FIRST AND SECOND ZAGREB INDICES AND POLYNOMIALS OF V-PHENYLENIC NANOTUBES AND NANOTORI

FIRST AND SECOND ZAGREB INDICES AND POLYNOMIALS OF V-PHENYLENIC NANOTUBES AND NANOTORI

Mohammad Reza Farahani ^*1, R. Pradeep Kumar ² and M.R. Rajesh Kanna ³

Department of Applied Mathematics ¹, Iran University of Science and Technology (IUST) Narmak, Tehran, 16844, Iran.

Department of Mathematics ², The National Institute of Engineering, Mysuru 570008, Karnataka, India.

Department of Mathematics ³, Maharani's Science College for Women, Mysore- 570005, Karnataka, India.

ABSTRACT: Let G be a simple molecular graph with vertex and edge sets V(G) and E(G), respectively. In 1972, Gutman and Trinajstić introduced the First and Second Zagreb topological indices of molecular graphs. These topological indices are useful in the study of anti-inflammatory activities of certain chemical instances, and in elsewhere. In this present study, compute the First Zagreb index Zg₁(G)=  and Second Zagreb index Zg₂(G)=  and the First Zagreb polynomial Zg₁(G,x)=  and the Second Zagreb polynomial Zg₂(G,x)=  of V-Phenylenic Nanotubes and Nanotori.

V-Phenylenic Nanotubes and Nanotori; Molecular graph; Zagreb index; Zagreb Polynomial.

INTRODUCTION: Let G=(V,E) be a simple connected graph. In chemical graphs, the vertices of the graph correspond to the atoms of molecules while the edges represent chemical bonds (V(G) and E(G) are the vertex and edge set of G). In mathematics chemistry, there exist many topological indices and connectivity indices in graph theory. A topological index is a numeric quantity from the structural graph of a molecule which is invariant under graph automorphisms. As usual, the distance between the vertices u and v of G is denoted by d(u,v) and it is defined as the number of edges in a minimal path connecting vertices u and v ^1-5.

One of the oldest graph invariants is the Wiener index W(G), introduced by the chemist Harold Wiener [5] in 1947. It is defined as the sum of topological distances d(u,v) between any two atoms in the molecular graph G

W(G)=½  

An important topological index introduced more than forty years ago by I. Gutman and N. Trinajstić is the Zagreb index Zg₁(G) (or, more precisely, the First Zagreb index, because there exists also a Second Zagreb index, Zg₂(G) ^6-8). First Zagreb index Zg₁(G) of the graph G is defined as the sum of the squares of the degrees of all vertices of G. the First and Second topological indices are defined as:

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

The First Zagreb polynomial Zg₁(G,x) and the Second Zagreb polynomial Zg₂(G,x) for these topological indices and are defined as:

In Refs ^6-34 these topological indices and their polynomials of some Nanotubes and Nanotorus are computed. In this paper, we focus on the First and Second Zagreb indices and their topological polynomials of V-Phenylenic Nanotubes and Nanotori.

Results and Discussion: The goal of this section is computing a closed formula of the First Zagreb index Zg₁(G)=  and Second Zagreb index Zg₂(G)=  and the First Zagreb polynomial Zg₁(G,x)=  and the Second Zagreb polynomial Zg₂(G,x)=  for V-Phenylenic Nanotubes and Nanotori. Following M.V. Diudea 35 we denote a V-Phenylenic Nanotubes and V-Phenylenic Nanotorus by G=VPHX[m,n] and H=VPHY[m,n], respectively (are shown in Fig. 1 and Fig. 2).

Molecular graphs V-Phenylenic Nanotubes VPHX[m,n] and V-Phenylenic Nanotorus VPHY[m,n] are two families of Nano-structures that their structure are consist of cycles with length four, six and eight by different compound. The novel Phenylenic and Naphthylenic lattices proposed can be constructed from a square net embedded on the toroidal surface. For a review, historical details and further bibliography see the references ^36-44. Before presenting the main results, let us introduce following definition.

Definition 1: ^{20, 21} Let G=(V;E) be a simple connected graph and d_v is degree of vertex vÎV(G) (Obviously 1≤ δ ≤d_v ≤ Δ≤n-1, such that δ=Min{ d_v|vÎ V(G)} and Δ=Max{d_v|vÎV(G)}). We divide the edge set E(G) and the vertex set V(G) of graph G to several partitions, as follow:

"k: δ≤k≤Δ, V_k={vÎV(G)| d_v=k}

"i: 2δ≤i≤2Δ, E_i={e=uvÎE(G)|d_u+d_v=i}

"j: δ²≤j≤Δ², E_j*={uvÎE(G)|d_u×d_v=j}.

FIG. 1: 2-DIMENSIONAL LATTICE OF THE V-PHENYLENIC NANOTUBES VPHX [m,n].

Theorem 1. Let G be the V-Phenylenic Nanotubes VPHX[m,n] ("m,nÎℕ). Then:

The First Zagreb polynomial of G is equal to

Zg₁(VPHX[m,n], x)=(9mn-5m)x ⁶+(4m)x ⁵

So the First Zagreb index of G is Zg₁(VPHX[m,n])=54mn-10m

The Second Zagreb polynomial of G is equal to

Zg₂(VPHX[m,n], x)=(9mn-5m)x ⁹+(4m)x ⁶

So the Second Zagreb index of G is Zg₂(VPHX[m,n])=81mn+3m

Proof: "m,nÎℕ, consider Nanotubes G=VPHX[m,n], where m and n be the number of hexagon in the first row and column in this Nanotubes. Thus the number of vertices in this Nanotubes is equal to |V(VPHX[m,n])|=6mn ("m,nÎℕ) Since |V₂|=m+m and |V₃|=6mn-2m and the number of edges of G=VPHX[m,n] is equal to

|E(VPHX[m,n])|= =9mn-m.

Now by using the structure of V-Phenylenic Nanotubes VPHX[m,n] in Fig. 1, we mark the edges of E₅, E₆* by red color and the edges of E₆, E₉* by black color.

Therefore, we have the number of 2m+2m edges in the edge partition E₅ (or E₆*) and 9mn-5m members in the edge partition E₆ (or E₉*) of G=VPHX[m,n], respectively.

Thus, by according to Definition 1 and using the definitions of Zagreb polynomials, one can see that

Finally, the First Zagreb polynomial of G=VPHX[m,n] is equal to

Zg₁(VPHX[m,n], x)=(9mn-5m)x ⁶+(4m)x ⁵ 

and the Second Zagreb polynomial of G=VPHX[m,n] is equal to

Zg₂(VPHX[m,n], x)=(9mn-5m)x ⁹+(4m)x ⁶

Now, it is easy to see that

Zg₁(VPHX[m,n])= =(9mn-5m) ´6+(4m) ´5=54mn-10m

And

Zg₂(VPHX[m,n])= =(9mn-5m) ´9+(4m) ´6=81mn+3m

Here, we complete the proof of Theorem 1.

Lemma 1. Let G be the V-Phenylenic Nanotorus H=VPHY[m,n] ("m,nÎℕ). Then:

The First Zagreb polynomial and its index of H are equal to

Zg₁(VPHY[m,n], x)=(9mn)x ⁶

Zg₁(VPHY[m,n])=54mn

The Second Zagreb polynomial and its index of H are equal to

Zg₂(VPHY[m,n], x)=(9mn)x ⁹

Zg₂(VPHY[m,n])=81mn

Proof: Consider V-Phenylenic Nanotori H=VPHY[m,n] ("m,nÎℕ), where m and n be the number of hexagon in the first row and column in H. From the structure of this V-Phenylenic Nanotori in Fig. 2, one can see that this Nanotorus is a member of Cubic graph families and all vertices have degree three. Therefore, H=VPHY[m,n] has 6mn vertices with degree three and alternatively number of edge in H is |E(VPHY[m,n])|=9mn.

Now, by according to definitions of Zagreb polynomials, we will have:

The First and Second Zagreb polynomials of V-Phenylenic Nanotori H=VPHY[m,n] are equal to

Zg₁(VPHY[m,n], x)=(9mn)x ⁶

Zg₂(VPHY[m,n], x)=(9mn)x ⁹

And obviously, the First and Second Zagreb indices of VPHY[m,n] are equal to

Zg₁(VPHY[m,n])=(9mn) ´6=54mn

And

Zg₂(VPHY[m,n])=(9mn) ´9=81mn.

FIG. 2: ⁴² 2-DIMENSIONAL LATTICE OF THE V-PHENYLENIC NANOTORI H=VPHY [m,n].

CONCLUSIONS: In this report, we study some properties of some of oldest topological indices and polynomials of (molecular) graphs that called the First and Second Zagreb topological indices,  First and Second Zagreb polynomials. In continue, closed analytical formulas for First and Second Zagreb topological indices 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-Phenylenic 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.

ACKNOWLEDGMENTS: The authors are thankful to Prof. Mircea V. Diudea from Faculty of Chemistry and Chemical Engineering Babes-Bolyai University (Romania) for their helpful comments and suggestions.

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    How to cite this article:

    Farahani MR, Kumar RP and Kanna MRR: First and second zagreb indices and polynomials of V-phenylenic nanotubes and nanotori. Int J Pharm Sci Res 2017; 8(1): 330-33.doi: 10.13040/IJPSR.0975-8232.8(1).330-33.

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