THE VERTEX SZEGED INDEX OF TITANIA CARBON NANOTUBES TiO2 (m,n)

THE VERTEX SZEGED INDEX OF TITANIA CARBON NANOTUBES TiO₂ (m,n)

Mohammad Reza Farahani^1*, R. Pradeep Kumar ², M. R. Rajesh Kanna ³ and Shaohui Wang ⁴

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

Department of Mathematics ², the National Institute of Engineering, Mysuru, India

Department of Mathematics ³, Maharani's Science College for Women, Mysore, India

Department of Mathematics ⁴, University of Mississippi, University, MS, USA

ABSTRACT: Let G=(V,E) be a simple connected molecular graph in chemical graph theory, where the vertex set and edge set of G denoted by V(G) and E(G) respectively and its vertices correspond to the atoms and the edges correspond to the bonds. A topological index of a graph G is a numeric quantity related to G which is invariant under automorphisms of G. In this paper, the vertex Szeged index of Titania carbon Nanotubes TiO₂(m,n) is computed.

Molecular graph,

Carbon Nanotubes, Titania Nanotubes, vertex Szeged index

INTRODUCTION: 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 ^1-8.

We first describe some notations which will be kept throughout. Let G be a simple molecular graph without directed and multiple edges and without loops, the vertex and edge-sets of which are represented by V(G) and E(G), respectively.

Suppose G is a connected molecular graph and x, y ∊V(G). The distance d(x,y) between x and y is defined as the length of a minimum path between x and y. Many topological indices there are in mathematical chemistry and several applications of them have been found in physical, chemical and pharmaceutical models and other properties of molecules. A topological index of a graph G is a numeric quantity related to G which is invariant under automorphisms of G. The oldest nontrivial topological index is the Wiener index which was introduced by Chemist Harold Wiener ⁹. The Wiener index is defined as

where d(u,v) be the distance between two vertices u and v.

In 1994, Ivan Gutman defined a new topological index and named it Szeged index (Sz) index and the Szeged index of the graph G is defined as ^{10, 11}.

where n_u(e|G) is the number of vertices of G lying closer to u than v and n_v(e|G) is the number of vertices of G lying closer to v than u. Notice that vertices equidistance from u and v are not taken into account.

The aim of this paper is to compute the vertex Szeged index of Titania carbon Nanotubes TiO₂(m,n). Throughout this paper, our notation is standard. For further study of some applications of Szeged indices in nanotechnology can be finding in the paper series ^12-18.

RESULTS AND DISCUSSION:

In this present section, the vertex Szeged index of Titania carbon Nanotubes TiO₂(m,n). The graph of the Titania Nanotubes TiO₂(m,n) is presented in Fig.1, where m denotes the number of octagons in a column and n denotes the number of octagons in a row of the Titania Nanotubes. We encourage the reader to consult papers ^19-30, for further study and more information of Titania Nanotubes TiO₂.

FIG. 1: THE TITANIA PLANAR NANOTUBES TiO₂(m,n) "m,nÎℕ.

Theorem 1: Let TiO₂(m,n) be the Titania Nanotubes for a non-negative integers m,n. Then vertex Szeged index of TiO₂(m,n) is equal to:

SZ_v(TiO₂(m,n))

Proof. Consider the Titania Planar Nanotubes TiO₂(m,n) for all m,n Îℕ with 12(m+1)(½n)+4(m+1) =6mn+4m+6n+4=2(3n+2)(m+1) vertices/atoms bonds (|V(TiO₂(m,n))|) and 10mn+6m+8n+4 edges/Chemical bonds (|E(TiO₂(m,n))|) where 6 +2+4(m-1) +0+7 +6 +1=2mn+4n+4 vertices have degree two, 2 +2 =2n vertices have degree four, 2(m) =2mn vertices have degree five and there are 3+2 +1+5(m-1)+4(m-1)  +3(m-1)+2 =2mn+4m vertices with degree 3.

Here by using the Cut Method and Orthogonal Cuts of the Titania Nanotubes TiO₂(m,n), we can determine all edge cuts (quasi-orthogonal) of the Titania Nanotubes TiO₂(m,n) in Table 1 and Fig. 1. The edge cut C(e) is an orthogonal cut, such that the set of all edges f∊E(G) are strongly co-distant to e (C(e):={ f∊E(G)|f is co-distant with e}). Also, for further research and study of the cut method and orthogonal cuts in some classes of chemical graphs, see ^{31, 32}. Some applications of the cut method include the Wiener, hyper-Wiener, weighted Wiener, Wiener-type, Szeged indices and classes of chemical graphs such as trees, Benzenoid graphs and phenylenes.

Now by using the Cut Method and finding Orthogonal Cuts, we can compute the quantities of n_u(e|TiO₂(m,n)) and n_v(e|TiO₂(m,n)), ∀e∊E (TiO₂ (m,n)), which are the number of vertices in two sub-graphs TiO₂(m,n)-C(e). In case the Titania Nanotubes TiO₂(m,n) ∀e=uv∊E(TiO₂(m,n)), we denote n_u(e|TiO₂(m,n)) as the number of vertices in the left component of TiO₂(m,n)-C(e) and alternatively n_v(e|TiO₂(m,n)) as the number of vertices in the right component of TiO₂(m,n)-C(e), since all edges in TiO₂(m,n) Nanotubes sheets are oblique or horizontal.

Thus, by according to the structure of the Titania Nanotubes TiO₂(m,n) for all integer numbers m,n>1, we have following results:

For the edge e₁=u₁v₁ that belong to the first square of TiO₂(m,n) Nanotubes (in the first column and row), we see that

n_u1(e₁|TiO₂(m,n))=2(m+1)

and

n_v1(e₂|TiO₂(m,n))=6mn+4m+6n+4-2(m+1)=6mn+2m+6n+2.

For the edge e₂=u₂v_2:

n_u2(e₂|TiO₂(m,n))=3×2(m+1)+1×2(m+1)=8(m+1)

and

n_v2(e₂|TiO₂(m,n))=6mn+4m+6n+4-8(m+1)=6mn-4m+6n-4.

For the edge e_n+1=u _n+1v _n+1:

n_{u n+1}(e _n+1|TiO₂(m,n))=(3n+1)×2(m+1)

and

n_{v n+1} (e _n+1|TiO₂(m,n))=6mn+4m+6n+4-(6mn+6n+2m+2)=2(m+1).

Thus, by a simple induction for i=1,2,…,n; we can see that for the edge e_i=u _iv _i:

n_ui(e _i|TiO₂(m,n))=2(m+1)×(3(i-1)+1)

and

n_vi(e _i|TiO₂(m,n))=6mn+4m+6n+4-(6mi+6i-4m-4)

=6m(n-i)+6(n-i)+8(m+1)

=6(m+1)(n-i)+8(m+1)

=2(m+1)(3(n-i)+4).

Let the edge f₁=u₁v₁∊E(TiO₂(m,n)) be the first oblique edge in the first square of TiO₂(m,n) Nanotubes (in the first column and row), we see that
n_u1(f₁|TiO₂(m,n))=m+1

and

n_v1(f₁|TiO₂(m,n))=6mn+4m+6n+4-(m+1)

=6mn+3m+6n+3

=6n(m+1)+3(m+1)

=(6n+3)(m+1).

For the edge f₂=u₂v_2:

n_u2(f₂|TiO₂(m,n))=(m+1)+3×2(m+1)=7(m+1)

and

n_v2(f₂|TiO₂(m,n))=6mn+4m+6n+4-7(m+1)=6n(m+1) -3 (m+1)=(6n-3)(m+1).

For the edge f_(n+1)=u v_:

n_u(n+1) (f_(n+1)|TiO₂(m,n))=(m+1)+3n×2(m+1)=(6n+1)(m+1)

and

n_v(n+1) (f _(n+1)|TiO₂(m,n))=(6n+4)(m+1)- (6n+1)(m+1)=3(m+1).

FIG. 2: CATEGORIES FOR EDGES OF THE TITANIA CARBON NANOTUBES TiO₂(m,n).

Therefore, by a simple induction for j=1,2,…,n+1; we can see that

For the edge f_j=u _jv _j:

n_uj(f _j|TjO₂(m,n))=3(j-1)×2(m+1)+(m+1)=(m+1)(6j-5)

and

n_vj(f_j|TjO₂(m,n))=(6n+4)(m+1)- (m+1)(6j-5)

=(m+1)(6n-6j+9)

=3(m+1)(2n+3-2j).

Let the edge g₁=u₁v₁∊E(TiO₂(m,n)) be the second oblique edge in the first square of TiO₂(m,n) Nanotubes (in the first column and row), so we have

n_u1(g₁|TiO₂(m,n))=2(m+1)+(m+1)

and

n_v1(g₁|TiO₂(m,n))=(6n+4)(m+1)-3(m+1)=(6n+1)(m+1)

For g₂=u₂v_2:

n_u2(g₂|TiO₂(m,n))=3×2(m+1)+2(m+1)+(m+1)=9(m+1)

and

n_v2(g₂|TiO₂(m,n))=(6n+4)(m+1)-9(m+1)=(6n-5)(m+1)

For g_n+1=u_n+1 v_n+1:

n_un+1 (g_n+1|TiO₂(m,n))=3n×2(m+1)+2(m+1)+(m+1)=(6n+3)(m+1)

and

n_vn+1 (g_n+1|TiO₂(m,n))=(6n+4)(m+1)- (6n+3)(m+1)=(m+1).

And these imply that ∀j=1,2,…,n+1

n_uj(g _j|TjO₂(m,n))=3j×2(m+1)+3(m+1)=(m+1)(6j+3)

and

n_vj(g_j|TjO₂(m,n))=(6n+4)(m+1)- (m+1)(6j+3)=(m+1)(6n-6j+1).

Finally, let h₁=u₁v₁ & l₂=u₂v₂∊E(TiO₂(m,n)) be the first and second oblique edges in the second square of the first row (or the first square in the second column) of TiO₂(m,n) Nanotubes, then

n_u1(h₁|TiO₂(m,n))=2×2(m+1)

and                                                         n_u2(l₂|TiO₂(m,n))=3×2(m+1)

n_v1(h₁|TiO₂(m,n))=(6n+4)(m+1)- 2(m+1)=(6n+2)(m+1)

and                                                         n_v2(l₂|TiO₂(m,n))=(6n+1)(m+1)

And by a simple induction on ∀i=1,2,…,n; for the edges h_i=u _iv _i and l_i=a _ib _i we have

n_ui(h_i|TiO₂(m,n))=3(i-1)×2(m+1)+2×2(m+1)=2(m+1)(3i-1)

and

n_{v i} (h_i|TiO₂(m,n))=(6n+4)(m+1)- 2(m+1)(3i-1)

=(m+1)(6n-6i+6)

=6(m+1)(n+1-i).

n_ai(l_i|TiO₂(m,n))=3(i-1)×2(m+1)+3×2(m+1)=6i(m+1)

and

n_bi (l_i|TiO₂(m,n))=(6n+4)(m+1)- 6i(m+1)

=(m+1)(6n-6i+4)

=3(m+1)(3n+2-2i).

On the other hands, by according to Fig. 2, we can see that the size of all orthogonal cuts for these edge categories in the Titania Nanotubes TiO₂(m,n) are equal to (∀i=1,2,…,n+1):

|C(e_i)|=|C(h_i)|=|C(l_i)|=2(m+1)

And

|C(f_i)|=|C(g_i)|=2m+1.

 Here by above mentions results of n_u(e|TiO₂(m,n)) and n_v(e|TiO₂(m,n)) (∀e∊E(TiO₂(m,n)), m,nÎℕ-{1}) and according to Fig. 2, we will have following computations for the vertex PI, Szeged indices of Titania Carbon Nanotubes TiO₂(m,n).

Sz_v(TiO₂(m,n))=

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    How to cite this article:

    Mod. Farahani R, Kumar RP, Rajesh Kanna MR and Wang S: The Vertex Szeged Index of Titania Carbon Nanotubes TiO₂ (m,n). Int J Pharm Sci Res 2016; 7(9): 3734-41.doi: 10.13040/IJPSR.0975-8232.7(9).3734-41.

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