Short communication
Computing Fourth Atom-bond Connectivity Index of V-Phenylenic Nanotubes and Nanotori
Mohammad Reza Farahani
Department of Applied Mathematics of Iran University of Science and Technology (IUST),
Narmak, Tehran 16844, Iran.
* Corresponding author: E-mail: Mr_Farahani@Mathdep.iust.ac.ir
Received: 17-12-2012
Abstract
Among topological descriptors connectivity topological indices are very important and they have a prominent role in chemistry. One of them is atom-bond connectivity (ABC) index defined as ABC(C)= V I— +it'—in which degree of
»--¿«.ill dA
vertex v denoted by dv. Recently, a new version of atom-bond connectivity (ABC4) index was introduced by M. Ghorba-ni et.al in 2010 and is defined as A3C4(G)= £2- where Su	,„,4 andNG(u) = {ve V(G)\uveE(G)}. In this
paper we compute this new topological index for V-Phenylenic Nanotube and Nanotori. Keywords: V-phenylenic, nanotube, nanotori, topological index, Atom bond connectivity index.
1. Introduction
Let G = (V;E) be a simple molecular graph without directed and multiple edges and without loops, the vertex and edge sets of it are represented by V=V(G) and E = E(G), respectively. In chemical graphs, the vertices correspond to the atoms of the molecule, and the edges represent to the chemical bonds. Also, if e is an edge of G, connecting the vertices u and v, then we write e = uv and say »u and v are adjacent«.
Mathematical chemistry is a branch of theoretical chemistry for discussion and prediction of the molecular structure using mathematical methods without necessarily referring to quantum mechanics. Chemical graph theory is a branch of mathematical chemistry which applies graph theory to mathematical modeling of chemical phenome-na.1-3 This theory had an important effect on the development of the chemical sciences.
In mathematical chemistry, numbers encoding certain structural features of organic molecules and derived from the corresponding molecular graph, are called graph invariants or more commonly topological indices.
Among topological descriptors, connectivity indices are very important and they have a prominent role in chemistry. In other words, if G be the connected graph, then
we can introduce many connectivity topological indices for it, by distinct and different definition. A connected graph is a graph such that there is a path between all pairs of vertices. One of the best known and widely used is the connectivity index, introduced in 1975 by Milan Randic 1, who has shown this index to reflect molecular branching and defined as follows:
Z(G)= I	(1)
e=K»s£(G) y«,«,,
where du denotes G degree of vertex u.
One of the important classes of connectivity indices is atom-bond connectivity (ABC) index defined as
Mc^m* I Jr^1'
»semi
(2)
where Qv is some quantity that in a unique manner can be associated with the vertex v of the graph G. The first member of this class was considered by E. Estrada et. al.,5 by setting Qv and Qu to be the degree of a vertex v and u:
ABC,(G)= I IS
The second and third members of this class were introduced by A. Graovac and M. Ghorbani6 and M. R. Fa-rahani,7-9 separately as follow:
¿¡¡cm- I
«.A
ABC,(G)= X J^
iivtEiGt V
+m -2
(4)
(5)
where nu denotes the number of vertices of G whose distances to vertex u are smaller than those to other vertex v of the edge e = uv (nu = {x|xe V(G),d(u,x)<d(x,v)}) and nv is defined analogously. And mu denotes the number of vertices of G whose distances to vertex u are smaller than those to other vertex v of the edge e = uv (mu = {ff e E(G),d(uf)<d(f,v)}) and mv is defined analogously.
Suppose Sv as the summation of degrees of all neighbors of vertex v in G. In other words, Su = 'ZveN^u)dv and NG(u) = {ve V(G)|uv in E(G)}. The fourth mentor of this class was considered by M. Ghorbani et al.10, by setting Qv to be the number Sv of vertex v:
ABC4(G) = £ J
wstlO) V
S, +5,-2 SBSV
(6)
In Refs11-22 some topological indices of V-phenyle-nic nanotube and V-phenylenic nanotori are computed. In this paper, we continue this work to compute the fourth atom-bond connectivity index of molecular graphs related to V-phenylenic nanotube and nanotori. Our notation is standard and mainly taken from Refs.1,2,23,24
2. Main Results and Discussion
The goal of this section is to computing the ABC4 index of V-phenylenic nanotube and nanotori. The novel phenylenic and naphthylenic lattices proposed can be constructed from a square net embedded on the toroidal surface. Phenylenes are polycyclic conjugated molecules, composed of four membered ring (= square) and six-membe-red rings (= hexagons) such that every four membered ring (4-membered cycle) is adjacent to two 6-membered cycles, and no two six-membered rings are mutually adjacent. Each four-membered ring lies between two six-membered rings, and each hexagon is adjacent only two four-membered rings. Because of such structural features phenylenes are very interesting conjugated species.25-30 The rapid development of the experimental study of phenylenes motivated a number of recent theoretical studies of thee conjugated n-electron systems.29
Following M. V. Diudea24 we denote a V-Phenylenic nanotube and V-Phenylenic nanotorus by G=VPHX[m,n] and H = VPHY[m,n], respectively. The general representation of these nano structures are shown in Figure 1 and Fi-
gure 2. For more information and background materials, refer to paper series11-30 again.
Now we have following theorems, immediately.
Theorem 1. Vm,neN, the fourth atom-bond connectivity index of V-Phenylenic Nanotube VPHX[m,n] is equal to
ABC\{VPHX [m,n]) = (4K + +
(7)
Proof. Consider the V-phenylenic nanotube G = VPHX[m,n] with 6mn vertices and 9mn-m edges (Figure 1). In V-phenylenic molecule, there are two partitions V2 = {v e V (G)|dv = 2} and V3 = {v e V (G)\dv = 3} of V(VPHX[m,n]), since the degree of an arbitrary vertex/atom of a molecular graph is equal to 2 or 3. Next, the two partitions of E(G) are E5 = {u,v e V (G)\du = 3&dv = 2} and E6 = {u,v e V (G)\du = dv = 3}.
Also, two adjacent vertices v1,v2 of a vertex veV2 have degree three, then Sv = 2 x 3 = 6 and two edges vv1 and vv2 belong to E5 (and |E5| = 2|V2| = 4m). Also, for all vertices u in first and end row of V-phenylenic nanotube with degree three, N(u) = {v1,v2,v3} such that v1e V2 and v2 v3e V3 (uv1eE5anduv3,uv2eE6), thus Su = 2 x 3 + 2 = 8. Finally, for other vertices Sw = 9, because all other vertices and their edges belong to V3 and E6, respectively. So, the fourth atom-bond connectivity index of VPHX[m,n] (m,n > 1) will be
,„ , /8+9-2	„ , ¡9+
9+9-2 (8)
Thus, the proof is completed.
Theorem 2. Vn,k e N the fourth atom-bond connectivity index of all k-regular graph K with n vertices is equal to n^t*2-').
2it
Proof. Let K be a k-regular graph with n vertices. Then clearly, all vertices and edges of K belong to Vk and
E2k, respectively. For every vertex v of K, S, = It, thus
Example 1. Let Kn be the complete graph on n vertices. Then \E(KJ\ = VMn-l) and implies that ABCJKJ
.4BCa(K„) nptjn-2)
2(n-l)
Figure 1. The Molecular Graph of V-Phenylenic Nanotube VPHX[m,n].
Example 2. Let Cn be the cycle of length n. Then for
every ve V(CJ) d, = 2 Sy = 4, So ABC J C J = " ^
Lemma 1. Let Qn be a Cubic graph such that all n
vertices of Qn have degree 3. Then its fourth atom-bond connectivity index is equal to 2"/3 .
Proof. The proof is clear, by using proof of Theorem 2. Theorem 2. Vm,n > 1, the fourth atom-bond connectivity index of V-Phenylenic Nanotori H = VPHY[m,n] is equal to ABC4(H) = 4 mn.
Proof. The proof is easily, since by considering the V-phenylenic nanotori H = VPHY[m,n] with 6 mn vertices and 9mn edges (Figure 1). We see that this nanotori is a Cubic graph and all vertices belong to V3 and Vv e V(VPHY[m,n]) Sv = 9. This implies that all edges belong
to E6, immediately. Thus Vm,n > 1m,n > 1, we have the following computations.
3. Conclusions
In this report, we study some properties of a new connectivity index of (molecular) graphs that called fourth atom-bond connectivity index. This connectivity index (ABC4) index was proposed by M. Ghorbani et.al recently and was defined as
Figure 2. The Molecular Graph of V-Phenylenic Nanotorus VPHY[m,n].
such that .....is the
summation of degrees of all neighbors of vertex (atom) v in G (NG(u) = {veV(G)\uveE(G}}.). In continue, closed analytical formulas for ABC4 of a physico chemical structure of Phenylenic nanotubes and Nanotorus are given. These nano structures are V-Phenylenic Nanotube VPHX[m,n] and V-Phenylenic Nanotorus VPHY[m,n]. The structures of V-Phenylenic Nanotube and V-Phenyle-nic Nanotorus consist of several C4C6C8 net. A C4C6C8 net is a trivalent decoration made by alternating C4, C6 and C8. Phenylenes are polycyclic conjugated molecules, composed of four-and six-membered rings such that every four membered ring (= square) is adjacent to two six-membered rings (= hexagons). In other words, a composition of four-, six-and eight-membered rings in the structures of VPHX[m,n] and VPHY[m,n] is a C4C6C8 net.
4. Acknowledgments
The author is thankful to Prof. Mircea V. Diudea and Dr. Katalin Kata from Faculty of Chemistry and Chemical Engineering Babes-Bolyai University (Romania) and Prof. Ali Reza Ashrafi from Department of Mathematics of Faculaty of Science of University of Kashan (Iran) for their helpful comments and suggestions.
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Povzetek
Med topološkimi deskriptorji so zaradi pomena v kemiji zelo pomembni topološki indeksi. Mednje sodi indeks atomske povezanosti (ABC), definiran kot /f RC(G)= T" j"—kjer vozel grafa označuje dY. Leta 2010 je uvedel Ghorbani s
»e£(G>l| <W	"	|_
sodelavci konektivnost atom-vez kot indeks (ABC4) in ga definiral kot jjjjjjjjjj jjjlljjjjjjjjj kjer j^Bjjjjjj jjlin
.'.it B °'
Ng(u) = {ve V(G)\uveE(G)}. V tem članku smo izračunali novi topološki indeks za V fenilensko nanocevko in nanozvitek.