On Wiener Index of One Heptagonal Nanocone - Ingenta Connect

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Current Nanoscience, 2012, 8, 180-185

On Wiener Index of One
Heptagonal Nanocone Ali Reza Ashrafi* and Zohreh Mohammad
Abadi Institute of Nanoscience and Nanotechnology, University of Kashan, Kashan 87317
51167, I. R. Iran; Department of Mathematics, Statistics and Computer Science, Faculty of Science, University of Kashan, Kashan 87317
51167, I. R. Iran Abstract: The Wiener index of a molecular graph G is defined as the summation of topological distances between all pair of atoms in G. In this paper an algorithm by Sandi Klavar [European J. Combin. 2006, 27, 68–73] is applied to calculate the Wiener index of one
heptagonal nanocone L[n]. It is proved that W(L[n]) = (238/5)n 5 + (238)n 4 + (2821/6)n 3 + (917/2)n 2 + (3311/15)n + 42.

Keywords: Carbon nanocone, Djokovic
Winkler relation, molecular graph, one
heptagonal nanocone, topological index, Wiener index. 1. INTRODUCTION Carbon nanocones form an interesting class of carbon nanomaterials which originally discovered by Ge and Sattler in 1994 [1]. These are constructed from a graphene sheet by removing a 60 wedge and joining the edges produces a cone with a single pentagonal defect at the apex. The inclusion of the heptagons in the hexagonal lattice leads to the appearance of negative curvature, Fig. (1). The single sevenfold in the plain graphene lattice was theoretically studied but this situation, unfortunately, has not been observed in the experiment yet. We now describe some notations which will be kept throughout. A graph is a pair (V, E) such that V   and E  P2(V), the set of all 2
element subsets of V. A molecular graph is a simple graph such that its vertices correspond to the atoms and the edges to the bonds. Note that hydrogen atoms are often omitted. An automorphism of a graph G is a permutation g of the vertex set V(G) with the property that, for any vertices u and v, g(u) and g(v) are adjacent if and only if u is adjacent to v. Suppose Graphs denotes the set of all molecular graphs. A map Top from Graphs into real numbers is called a topological index, if G  H implies that Top(G) = Top(H) [2,3]. Topological indices are graph invariants and are used for Quantitative Structure-Activity Relationship (QSAR) and Quantitative Structure-Property Relationship (QSPR) studies. The QSAR and QSPR studies are the active areas of chemical research that focus structure-dependent chemical behavior of molecules [4,5]. Obviously, the maps Top1 and Top2 defined as the number of edges and vertices, respectively, are topological index. The distance d(u,v) between two vertices u and v of a graph G is defined as the length of a shortest path connecting them. The summation of these numbers over all edges of G is called the Wiener index of G [6]. A topological index defined by the distance function d(
,
) is called a distance
based topological index. The Wiener index is the first distance
based topological index introduced by Harold Wiener. We encourage the interested readers to consult Refs. [7-11] and references therein, for further study on the topic. The importance of distance
based topological indices was discovered by Diudea. He computed the Wiener index of some nanotubes and tori [12-18]. One of us (ARA) continued the pioneering work of Diudea to compute some distance
based topological indices of nanocones [19-24]. *Address correspondence to these authors at the Institute of Nanoscience and Nanotechnology, University of Kashan, Kashan 87317
51167, I. R. Iran; Tel: +98 (361) 591 23 67; Fax: +98 (361) 555 29 30; E-mail: [email protected]

17-/12 $58.00+.00

Klavar [25-27] presented a powerful method for computing Wiener index of graphs and Ariza and Ortiz [28] presented an application of the theory of discrete dislocations to the analysis of dislocations in graphene. In this paper, in a similar way as [28,29] and by applying Klavar's method, we compute the Wiener index of L[n] = CNC7[n] nanocone, Fig. (1).

Fig. (1). The One Heptagonal Nanocone CNC7[4].

Throughout this paper, our notation is standard and taken from the standard book of graph theory. The main result of this paper is as follows: Theorem A. Suppose L[n] = CNC7[n] is an n
layers one heptagonal carbon nanocone. Then

W(L[n]) = (238/5)n 5 + (238)n 4 + (2821/6)n 3 + (917/2)n 2 + (3311/15)n + 42. 2. COMPUTATIONAL DETAILS The aim of this section is describing the efficient method of Klavar for computing Wiener index of graphs. To do this, we assume that G is a connected graph. The edges e = xy and f = uv of G are called to be in the Djokovic
Winkler relation  if dG(x,u) + dG(y,v)
dG(x,v) + dG(y,u). This relation is reflexive and symmet-

© 2012 Bentham Science Publishers

On Wiener Index of One
Heptagonal Nanocone

ric, but not usually transitive. It is well-known that  is transitive for partial cubes and so it partitions the edge set of the graph into equivalence classes, called 
classes. In general, we consider the transitive closure * of  and assume that E1, E2, …, Ek be the *
equivalence classes, *
classes for short. A graph G together with a weight function w: V(G)R is called a weighted graph. Following Klavar and Gutman [27], the weighted Wiener index W(G,w) of a weighted graph (G,w) is defined as W(G,w) = 1/2{u,v}V(G) w(u)w(v)dG(u,v). Clearly, if all the weights are 1 then W(G,w) = W(G). Another concept that we need for describing Klavar’s method is the canonical metric representation of a graph due to Graham and Winkler [30]. The canonical metric representation  of a connected graph G is defined as follows: 1- Let G be a connected graph and F1, …, Fk its *
classes. Define quotient graphs G/Fi, 1  i  k, to be the graph with the connected components of G
Fi, as vertices. Two vertices C

Fig. (2). The *
Classes of L[1].

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and C are adjacent if there exist vertices x  C and y  C  such that xy  Fi. Define  : G1  i  k G/Fi with  : u (1(u), …, k(u)), where i(u) is the connected component of G
Fi that contains u. 3- Let G be an arbitrary connected graphs and  : G  1  i  k G/Fi the canonical metric representation of G. Let (G/Fi,wi) be "natural" weighted graphs: the weight of a vertex of G/Fi is the number of vertices in the corresponding connected component of G
Fi. In [26], Klavar applied above notations to prove the following important algorithm for calculation of Wiener index:

2-

Theorem B. For any connected graph G, W(G) = 1  i  k W(G/Fi,wi). Notice that above theorem is also a particular instance of the cut method. In this case the cuts are the *
classes, and we derive the

182 Current Nanoscience, 2012, Vol. 8, No. 1

Ashrafi and Mohammad
Abadi

Fig. (3). The *
Classes of L[2].

Wiener index of G from the connected components of the graphs G
Fi via the weighted Wiener index. 3. MAIN RESULTS In this section the Wiener index of L[n] = CNC7[n] nanocone is computed by Klavar’s method. In [10,11] another general method

was presented which is useful for computing Wiener index of graphs. Our calculations here show that in the case of nanocones, the Klavar’s method is more efficient. To compute the Wiener index of L[n] = CNC7[n], we first consider some small cases of n. In Figs. (2-5), the canonical representation of L[1] = CNC7[1], L[2] = CNC7[2], L[3] = CNC7[3] and L[4] = CNC7[4] are depicted. In

On Wiener Index of One
Heptagonal Nanocone

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183

Fig. (4). The
*
Classes of L[3].

each case the corresponding weighted graphs are determined. Apply these representations and Theorem B, one can easily from these
*
classes seen that: W(L[1]) = 7[5.23]+7[4.4.1+4.4.2+4.4.3] = 1477, W(L[2]) = 7[7.56+16.47]+7[9.9.1+9.9.2+9.9.3] = 11410,

W(L[3]) = 7[9.103+20.92+33.79]+7[16.16.1+16.16.2+16.16.3] = 48370, W(L[4]) = 7[11.164+24.151+39.136+56.119]+7[25.25.1+ 25.25.2+25.25.3] = 148022.

Ashrafi and Mohammad
Abadi

184 Current Nanoscience, 2012, Vol. 8, No. 1

Fig. (5). The
*
Classes of L[4].

By an inductive argument and considering the
*
classes of one
heptagonal carbon nanocones, one can easily seen that the Wiener index of L[n] can be computed from the following formula:

W(L[n]) = (238/5)n 5 + (238)n 4 + (2821/6)n 3 + (917/2)n 2 + (3311/15)n + 42.

On Wiener Index of One
Heptagonal Nanocone

Current Nanoscience, 2012, Vol. 8, No. 1

4. CONCLUSION In this paper the Wiener index of one – pentagonal carbon nanocone is computed by Klavar’s method. It is shown that the method for one
heptagonal nanocone is efficient. CONFLICT OF INTEREST None.

[13] [14] [15] [16] [17]

ACKNOWLEDGEMENTS None.

[18] [19]

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Received: January 12, 2011

Revised: August 11, 2011

Accepted: August 11, 2011

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