1. Introduction

Let G be a simple connected graph and e=uv be an edge of G in graph theory. A graph is a collection of points and lines connecting a subset of them. The points and lines of a graph also called vertices and edges of the graph, respectively. Also in chemical graph theory, the vertices and edges of a graph also correspond to the atoms and bonds of the molecular graph, respectively. If e is an edge/bond of G, connecting the vertices/atoms u and v, then we write e=uv and say "u and v are adjacent".

Chemical graph theory is an important branch of graph theory and Mathematical chemistry, which applies graph theory to mathematical modeling of chemical phenomena [1-6]. A chemical topological index is a numeric quantity from the structural graph of a molecule and is invariant on the automorphism of the graph. One of the important topological connectivity index in chemical graph theory is the Geometric-Arithmetic index GA considered by D. Vukičević and B. Furtula [7] as

where d_u and d_v are the degrees of the vertices u and v, respectively.

Another important connectivity index is the Atom-Bond Connectivity index ABC and was introduced by B. Furtula et al [8] and is defined as follows:

For a comprehensive survey of the mathematical properties and chemical properties of these indices see papers series and books [9-33].

In this paper, we focus on the Atom-Bond Connectivity index ABC and the Geometric-Arithmetic index GA and compute some results about these connectivity topological indices for two kind of Nano-structures “V-Phenylenic Nanotubes G=VPHX[m,n] and V-Phenylenic Nanotorus H=VPHY[m,n]”.

2. Main Results

The goal of this section is to compute the Atom-Bond Connectivity ABC and the Geometric-Arithmetic GA indices of two carbon Nano-structures. The structure of these two carbon Nano-structures “V-Phenylenic Nanotubes and Nanotorus” are consist of cycles with length four (C₄), six (C₆) and eight (C₈). In other words, in the structure of these nanotubes a C₄C₆C₈ net is a trivalent decoration made by alternating C_4, C₆ and C₈. These can cover either a cylinder or a torus. In Figure 1 and Figure 2, a 2-Dimensional lattice of V-Phenylenic Nanotubes G=VPHX[m,n] and V-Phenylenic Nanotorus H=VPHY[m,n] are shown (∀m,n>1).

Let we denote the number of hexagon in the first row of the 2D-lattice of V-Phenylenic G and H by m and alternatively we denote the number of hexagon in the first column by n. From Figures 1 and 2, we see that in each period, there are 6 vertices and we have mn repetition. Thus the number of vertices in V-Phenylenic G and H is equal to |V(VPHX[m,n])|=|V(VPHY[m,n])|=6mn. Since there are m vertices with degree two in the first row and m vertices with degree two in the last row, too and other vertices have degree three, then the number of edges in V-Phenylenic Nanotubes G=VPHX[m,n] (∀m,n>1) is equal to

On other hands, since all vertices in V-Phenylenic Nanotorus H=VPHY[m,n] (∀m,n>1), have degree three, thus,

|E(VPHY[m,n])|=½×3(6mn)=9mn.

For a review, historical details and further bibliography see references [33-42]. Now we compute the ABC and GA indices of the V-Phenylenic Nanotubes and V-Phenylenic Nanotorus by G=VPHX[m,n] and H=VPHY[m,n] (∀m, n∈ℕ-{1}),

Theorem 1. Let G be the V-Phenylenic Nanotubes VPHX[m,n] and H be the V-Phenylenic Nanotorus VPHY[m,n] for every m,n∈ℕ-{1}, then

• Geometric-Arithmetic index of G is equal to

• Atom-Bond Connectivity index of G is equal to

• Geometric-Arithmetic index of H is equal to

GA(H)=|E(VPHY[m,n])|=9mn

• Atom-Bond Connectivity index of H is equal to

ABC(H)=|V(VPHY[m,n])|=6mn

Before prove the main results in Theorem 1, let us introduce following definition.

Definition 1. [32] For a connected graph G=(V(G),E(G)) with the minimum and maximum of degrees δ=Min{d_v|v∈V(G)} and Δ=Max{d_v|v∈V(G)}, respectively, there exist vertex.atom and edge/bond partitions as follow

∀k: δ≤k≤Δ, V_k={v∈V(G)| d_v=k}

∀i: 2δ≤i≤2Δ, E_i={e=uv∈E(G)|d_u+d_v=i}

∀j: δ²≤j≤Δ², E_j*={uv∈E(G)|d_u×d_v=j}.

Thus, the edge set of V-Phenylenic Nanotubes G=VPHX[m,n] can be dividing to two partitions, e.g. E₅ and E₆, as follow:

• For every e=uv belong to E₆, d_u=d_v=3 or

E₅=E₆*={uv∈E(VPHX[m,n])|d_u+d_v=5 & d_u×d_v=6}

• For every e=uv belong to E₅, then d_u=2 and d_v=3 or

E₆=E₉*={uv∈E(VPHX[m,n])|d_u+d_v=6 & d_u×d_v=9}.

Also, all member in edge set of V-Phenylenic Nanotorus VPHY[m,n] exist in a following partition E₆ or E₉* since all vertices/atoms have degree three.

E₆=E₉*={uv∈E(VPHY[m,n])|d_u+d_v=6& d_u×d_v=9}

=E(VPHY[m,n])

Figure 1. The 2-Dimensional lattice of V-Phenylenic Nanotubes G=VPHX[m,n] and V-Phenylenic Nanotorus H=VPHY[m,n]

Proof. Consider V-Phenylenic Nanotubes G=VPHX[m,n] with 6mn vertices and 9mn-m edges. By according to the 2-Dimensional lattice of G=VPHX[m,n] in Figure1, we mark all edges from E₅ or E₆* by yellow color and all member from E₆ or E₉* by blue color then we see that in V-Phenylenic Nanotubes G=VPHX[m,n] |E₅|=|E₆*| = 2m+2m and also |E₆|=|E₉*|=9mn-5m.

Now, we have following computations for the Atom-Bond Connectivity index ABC and the Geometric-Arithmetic index GA of V-Phenylenic Nanotubes VPHX[m,n] as follow:

Finally, Consider V-Phenylenic Nanotorus H=VPHY[m,n] with 6mn vertices and 9mn edges. And by according to the 2-Dimensional lattice of H in Figure1, we see that all member of single edge partition E₆ (or E₉*) are mwrked by black color, such that in H=VPHY[m,n] (∀m,n∈ℕ-{1}), |E₆|=|E₉*|=9mn=|E(VPHY[m,n])|. Therefore the Atom-Bond Connectivity index ABC and the Geometric-Arithmetic index GA of V-Phenylenic Nanotorus H=VPHY[m,n] are equal to:

And these completed the proof of theorem.

ACKNOWLEDGEMENTS

Authors are also thankful to the University Grants Commission, Government of India, for the financial support under the Grant MRP(S)-0535/13-14/KAMY004/ UGC-SWRO.