Research Article
Szeged Index of HAC5C6C7[k, p] Nanotube
Department of Mathematics, Tarbiat Modares University, P.O. Box 14115-137, Tehran, Iran
Y. Pakravesh
Department of Mathematics, Tarbiat Modares University, P.O. Box 14115-137, Tehran, Iran
A graph G consists of a set of vertices V(G) and a set of edges E(G). In chemical graphs, each vertex represented an atom of the molecule and covalent bonds between atoms are represented by edges between the corresponding vertices. This shape derived form a chemical compound is often called its molecular graph and can be a path, a tree or in general a graph.
A topological index is a single number, derived following a certain rule, which can be used to characterize the molecule (Khadikar and Karmakar, 2002). Usage of topological indices in biology and chemistry began in 1947 when chemist Harold Wiener (Wiener, 1947) introduced Wiener index to demonstrate correlations between physico-chemical properties of organic compounds and the index of their molecular graphs. Wiener originally defined his index (W) on trees and studied its use for correlation of physico chemical properties of alkenes, alcohols, amines and their analogous compounds. A number of successful QSAR studies have been made based in the Wiener index and its decomposition forms (Agrawal et al., 2000).
In a series of papers, the Wiener index of some nanotubes is computed (Deng, 2007; Diudea et al., 2004; Randic, 2002; Stefu and Diudea, 2004; Yousefi and Ashrafi, 2006). Another topological index was introduced by Gutman and called the Szeged index, abbreviated as Sz (Gutman, 1994).
Let e be an edge of a graph G connecting the vertices u and v. Define two sets N1(e|G) and N2(e|G) as N1(e|G) = {x∈V(G)|d(u,x)<d(v,x)} and N2(e|G) = {x∈V(G)|d(x,v)<d(x,u)}. The number of elements of N1(e|G) and N2(e|G) are denoted by n1(e|G) and n2(e|G), respectively. The Szeged index of the graph G is defined as Sz(G) = Sz = Σe∈e(G)n1(e|G)n2(e|G). The Szeged index is a modification of Wiener index to cyclic molecules. The Szeged index was conceived by Gutman at the Attila jozsef University in Szeged. This index received considerable attention. It has attractive mathematical characteristics. In (Iranmanesh and Khormali, 2007a; Iranmanesh et al., 2007) Szeged index and in (Ashrafi and Loghman, 2006; Deng, 2007; Iranmanesh and Ashrafi, 2007; Iranmanesh and Khormali, 2007b; Iranmanesh and Pakravesh, 2007; Iranmanesh and Solemani, 2007), another topological index of some nanotubes is computed. In this study, we computed the Szeged index of HAC5C6C7[k, p] nanotube. In Fig. 1, an HAC5C6C7[2, 2] lattice is shown.
We denote the number of pentagons in one row by p, the number of the periods by k and each period consist of three rows as in Fig. 2, which shows the m-th period, 1≤m≤k.
Fig. 1: | HAC5C6C7[2, 2] nanotube, p = 2, k = 2 |
Fig. 2: | m-th period of HAC5C6C7[k, p] |
THE SZEGED INDEX OF HAC5C6C7[k, p] NANOTUBE
Let e be an edge in Fig. 1. Denote:
And the number of vertices in each period of this nanotube is equal to 16p. In the hole of paper, the notation [f] is the greatest integer function. For computing the Szeged index, we must discus in two cases:
Case1:p is even.
Let e = uv be an edge denoted in Fig. 3.
a) | If e ∈ E1, then according to Fig. 3, the region R has vertices belong to N1(e|G) and the region of R has vertices belong to N2(e|G) (The notations n1(e|G) and n2(e|G) are indicated with ne(u) and ne(v), respectively). Then |
In this study for simplicity we define:
If | m≤A(-4)+1 | then |
Fig. 3: | e = uv belong to E1 |
If m>A(-4)+1, then
b) | If e ∈ E2, then according to Fig. 4, the region R has the vertices that belongs to N1(e|G) and the region of R has vertices belong to N2(e|G). Then |
Fig. 4: | e = uv belong to E2 |
c) | If e ∈ E3, then according to Fig. 5, the region R has the vertices that belongs to N1(e|G) and the region of R has vertices belong to N2(e|G). |
In Fig. 5, 6 and 10 the symbol B means that, the vertex that assigned with this symbol, have the same distance from u and v. Then
Fig. 5: | e = uv belong to E3 |
Fig. 6: | e = uv belong to E4 |
d) | If e ∈ E4, then according to Fig. 6, the region R has the vertices that belongs to N1(e|G) and the region of R has vertices that belongs to N2(e|G). Then: |
e) | If e ∈ E5, then according to Fig. 7, the region R has the vertices that belongs to N1(e|G) and the region of R has vertices that belongs to N2(e|G). Then: ne(v) = 16p(k-m)-1 |
f) | If e ∈ E6, then according to Fig. 8, the region R has the vertices that belongs to N1(e|G) and the region of R has vertices that belongs to N2(e|G). Then: |
Fig. 7: | e = uv belong to E5 |
Fig. 8: | e = uv belong to E6 |
Fig. 9: | e = uv belong to E7 |
g) | If e ∈ E7, then according to Fig. 9, the region R has the vertices that belongs to N1(e|G) and the region of R has vertices that belongs to N2(e|G). Then: |
Fig. 10: | e = uv belong to E8 |
h) | If e ∈ E8, then according to Fig. 10, the region R has the vertices that belongs to N1(e|G) and the region of R has vertices that belongs to N2(e|G). Then: |
And for ne(u) we have:
For simplicity we define
Case 2: p is odd.
if we subsitute to then we obtain the Szeged index of In fact
The Szeged index of nanotube for as given as follows:
Therefore the Szeged index of above nanotube is computed.
This research is partially supported by Iran National Science Foundation (INSF) (Grant No. 83120).