#### 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)*=*

**REFERENCES:**

- B. West. An Introduction to Graph Theory. Prentice-Hall. (1996).
- Todeschini and V. Consonni, Handbook of Molecular Descriptors,
*Wiley, Weinheim*, (2000). - Trinajstić, Chemical Graph Theory, CRC Press,
*Boca Raton*, FL. (1992). - Wang, B. Wei, Multiplicative Zagreb indices of Cacti, Discrete Math. Algorithms. Appl. . 2016, In press.
- Wang, S. Wang, B. Wei, Cacti with Extremal PI Index, Transactions on Combinatorics. 5(4), (2016), 1-8.
- Wang, B. Wei, Multiplicative Zagreb indices of k-trees, Discrete Appl. Math., 180 (2015), 168-175.
- Wang, J.B. Liu, C. Wang, S. Hayat, Further results on computation of topological indices of certain networks. 2016, In press.
- Wang, M.R. Farahani, M.R. Kanna, M.K. Jamil, R.P. Kumar, The Wiener Index and the Hosoya Polynomial of the Jahangir Graphs, Applied and Computational Mathematics. 2016, In press.
- Wiener, Structural determination of paraffin boiling points,
*J. Am. Chem. Soc.*69, 17, 1947. - Gutman, A formula for the Wiener number of trees and its extension to graphs containing cycles,
*Graph Theory Notes of New York*27 (1994) 9–15. - Gutman, P. V. Khadikar, P. V. Rajput, S. Karmarkar, The Szeged index of polyacenes,
*J. Serb. Chem. Soc.*60 (1995) 759–764.

- R. Ashrafi, B. Manoochehrian, H. Yousefi-Azari, On Szeged polynomial of a graph,
*Bull. Iranian Math. Soc*. 33 (2006) 37–46. - Iranmanesh, A. R. Ashrafi, Balaban index of an armchair polyhex TUC4C8 (R) and TUC4C8 (S) nanotorus,
*J. Comput. Theor. Nanosci.*4 (2007) 514–517. - Iranmanesh, B. Soleimani, PI index of TUC4C8(R) nanotubes,
*MATCH Commun. Math. Comput. Chem.*57 (2007) 251–262. - Ghorbani and M. Jalali. The Vertex PI, Szeged and Omega Polynomials of Carbon Nanocones CNC4[n]. MATCH Commun. Math. Comput. Chem. 62 (2009) 353-362.
- R. Farahani, The Application of Cut Method to Computing the Edge Version of Szeged Index of Molecular Graphs. Pacific Journal of Applied Mathematics. 6(4), 2014, 249-258.
- R. Farahani, Computing Edge-PI index and Vertex-PI index of Circumcoronene series of Benzenoid
*H*by use of Cut Method. Int. J. Mathematical Modeling and Applied Computing. 1(6), September (2013), 41-50._{k} - R. Farahani, M.R. Rajesh Kanna and Wei Gao. The Edge-Szeged index of the Polycyclic Aromatic Hydrocarbons
*PAH*. Asian Academic Research Journal of Multidisciplinary 2(7), 2015, 136-142._{k} - Ramazani, M. Farahmandjou, T. P. Firoozabadi,
*Effect of Nitric acid on particle morphology of the TiO*Int. J. Nanosci. Nanotechnol. 11(1) (2015) 59-62._{2}, - A. Evarestoy, Y. F. Zhukovskii, A. V. Bandura, S. Piskunov (2011), Symmetry and models of single-walled TiO2 Nanotubes with rectangular morphology Open Physics. 9(2), 492-501. DOI: 10.2478/s11534-010-0095-8.

- A. Evarestov, Yu. F. Zhukovskii, A. V. Bandura, S. Piskunov, M. V. Losev. Symmetry and Models of Double-Wall BN and TiO 2 Nanotubes with Hexagonal Morphology The Journal of Physical Chemistry, 2011, 115 (29),14067- 14076.
- A. Evarestov, Yu. F. Zhukovskii, A. V. Bandura, S. Piskunov. Symmetry and Models of Single-Wall BN and
*TiO*Nanotubes with Hexagonal Morphology. The Journal of Physical Chemistry, 2010, 114 (49), 21061–21069._{2} - Imran, S. Hayat, M.Y.H. Mailk,
*Appl. Math. Comput*. 2014,*244,*936–951. - A. Malik, M. Imran,
*On multiple Zagreb indices of TiO*Acta Chem. Slov. 62 (2015) 973-976._{2}Nanotubes, - R. Farahani, M.K. Jamil, M. Imran,
*Vertex PI*Applied Mathematics and Nonlinear Sciences, 1(1) (2016) 170-176._{v}topological index of Titania Nanotubes, - Gao, M.R. Farahani, M.K. Jamil, M. Imran. Certain topological indices of Titania TiO2(m,n). Journal of Computational and Theoretical Nanoscience. 2016, In press.
- Yan, Y. Li, M.R. Farahani, M. Imran, M.R. Rajesh Kanna. Sadhana and Pi polynomials and their indices of an infinite class of the Titania Nanotubes TiO2(m,n). Journal of Computational and Theoretical Nanoscience. 2016, In press.
- Li, L. Yan, M.R. Farahani, M. Imran, M.K. Jamil. Computing the Theta Polynomial Q(G,x) and the Theta Index Q(G) of Titania Nanotubes TiO2(m,n). Journal of Computational and Theoretical Nanoscience. 2016, In press.
- Yan, Y. Li, M.R. Farahani, M.K. Jamil. The Edge-Szeged index of the Titania Nanotubes TiO2(m,n). International Journal of Biology, Pharmacy and Allied Sciences. 2016, In press.
- R. Farahani, M.R. Kanna, R.P. Kumar, M.K. Jamil. Computing Edge Co-Padmakar-Ivan Index of Titania TiO2(m,n). Journal of Environmental Science, Computer Science and Engineering & Technology. 2016, 5(3), 285-295.
- Klavžar. A Bird's Eye View of the Cut Method and A Survey of Its Applications In Chemical Graph Theory.
*MATCH Commun. Math. Comput. Chem.,*2008, 60, 255-274. - E. John, P.V. Khadikar and J. Singh. A method of computing the PI index of Benzenoid hydrocarbons using orthogonal cuts.
*J. Math. Chem*., 2007,*42(1)*, 27-45.

**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

649

724

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