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

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**THE VERTEX SZEGED INDEX OF TITANIA CARBON NANOTUBES TiO_{2 }(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 *TiO _{2}(m,n)* is computed.

Keywords: |

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 *TiO _{2}(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 *TiO _{2}(m,n).* The graph of the Titania Nanotubes

*TiO*is presented in

_{2}(m,n)**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

*TiO*

_{2}.**FIG. 1:** **THE TITANIA PLANAR NANOTUBES TiO_{2}(m,n) **

*"*

*m,n*

*Î*

*ℕ***.**

**Theorem 1:** Let *TiO _{2}(m,n)* be the Titania Nanotubes for a non-negative integers

*m,n*. Then vertex Szeged index of

*TiO*is equal to:

_{2}(m,n)*SZ _{v}(*

*TiO*

_{2}(m,n)*)*

** Proof.** Consider the Titania Planar Nanotubes

*TiO*for all

_{2}(m,n)*m,n*

*Î*

*ℕ*with

*12(m+1)(½n)+4(m+1) =6mn+4m+6n+4*

*=2(3n+2)(m+1)*vertices/atoms bonds (|V(

*TiO*)|) and 10

_{2}(m,n)*mn+6m+8n+4*edges/Chemical bonds (|E(

*TiO*)|) where 6

_{2}(m,n)*+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 *TiO _{2}(m,n)*

*,*we can determine all edge cuts (quasi-orthogonal) of the Titania Nanotubes

*TiO*in

_{2}(m,n)**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|*

*TiO*) and

_{2}(m,n)*n*

_{v}(e|*TiO*), ∀

_{2}(m,n)*e*

*∊*

*E (TiO*)

_{2 }(m,n)*,*which are the number of vertices in two sub-graphs

*TiO*

_{2}(m,n)*-C(e).*In case the Titania Nanotubes

*TiO*∀

_{2}(m,n)*e=uv*

*∊*

*E(TiO*), we denote

_{2}(m,n)*n*

_{u}(e|*TiO*) as the number of vertices in the left component of

_{2}(m,n)*TiO*

_{2}(m,n)*-C(e)*and alternatively

*n*

_{v}(e|*TiO*) as the number of vertices in the right component of

_{2}(m,n)*TiO*

_{2}(m,n)*-C(e)*, since all edges in

*TiO*Nanotubes sheets are oblique or horizontal.

_{2}(m,n)Thus, by according to the structure of the Titania Nanotubes *TiO _{2}(m,n)* for all integer numbers

*m,n>1,*we have following results:

For the edge *e _{1}=u_{1}v_{1} *that belong to the first square of

*TiO*Nanotubes (in the first column and row), we see that

_{2}(m,n)*n _{u1}(*

*e*)=

_{1}|TiO_{2}(m,n)*2(m+1)*

and

*n _{v1}(e_{2}|*

*TiO*)=

_{2}(m,n)*6mn+4m+6n+4-*

*2(m+1)=6mn+2m+6n+2.*

For the edge *e _{2}=u_{2}v_{2:}*

*n _{u2}(*

*e*

_{2}*|*

*TiO*)=3×

_{2}(m,n)*2(m+1)+1×2(m+1)*=

*8(m+1)*

and

*n _{v2}(e_{2}|*

*TiO*)=

_{2}(m,n)*6mn+4m+6n+4-*

*8(m+1)=6mn-4m+6n-4.*

For the edge *e _{n+1}=u_{ n+1}v_{ n+1:}*

*n _{u}*

_{ n+1}*(*

*e*

_{ n+1}*|*

*TiO*)=(3n+1)×

_{2}(m,n)*2(m+1)*

and

*n _{v}*

_{ n+1}*(e*

_{ n+1}*|*

*TiO*)=

_{2}(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_{ i}v_{ i:}*

*n _{u}*

_{i}*(*

*e*

_{ i}*|*

*TiO*

_{2}(m,n))*=2(m+1)×(3(i-1)+1)*

and

*n _{v}*

_{i}*(e*

_{ i}*|*

*TiO*

_{2}(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 f_{1}=u_{1}v_{1}*∊**E(TiO _{2}(m,n)*) be the first oblique edge in the first square of

*TiO*Nanotubes (in the first column and row), we see that

_{2}(m,n)

*n*

_{u1}(*f*)=

_{1}|TiO_{2}(m,n)*m+1*

and

*n _{v1}(f_{1}|*

*TiO*)=

_{2}(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 *f _{2}=u_{2}v_{2:}*

*n _{u2}(*

*f*

_{2}*|*

*TiO*)=

_{2}(m,n)*(m+1)+3×2(m+1)*=

*7(m+1)*

and

*n _{v2}(f_{2}|*

*TiO*)=

_{2}(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)}*|*

*TiO*)=

_{2}(m,n)*(m+1)+3n×2(m+1)*=

*(6n+1)(m+1)*

and

*n _{v}*

_{(n+1) }*(f*

_{ (n+1)}*|*

*TiO*)=

_{2}(m,n)*(6n+4)(m+1)*

*-*

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

**FIG. 2:**** CATEGORIES FOR EDGES OF THE TITANIA CARBON NANOTUBES TiO_{2}(m,n).**

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

For the edge *f _{j}=u_{ j}v_{ j:}*

*n _{u}*

_{j}*(*

*f*

_{ j}*|*

*TjO*

_{2}(m,n))*=3(j-1)×2(m+1)+(m+1)=(m+1)(6j-5)*

and

*n _{v}*

_{j}*(f*

_{j}*|*

*TjO*

_{2}(m,n))*=(6n+4)(m+1)*

*-*

*(m+1)(6j-5)*

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

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

Let the edge g_{1}=u_{1}v_{1}*∊**E(TiO _{2}(m,n)*) be the second oblique edge in the first square of

*TiO*Nanotubes (in the first column and row), so we have

_{2}(m,n)*n _{u1}(*

*g*)=

_{1}|TiO_{2}(m,n)*2(m+1)*+

*(m+1)*

and

*n _{v1}(g_{1}|*

*TiO*)=

_{2}(m,n)*(6n+4)(m+1)*

*-*

*3(m+1)*=

*(6n+1)(m+1)*

For g_{2}*=u _{2}v_{2:}*

*n _{u2}(*

*g*

_{2}*|*

*TiO*)=

_{2}(m,n)*3×2(m+1)*+

*2(m+1)*+

*(m+1)*=

*9(m+1)*

and

*n _{v2}(g_{2}|*

*TiO*)=

_{2}(m,n)*(6n+4)(m+1)*

*-*

*9(m+1)*=

*(6n-5)(m+1)*

For g_{n+1}*=u _{n+1 }v_{n+1:}*

*n _{u}*

_{n+1 }*(*

*g*

_{n+1}*|*

*TiO*)

_{2}(m,n)*=3n×2(m+1)*+

*2(m+1)*+

*(m+1)*=

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

and

*n _{v}*

_{n+1 }*(g*

_{n+1}*|*

*TiO*)=

_{2}(m,n)*(6n+4)(m+1)*

*-*

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

*(m+1).*

And these imply that ∀*j=1,2,…,n+1*

*n _{u}*

_{j}*(*

*g*

_{ j}*|*

*TjO*)=3j×

_{2}(m,n)*2(m+1)*+

*3(m+1)=(m+1)(6j+3)*

and

*n _{v}*

_{j}*(g*

_{j}*|*

*TjO*)=

_{2}(m,n)*(6n+4)(m+1)*

*-*

*(m+1)(6j+3)=(m+1)(6n-6j+1).*

Finally, let *h _{1}=u_{1}v_{1 }& l_{2}=u_{2}v_{2}*

*∊*

*E(TiO*be the first and second oblique edges in the second square of the first row (or the first square in the second column) of

_{2}(m,n))*TiO*Nanotubes, then

_{2}(m,n)*n _{u1}(*

*h*)=

_{1}|TiO_{2}(m,n)*2×2(m+1)*

and *n _{u2}(*

*l*)=

_{2}|TiO_{2}(m,n)*3×2(m+1)*

*n _{v1}(h_{1}|*

*TiO*)=

_{2}(m,n)*(6n+4)(m+1)*

*-*

*2(m+1)*=

*(6n+2)(m+1)*

and *n _{v2}(l_{2}|*

*TiO*)=

_{2}(m,n)*(6n+1)(m+1)*

And by a simple induction on ∀*i=1,2,…,n; *for the edges *h _{i}=u_{ i}v_{ i }*and

*l*we have

_{i}=a_{ i}b_{ i}*n _{u}*

_{i}*(*

*h*

_{i}*|*

*TiO*

_{2}(m,n))*=3(i-1)×2(m+1)+2×2(m+1)=2(m+1)(3i-1)*

and

*n _{v}*

_{ i}*(h*

_{i}*|*

*TiO*

_{2}(m,n))*=(6n+4)(m+1)*

*-*

*2(m+1)(3i-1)*

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

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

*n _{a}*

_{i}*(*

*l*

_{i}*|*

*TiO*

_{2}(m,n))*=3(i-1)×2(m+1)+3×2(m+1)=6i(m+1)*

and

*n _{b}*

_{i}*(l*

_{i}*|*

*TiO*

_{2}(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* TiO _{2}(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|*

*TiO*and

_{2}(m,n))*n*

_{v}(e|*TiO*

_{2}(m,n))*(*

*∀*

*e*

*∊*

*E(TiO*

_{2}(m,n)), m,n*Î*

*ℕ*

*-{1}*

*)*and according to

**Fig. 2**, we will have following computations for the vertex PI, Szeged indices of Titania Carbon Nanotubes

*TiO*

_{2}(m,n).*Sz _{v}(*

*TiO*)

_{2}(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

*TiO*. Int J Pharm Sci Res 2016; 7(9): 3734-41.doi: 10.13040/IJPSR.0975-8232.7(9).3734-41._{2 }(m,n)

All © 2013 are reserved by International Journal of Pharmaceutical Sciences and Research. This Journal licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

# Article Information

21

3734-41

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English

IJPSR

Mohammad Reza Farahani, R. Pradeep Kumar , M. R. Rajesh Kanna and Shaohui Wang

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

mrfarahani88@gmail.com

18 April, 2016

20 May, 2016

21 July, 2016

10.13040/IJPSR.0975-8232.7(9).3734-41

01 September 2016