#### THE EDGE-PADMAKAR-IVAN INDEX OF THE TITANIA NANOTUBES TiO2(m,n)

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**THE EDGE-PADMAKAR-IVAN INDEX OF THE TITANIA NANOTUBES TiO_{2 }(m,n)**

Jialin Zheng ^{1}, Mohammad R. Farahani ^{3}, Muhammad K. Jamil ^{4 }Muhammad K. Siddiqui^{ 5 }and Xiujun Zhang^{* 1, 2}

School of Information Science and Engineering ^{1}, Key Laboratory of Pattern Recognition and Intelligent Information Processing Institutions of Higher Education of Sichuan Province ^{2}, Chengdu University, Chengdu, 610106, China.

Department of Applied Mathematics ^{3}, Iran University of Science and Technology (IUST), Narmak, Tehran 16844, Iran.

Department of Mathematics ^{4}, Riphah Institute of Computing and Applied Sciences (RICAS), Riphah International University, Lahore, Pakistan.

Department of Mathematics ^{5}, Comsats Institute of Information Technology, Sahiwal, Pakistan.

**ABSTRACT:** Let *G(V,E)* be a simple connected graph. For an edge *e=uv*, *m _{u}(e)* is the number of edges lying closer to the vertex

*u*than

*v*, analogously define

*m*. The edge version of

_{v}(e)*PI*index of a graph

*G*is defined as . Nano-structured

*TiO*has been widely used in various applications such as biosensors, solar cells and biomaterials. Synthesis of nano-structured Titanium dioxide (

_{2}*TiO*) such as Nanotubes, Nano-wires and nano-fibers has raised interest lately due to their high surface to volume ratio and the ability of provoke a greater degree of biological plasticity compared to conventional microstructures. Nano-structured

_{2}*TiO*, has been widely used in various applications such as biosensors, solar cells, photocatalysis, phoelectrolysis and biomaterials. In this paper, we compute the edge-PI index of Titania Nanotubes

_{2}*TiO*.

_{2}Keywords: |

Nano-structured, Orthogonal cuts, Topological indices, Edge-PI index, Titania Nanotubes *TiO _{2}*(

*m,n*).

**INTRODUCTION:** Let *G* be a simple connected graph with vertex set *V(G) *and edge set *E(G)*, respectively. The distance between two vertices *u,v V(G)* is defined as the number of edges in a minimal path connecting the vertices *u,v*, and is denoted as *d(u,v).* If *e=uv*, is an edge and *y* is a vertex of the connected graph *G, *then the distance between *e* and *y *is equal to

For an edge *e=uv E(G)*, *n _{u}(e) *is the number of vertices of graph

*G*whose distance to the vertex

*u*is smaller than the distance to the vertex

*v*in

*G*; analogously

*n*is the number of vertices of

_{v}(e)*G*whose distance to the vertex

*v*in

*G*is smaller than the distance to the vertex

*u.*Similarly,

*m*is the number of edges of

_{u}(e)*G*whose distance to the vertex

*u*is smaller than the distance to the vertex

*v,*analogously

*m*denotes the number of edges of

_{v}(e)*G*whose distance to the vertex

*v*is smaller than the distance to the vertex

*u.*

A topological index is a real number related to a graph. It must be a structural invariant, *i.e.*, it preserves by every graph automorphisms. There are several topological indices have been defined and

many of them have found applications as means to model chemical, pharmaceutical and other properties of molecules.

The vertex-PI index was introduced by Ashrafi *et al.*, ^{1} as:

Khadikar* et al.,* ^{2 }introduced the edge-PI index as:

The mathematical properties of the *PI* and its applications in chemistry and nano-sciences are well studied for details see ^{3 - 14}.

Synthesis of nano-structured Titanium dioxide (*TiO _{2}*) such as Nanotubes, Nano-wires and nano-fibers has raised interest lately due to their high surface to volume ratio and the ability of provoke a greater degree of biological plasticity compared to conventional microstructures. Nano-structured

*TiO*has been widely used in various applications such as biosensors, solar cells, photocatalysis, phoelectrolysis and biomaterials. A 2-dimensional lattice of the Titania Nanotubes,

_{2},*TiO*[

_{2}*m,n*], is shown in

**Fig. 1**and for more chemical properties of

*TiO*Nanotubes and

_{2}*TiO*nano-composite, see

_{2}^{15- 18}. In

*TiO*[

_{2}*m,n*],

*m*and

*n*denotes the number of octagons in a column and the number of octagons in a row

^{19 - 37}. In this paper, we computed the edge-PI index of the Titania Nanotubes.

**RESULTS AND DISCUSSION:** In this section, we will compute the edge-PI index of the Titania Nanotubes with the help of cut method and orthogonal cuts ^{38, 39}.

**Theorem 1: **The edge-PI index of the Titania Nanotubes *TiO _{2}[m,n*]

*( )*is given by

**FIG. 1: ****2-DIMENSIONAL LATTICE OF THE TITANIA NANOTUBES**, *TiO _{2}*[

*m,n*]

**Proof: **A graphical representation of Titania Nanotubes *TiO _{2}[m,n]* is shown in

**Fig. 1**. This graph has

*2(3n+2)(m+1)*vertices and

*10mn+6m +8n+4*edges.

By using the Cut Method and finding Orthogonal Cuts of the Titania Nanotubes *TiO _{2}(m,n)*, we can determine all edge cuts (quasi-orthogonal) of

*TiO*and compute all

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

_{u}(e|*TiO*) and

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

_{v}(e|*TiO*), ∀

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

*∊*

*E(TiO*)

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

Here in this paper (see **Fig. 2**) ∀*e=uv**∊**E (TiO _{2}(m,n)*), we denote

*m*

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

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

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

*m*

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

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

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

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

**Fig. 2**, we see that there are

*2n+3(n+1)=5n+3*vertical cuts for all oblique or horizontal edges in

*TiO*∀

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

*Î*

*ℕ*and obviously all these orthogonal cuts are vertical. Now on based an edge e is an oblique edge or a horizontal edge, we denote its orthogonal cut by

*C*or

_{i}*F*for all

_{j }*i=1,…,C=2(n+1)*and

*j=1,…, F= 2n+ n+1*(obviously C+F=

*5n+3).*

Again by according to the structure of *TiO _{2}(m,n)* in

**Fig. 2**, we can see that the size of all orthogonal cuts

*C*are equivalence and is

_{i }*2m+1=|C*and the size of all orthogonal cuts

_{i}|*F*are equivalence, too, and is

_{i }*2(m+1)=|F*Thus for all orthogonal cuts

_{i}|.*C*and

_{i }*F*we have following results. In case the orthogonal cuts

_{i},*C*see

_{i }(i=1,…,2(n+1)) ,**Fig. 2**:

*For C*_{1}: m_{u}(e_{1}|*TiO*and_{2}(m,n))=0*m*_{v}(e_{1}|*TiO*_{2}(m,n))*=|**E(TiO*_{2}(m,n))*|-|C*_{1}|=10*mn+6m+8n+4-(**2m+1**) =10mn+4m +8n+3.**For C*_{2}: m_{u}(e_{2}|*TiO*_{2}(m,n))=*|C*and_{1}|+|F_{1}|=2m+1+2m+2=4m+3*m*_{v}(e_{2}|*TiO*_{2}(m,n))*=|**E(TiO*_{2}(m,n))*|- (|C*_{1}|+|F_{1}|+|C_{2}|)=10*mn+6m+8n+4-(**6m+4**)=**10mn+8n.**For C*_{3}: m_{u}(e_{3}|*TiO*_{2}(m,n))=*2|C*and_{1}|+3|F_{1}|=10m+8*m*_{v}(e_{3}|*TiO*_{2}(m,n))*=|**E(TiO*_{2}(m,n))*|-(3|C*_{1}|+3|F_{1}|) =10*mn+6m+8n+4-(**12m+9**)=10mn+8n-6m-5.**For C*_{4}: m_{u}(e_{4}|*TiO*_{2}(m,n))=*3|C*and_{1}|+4|F_{1}|=14m+11*m*_{v}(e_{4}|*TiO*_{2}(m,n))*=|**E(TiO*_{2}(m,n))*|**-(4|C*_{1}|+4|F_{1}|) =10*mn+6m+8n+4-(**16m+12**)=10mn+8n-10m-8.**For C*_{(2h-1)}:

* m _{u}(e_{(2h-1)}|*

*TiO*

_{2}(m,n))=*(2h-2)|C*

_{1}|+(3h-3)|F_{1}|=(2h-2)(2m+1)+(3h-3) (2m+2)=(10m+8)(h-1)* *and *m _{v}(e_{(2h-1)}|*

*TiO*

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

*E(TiO*

_{2}(m,n))*|-((2h-1)|C*

_{1}|+(3h-3)|F_{1}|)=10*mn+6m+8n+4-*

*(10m+8)(h-1)-(2m+1)*

*For C*_{(2h)}:

* m _{u}(e_{(2h)}|*

*TiO*

_{2}(m,n))=*(2h-1)|C*

_{1}|+(3h-2)|F_{1}|=(2h-1)(2m+1)+(3h-2) (2m+2)=10hm+8h-6m-5and *m _{v}(e_{(2h)}| *

*TiO*

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

*E(TiO*

_{2}(m,n))*|-(2h|C*

_{1}|+(3h-2)|F_{1}|)=10m*(n-h)+10m+8(n-h)+8.*

*For C*_{2n+2}:

* m _{u}(e_{2n+2}|*

*TiO*

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

_{1}|+(3n+1)|F_{1}|=(2n+1)(2m+1)+(3n+1) (2m+2)=10nm+8n+4m+3* *and* **m _{v}(e_{2n+2}|*

*TiO*

_{2}(m,n))*=0.*

**FIG. 2:** **CUTTING OF EDGES BY ORTHOGONAL CUTS/CUT METHOD OF TITANIA NANOTUBE**

In case the orthogonal cuts *F _{j }(*j=1,…,3n+1

*)*, see

**Fig. 2**:

*For F*_{1}: m_{u}(e_{1}|*TiO*_{2}(m,n))=*2m+1=|C*and_{i}|*m*_{v}(e_{1}|*TiO*_{2}(m,n))*=|**E(TiO*_{2}(m,n))*|-(|C*_{1}|+|F_{1}|)= 10*mn+ 6m+8n+4-(**4m+3**)=10mn+8n+2m+1.**For F*_{2}: m_{u}(e_{2}|*TiO*_{2}(m,n))=*2|C*and_{1}|+|F_{1}|=6m+4*m*_{v}(e_{2}|*TiO*_{2}(m,n))*=|**E(TiO*_{2}(m,n))*|-(2|C*_{1}|+2|F_{1}|) =10*mn+6m+8n+4-(**8m+6**)=10mn+8n-2m-2.**For F*_{3}: m_{u}(e_{3}|*TiO*_{2}(m,n))=*2|C*and_{1}|+2|F_{1}|=8m+6*m*_{v}(e_{3}|*TiO*_{2}(m,n))*=|**E(TiO*_{2}(m,n))*|-(2|C*_{1}|+3|F_{1}|)

*= 10**mn+6m+8n+4-(**10m+8**)=10mn+8n-4m-4.*

*For F*_{4}: m_{u}(e_{4}|*TiO*_{2}(m,n))=*3|C*and_{1}|+3|F_{1}|=12m+9*m*_{v}(e_{4}|*TiO*_{2}(m,n))*=|**E(TiO*_{2}(m,n))*|-(3|C*_{1}|+4|F_{1}|)

*= 10**mn+6m+8n+4-(**14m+11**)=10mn+8n-8m-7.*

*For F*_{5}: m_{u}(e_{5}|*TiO*_{2}(m,n))=*4|C*and_{1}|+4|F_{1}|=16m+12*m*_{v}(e_{5}|*TiO*_{2}(m,n))*=|**E(TiO*_{2}(m,n))*|-(4|C*_{1}|+5|F_{1}|) =10*mn+6m+8n+4-(**18m+14**)=10mn+8n-12m-10.**For F*_{6}: m_{u}(e_{6}|*TiO*_{2}(m,n))=*4|C*and_{1}|+5|F_{1}|=18m+14*m*_{v}(e_{6}|*TiO*_{2}(m,n))*=|**E(TiO*_{2}(m,n))*|-(4|C*_{1}|+6|F_{1}|) =10*mn+6m+8n+4-(**20m+16**)=10mn+8n-14m-12.**For F*_{7}: m_{u}(e_{7}|*TiO*_{2}(m,n))=*5|C*and_{1}|+6|F_{1}|=22m+17*m*_{v}(e_{7}|*TiO*_{2}(m,n))*=|**E(TiO*_{2}(m,n))*|-(5|C*_{1}|+7|F_{1}|) =10mn+6m+8n+4-(24m+19)*.**For F*_{8}: m_{u}(e_{8}|*TiO*_{2}(m,n))=*6|C*and_{1}|+7|F_{1}|=26m+20*m*_{v}(e_{8}|*TiO*_{2}(m,n))*=|**E(TiO*_{2}(m,n))*|-(6|C*_{1}|+8|F_{1}|) =10mn+6m+8n+4-(28m+22)*.**For F*_{3h+1}(h=0,…,n):

* m _{u}(F_{3h+1}|*

*TiO*

_{2}(m,n))=*(2h+1)|C*

_{1}|+(3h)|F_{1}|=(2h+1)(2m+1)+(3h) (2m+2)=10hm+2m+8h+1.* m _{v}(F_{3h+1}|*

*TiO*

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

*E(TiO*

_{2}(m,n))*|-(10hm+2m+8h+1)=(10m+8)(n-h)+4m+3.*

*For F*_{3h-1}(h=1,…,n):

* m _{u}(F_{3h-1}|*

*TiO*

_{2}(m,n))=*(2h)|C*

_{1}|+(3h-2)|F_{1}|=(2h)(2m+1)+(3h-2) (2m+2)* =(10m+8)h-2|F _{1}|=10hm-4m+ 8h-4.*

* m _{v}(F_{3h-1}|*

*TiO*

_{2}(m,n))=*(10*

*mn+6m+8n+4*

*)-(10hm-4m+8h-4)=(10m+8)(n-h)+10m+8.*

*For F*_{3h}(h=1,…,n):

* m _{u}(F_{3h}|*

*TiO*

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

_{u}(F_{3h-1}|*TiO*

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

_{1}|=2h|C_{1}|+(3h-1)|F_{1}|* =(10m+8)h-|F _{1}|=(10m+8)h-2m-2.*

* m _{v}(F_{3h}|*

*TiO*

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

_{v}(F_{3h-1}|*TiO*

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

_{1}|=(10m+8)(n-h)+8m+6.Here, we can compute the edge Szeged index of the Titania Nanotubes *TiO _{2}(m,n)*

*(*

*∀*

*m,n>1)*as:

**CONCLUSION: **In this paper, we computed the closed formulas of the edge-PI index of Titania Nanotubes *TiO _{2}*. Nano-structured

*TiO*has been widely used in various applications such as biosensors, solar cells and biomaterials. Synthesis of nano-structured Titanium dioxide (

_{2}*TiO*) such as Nanotubes, Nano-wires and nano-fibers has raised interest lately due to their high surface to volume ratio and the ability of provoke a greater degree of biological plasticity compared to conventional microstructures. Khadikar

_{2}*et al.,*proposed a topological index named the edge-PI index (shortly PI

_{e}) as:

where *m _{u}(e)* is the number of edges of

*G*whose distance to the vertex

*u*is smaller than the distance to the vertex

*v,*analogously

*m*denotes the number of edges of

_{v}(e)*G*whose distance to the vertex

*v*is smaller than the distance to the vertex

*u.*

**ACKNOWLEDGEMENT: **This work was supported by Key Project of Sichuan Provincial Department of Education under grant 17ZA0079 and Automotive Creative Design Pilot Area of Chengdu University and Longquanyi District under grant 2015-CX00-00010-ZF.

**CONFLICTS OF INTEREST: ** The authors have declared no conﬂict of interest.

**REFERENCES:**

- Ashrafi AR, Khalifeh MH and Azari HY: Vertex and edge PI indices of Cartesian product graphs, Discrete Applied Mathematics. 2008; 156: 1780-1789.
- Khadikar PV, Karmarkar S and Agrawal VK: A novel PI index and its application to QSPR/QSAR studies, Chem. Inf. Comput. Sci. 2001; 41:934-949.
- Ashrafi AR and Loghman A: PI index of zig-zag polyhex Nanotubes, Match Commun. Math. Comput. Chem. 2006; 55: 447-452.
- Ashrafi AR and Loghman A: PI index of armchair polyhex Nanotubes, Ars Combin. 2006; 80:193-199.
- Ashrafi AR and Rezaei F: PI index of polyhex nanotori, Match Commun. Math. Comput. Chem. 2007; 57: 243-250.
- Barriere L, Comellas F, Dalfo C and Fiol MA: The hierarchical product of graph, Discrete Appl. Math. 2009; 157: 36-48.
- Barriere L, Dalfo C, Fiol MA and Mitjana M: The generalized hierarchical product of graphs, Discrete Math. 2009; 309: 3871-3881.
- Deng H, Chen S and Zhang J: The PI index of phenylenes, Math. Chem. 2007; 41: 63-69.
- Hoji M, Luo Z and Vumar E: Wiener and vertex PI indices of Kronecker products of graphs, Discrete Appl. Math. 2010; 158: 1848-1855.
- Khadikar PV: On a novel structural descriptor PI, Acad. Sci. Lett. 2000; 23:113-118.
- Klavžar S: On the PI index: PI partitions and Cartesian product graphs, Match Commun. Math. Comput. Chem. 2007; 57:573-586.
- Azari HY, Manoochehrian B, and Ashrafi AR: The PI index of product graphs, Math. Lett. 2008; 21: 624-527.
- Farahani MR and Kanna MR: The edge-PI index of the polycyclic aromatic hydrocarbons, Indian Journal of Fundamental and Applied Life Sciences. 2015; 5: 614-617.
- Farahani MR: Computing edge-PI index and vertex-PI index of circumcoronene series of benzenoid H
_{k}by use of cut method, International Journal of Mathematical Modeling and Applied Computing. 2013; 1(6):41-50. - Ramazani M, Farahmandjou M and Firoozabadi TP: Effect of Nitric acid on particle morphology of the
*TiO*J. Nanosci. Nanotechnol. 2015; 11(1): 59-62._{2}, - Evarestoy RA, Zhukovskii YF, Bandura AV and Piskunov S: Symmetry and models of single-walled
*TiO*Nanotubes with rectangular morphology Open Physics. 2011; 9(2): 492-501. DOI: 10.2478/s11534-010-0095-8._{2} - Evarestov RA, Zhukovskii YF, Bandura AV, Piskunov S, and Losev MV: Symmetry and Models of Double-Wall BN and
*TiO*Nanotubes with Hexagonal Morphology the Journal of Physical Chemistry. 2011; 115(29): 14067- 14076. http://dx.doi.org/10.1021/jp2027737_{2} - Evarestov RA, Zhukovskii YF, Bandura AV and Piskunov S: Symmetry and Models of Single-Wall BN and
*TiO*Nanotubes with Hexagonal Morphology. The Journal of Physical Chemistry. 2010; 114(49): 21061–21069._{2} - Evarestov RA, Zhukovskii YF, Bandura AV and Piskunov S: Cent. Eur. J. Phys. 2011; 9: 492–501.
- Ramazani M, Farahmandjou M and Firoozabadi TP: Effect of Nitric acid on Particle Morphology of the Nano-
*TiO*. Int. J. Nanosci. Nanotechnol. 2015; 11(2):115-122._{2} - Subramaniyan A. and Ilangovan R: Thermal Conductivity of
*Cu*Composite-Nanofluid Based on Maxwell model. Int. J. Nanosci. Nanotechnol. 2015; 11(1): 59-62._{2}O-TiO_{2} - Gao W, Farahani MR and Imran M: About the Randić connectivity, modify Randić connectivity and sum-connectivity indices of Titania nanotubes
*TiO*. Acta Chim. Slov. 2017; 64(1): 256–260._{2}(m,n) - Farahani MR, Pradeep RK, Rajesh MRK and Wang S: The vertex Szeged index of Titania Carbon Nanotubes
*TiO*. International Journal of Pharmaceutical sciences and Research. 2016; 7(9): 3734-3741._{2}(m,n) - Farahani MR, Jamil MK, Pradeep RK, Rajesh MRK: Computing Edge Co-Padmakar-Ivan Index of Titania
*TiO*. Journal of Environmental Science, Computer Science and Engineering and Technology. 2016; 5(3): 326-334._{2}(m,n) - Farahani MR, Jamil MK and Imran M: Vertex PI
_{v}topological index of Titania carbon Nanotubes, Applied Mathematics and Nonlinear Sciences, 2016; 1(1): 170-176. - Huo Y, Liu JB, Imran M, Saeed M, Farahani MR, Iqbal MA and Malik A: On Some Degree-Based Topological Indices of Line Graphs of
*TiO*J. Comput. Theor. Nanosci. 2016; 13(12): 9131–9135._{2}(m,n) - Jiang H, Jamil MK, Siddiqui MK, Farahani MR and Shao Z: Edge-Vertex Szeged Index of Titania Nanotube
*TiO*,_{2}(m,n)*m,n*>1. International Journal of Advanced Biotechnology and Research. 2017: 8(2):1590-1597. - Gao W, Liu JB, Siddiqui MK, Farahani MR: Computing three topological indices for Titania Nanotubes
*TiO*. AKCE International Journal of Graphs and Combinatorics, 2016; 13(3):255–260._{2}[m;n] - Gao W, Farahani MR, Jamil MK and Siddiqui MK: The Redefined First, Second and Third Zagreb Indices of Titania Nanotubes
*TiO*The Open Biotechnology Journal. 2016; 10:272-277._{2}[m,n]. - Gao W, Jamil MK, Farahani MR and Imran M: Certain topological indices of Titania
*TiO*. J. Comput. Theor. Nanosci. , 2016; 13(10):7324–7328._{2}(m,n) - Li Y, Yan L, Farahani MR, Imran M and Jamil MK: Computing the Theta Polynomial Q(G,x) and the Theta Index Q(G) of Titania Nanotubes
*TiO*. Journal of Computational and Theoretical Nanoscience. 2017; 14(1): 715–717._{2}(m,n) - Yan L, Li Y, Farahani MR, Imran M: Sadhana and Pi polynomials and their indices of an infinite class of the Titania Nanotubes
*TiO*. Journal of Computational and Theoretical Nanoscience. 2016; 13(11): 8772-8775._{2}(m,n) - Yan L, Li Y, Farahani MR, Jamil MK: The Edge-Szeged index of the Titania Nanotubes
*TiO*. International Journal of Biology, Pharmacy and Allied Sciences. 2016; 5(6): 1260-1269._{2}(m,n) - Yan L, Li Y, Hayat S, Afzal HMS, Imran M, Ahmad S and Farahani MR: On degree-based and frustration related topological indices of single-walled Titania nanotubes. Journal of Computational and Theoretical Nanoscience. 2016; 13(11): 9027–9032.
- Liu Y, Rezaei M, Husin MN, Farahani MR and Imran M: The Omega polynomial and the Cluj-Ilmenau index of an infinite class of the Titania Nanotubes
*TiO*. J. Comput. Theor. Nanosci. 2017; 14(7): 3429-3432._{2}(m,n) - Rezaei M, Farahani MR, Jamil MK, Ali K and Lee DW: Vertex Version of Co-PI index of Titania Nanotubes Advances and Applications in Mathematical. 2016; 15(8): 255-262.
- Malik MA and Imran M: On multiple Zagreb indices of
*TiO*Nanotubes, Acta Chem. Slov. 2015; 62:973-976._{2} - Klavžar S: 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.
- John PE, Khadikar PV and Singh J: A method of computing the PI index of Benzenoid hydrocarbons using orthogonal cuts. Math. Chem. 2007; 42(1): 27-45.

**How to cite this article:**Zheng J, Farahani MR, Jamil MK, Siddiqui MK and Zhang X: The edge-Padmakar-Ivan index of the Titania nanotubes

*TiO*(_{2}*m,n*). Int J Pharm Sci Res 2018; 9(3): 1274-80.doi: 10.13040/IJPSR.0975-8232.9(3).1274-80.

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

55

1274-1280

635

856

English

IJPSR

J. Zheng, M. R. Farahani, M. K. Jamil, M. K. Siddiqui and X. Zhang*

School of Information Science and Engineering, Chengdu University, Chengdu, China.

woodszhang@cdu.edu.cn

08 June, 2017

10 August, 2017

29 August, 2017

10.13040/IJPSR.0975-8232.9(3).1274-80

01 March, 2018