
|) (
| 2
|) (
|
) (
|) (
| )
(
|) (
| )
(
|) (
| ) (
) (
|) (
|
2 )
1 ( 2
| ) ( 2 |
1
| ) ( x) |
PI(T,
4 4 4 3 4
4 4 2 1
For every e in C, we have three cases:
Case 1. q≤p. In this case, we have:
) ( 2 6 )
( )|
(
| )
( )|
(
| 11
2
pq p qY f
E N T E Y
f
f N T
E
x pqx
x
− +∈
−
∈
−
= ∑ =
∑
Case 2. p<q<2p. In this case, we have:
) 2 1 2
) 1 (
( 2 2
) 2 2
( 2
) ...
1 ( 4
) 2 2
( 2
) ...
( 4
2 ) 1 ( 2 ) ( 2 6
) ( 2 ) ( )|
(
|
) ( 2 4
2 )
( )|
(
|
) ( )|
(
|
) ( )|
( ) |
( )|
(
| ) ( )|
(
| )
( )|
(
|
11 11
) 1 ( 1
) 1 ( 12 1
11
−
−
− +
= −
−
− +
+ + + +
=
−
− +
+ + +
∑ =
+ + −
−
−
−
−
−
−
−
−
−
−
− −
−
∈
−
+
−
+
−
q x p
px x
x q p p
x x
x px
x q p p
x x
x p x
p q q
p pq
p q E N T E
p q E
N T E
E N T E
E N T E E
N T E E N T E Y
f
f N T E
p q
p q
Case 3. q≥2p. In this case by Figure 1, we have:
) 1 2
) 1 (
( 2 2
) 2 ( 2 ) ...
1 ( 4
) 2 ( 2 ) ...
( 4
2 2 2 ) 1 ( 6
) 1 _ ( 2 ) ( )|
(
| )
1 ( 2 4
2 )
( )|
(
|
) ( )|
(
| )
( )|
( ) |
( )|
(
| ) ( )|
(
| )
( )|
(
|
11 11
1 12 1
11
p x q
px x
x p q p x
x x px
x p q p x
x x
p x
p q
p
p E N T E p
E N T E
E N T E E
N T E E
N T E E
N T E Y
f
f N T
E p p
−
− +
= −
− +
+ + + +
=
− +
+ + +
∑ =
+
−
−
−
−
−
−
−
−
−
− −
−
∈
−
which completes the proof.
Corollary 1. The PI index of TUC4C8(S)[4p,q] nanotube is as follows:
PI(TUC4C8(S)[4p,q])
⎩ ⎨
⎧
≥
= ≤
=
=p q Y
p q X
1
x)
xPI(T, dx
d
,where X = 36p2q2 – 28p2q +8 p2 - 8pq2 and Y = 36p2q2 – 36p2q – 4pq2 + 4pq + 4p3 + 4p2.
3. PI Polynomial of C4C8(S) nanotorus
In this section, the PI polynomial of the graph T' = T[2p,2q] were computed. From Figures 2(a) and 2(b), we can see that |E(T)| = 6pq. In the following theorem we compute the PI polynomial of the C4C8(S) nanotorus.
Theorem 2. The PI polynomial of C4C8(S) nanotorus is computed as follows:
⎩⎨
⎧
≤ +
+ +
≥ +
+
= −− + −− −− ++
p q A x
x x
pq
p q A x
x x
pq
p q q
p p
q
q p q
p p
q
) (
2
) (
x) 2 ,
PI(T 2 (3 1) 2 (3 1) 6 ( 1) 2
2 ) 1 ( 6 ) 1 3 ( 2 ) 1 3 ( 2
' .
where
| ( ) |
.2 1
| ) (
A | V T E T
⎟⎟ −
⎠
⎜⎜ ⎞
⎝
⎛ +
=
Proof. To compute the PI polynomial of T', it is enough to calculate N(e). By Lemma 2, 3 and 4 in [6] we have:
N(e)=
⎪ ⎪
⎩
⎪ ⎪
⎨
⎧
≤
−
≥
−
p q and oblique an
is e 2
6
p q and oblique an
is e 2
6
horizontal is
e 2
vertical is
e 2
if q
if p
if q
if p
Let X, Y and Z are the set of all horizontal, vertical and oblique edges of T'. It is easy to see that
|X|=|Y|=|Z|=2pq. Then Since T' is symmetric, we have:
⎩ ⎨
⎧
≤ +
+ +
≥ +
+
= +
⎩ ⎨
⎧
≥ + ≤
+ +
=
∑ +
∑ +
∑ +
=
+
−
−
−
+
−
−
−
+
− +
− −
−
∈
−
∈
−
∈
−
p q A x
x x
pq
p q A x
x x
pq
p q pqx
p q A pqx
pqx pqx
A x
x x
p q q
p p
q
q p q
p p
q
q p
p q q
p p
q
Z f
f N G E Y
f
f N G E X
f
f N G E
) (
2
) (
2
2 2 2
2 x) PI(G,
2 ) 1 ( 6 ) 1 3 ( 2 ) 1 3 ( 2
2 ) 1 ( 6 ) 1 3 ( 2 ) 1 3 ( 2
2 ) 1 ( 6
2 ) 1 ( 6 )
1 3 ( 2 )
1 3 ( 2
) ( )|
(
| )
( )|
(
| ) ( )|
(
|
which completes the proof.
Corollary 2. Suppose T' is a C4C8(S) nanotorus. Then we have:
⎩ ⎨
⎧
<
+
−
−
≥ +
−
= −
=
=p q pq pq
q p q
p
p q pq pq
q p q p dx
d
4 4
20 36
4 10
8 x) 36
, PI(T )
PI(T
2 2 2 22 2
2 2 1
x ' '
Acknowledgement
The author thanks Professor A.R. Ashrafi very much for his valuable information about the advances on study of molecular graph and topological index.
References
[1] A. R. Ashrafi, A. Loghman, PI Index of Zig-Zag Polyhex Nanotubes, MATCH Commun.
Math. Comput. Chem., 55(2), 447 (2006).
[2] A. R. Ashrafi, A. Loghman, Padmakar-Ivan Index of TUC4C8(S) Nanotubes, J. Comput.
Theor. Nanosci. 3(3), 378 (2006).
[3] A.R. Ashrafi, A. Loghman, PI Index of Armchair Polyhex Nanotubes, Ars Combinatoria, 80, 193 (2006).
[4] A.R. Ashrafi, A. Loghman, Computing Padmakar-Ivan Index of TUC4C8(R) Nanotorus, J. Comput. Theor. Nanosci. 5, 1431 (2008).
[5] A.R. Ashrafi, B. Manoochehrian, H. Yousefi-Azari, On the PI Polynomial of a Graph, Util. Math., 71, 97 (2006).
[6] A.R. Ashrafi, F. Rezaei, A. Loghman, PI Index of of the C4C8(S)−Nanotorus, Revue Roumaine de Chimie (In press).
[7] L. Badakhshian, A. Loghman, PI Polynomial of Armchair Polyhex Nanotubes and Nanotorus, Digest Journal of Nanomaterials and Biostructures. 4(1), 183 (2009).
[8] H. Deng, Extremal Catacondensed Hexagonal Systems with Respect to the PI Index, MATCH Commun. Math. Comput. Chem., 55(2), 453 (2006).
[9] A. Graovac, T. Pisanski, On the Wiener index of a graph, J. Math. Chem. 8, 53 (1991).
[10] P.V. Khadikar, On a Novel Structural Descriptor PI, Nat. Acad. Sci. Lett., 23, 113 (2000).
[11] P.V. Khadikar, P.P. Kale, N.V. Deshpande, S. Karmarkar and V.K. Agrawal, Novel PI Indices of Hexagonal Chains, J. Math. Chem., 29, 143 (2001).
[12] P.V. Khadikar, S. Karmarkar and R.G. Varma, The Estimation of PI Index of Polyacenes, Acta Chim. Slov. 49, 755 (2002).
[13] A. Loghman, L. Badakhshian, PI Polynomial of Zig-Zag Polyhex Nanotubes, Digest Journal of Nanomaterials and Biostructures., 3(4), 299 (2008).
[14] R. Todeschini, V. Consonni, Handbook of Molecular De- scriptors, Wiley, Weinheim, 2000.
[15] N. Trinajstic, Chemical Graph Theory, CRC Press, Boca Raton, FL. 1992.
[16] H. Wiener, Structural determination of the paraffin boiling points, J. Am. Chem. Soc.
69, 17 (1947).
