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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 TiO2 (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, mu(e) is the number of edges lying closer to the vertex u than v, analogously define mv(e). The edge version of PI index of a graph G is defined as . Nano-structured TiO2 has been widely used in various applications such as biosensors, solar cells and biomaterials. Synthesis of nano-structured Titanium dioxide (TiO2) 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 TiO2, 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 TiO2.

Keywords:

Nano-structured, Orthogonal cuts, Topological indices, Edge-PI index, Titania Nanotubes TiO2(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), nu(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 nv(e) is the number of vertices of G whose distance to the vertex v in G is smaller than the distance to the vertex u. Similarly, ­mu(e) is the number of edges of G whose distance to the vertex u is smaller than the distance to the vertex v, analogously mv(e) denotes the number of edges of 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 (TiO2) 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 TiO2, has been widely used in various applications such as biosensors, solar cells, photocatalysis, phoelectrolysis and biomaterials. A 2-dimensional lattice of the Titania Nanotubes, TiO­2[m,n], is shown in Fig. 1 and for more chemical properties of TiO2 Nanotubes and TiO2 nano-composite, see 15- 18. In TiO2[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 TiO2[m,n] ( ) is given by


FIG. 1: 2-DIMENSIONAL LATTICE OF THE TITANIA NANOTUBES, TiO­2[m,n]

Proof: A graphical representation of Titania Nanotubes TiO2[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 TiO2(m,n), we can determine all edge cuts (quasi-orthogonal) of TiO2(m,n) and compute all mu(e|TiO2(m,n)) and mv(e|TiO2(m,n)), ∀e∊E(TiO2(m,n)).

Here in this paper (see Fig. 2) ∀e=uv∊E (TiO2(m,n)), we denote mu(e|TiO2(m,n)) as the number of edges in the left component of TiO2(m,n)-C(e) and alternatively mv(e|TiO2(m,n)) as the number of edges in the right component of TiO2(m,n)-C(e).

Thus by according to the structure of Titania Nanotubes TiO2(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 TiO2(m,n), ∀ m,nÎℕ 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 Ci or Fj for all 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 TiO2(m,n) in Fig. 2, we can see that the size of all orthogonal cuts Ci are equivalence and is 2m+1=|Ci| and the size of all orthogonal cuts Fi are equivalence, too, and is 2(m+1)=|Fi|. Thus for all orthogonal cuts Ci and Fi, we have following results. In case the orthogonal cuts Ci ­(i=1,…,2(n+1)) , see Fig. 2:

  • For C1: mu(e1|TiO2(m,n))=0 and mv(e1|TiO2(m,n))=|E(TiO2(m,n))|-|C1|=10mn+6m+8n+4-(2m+1) =10mn+4m +8n+3.
  • For C2: mu(e2|TiO2(m,n))=|C1|+|F1|=2m+1+2m+2=4m+3 and mv(e2|TiO2(m,n))=|E(TiO2(m,n))|- (|C1|+|F1|+|C2|)=10mn+6m+8n+4-(6m+4)= 10mn+8n.
  • For C3: mu(e3|TiO2(m,n))=2|C1|+3|F1|=10m+8 and mv(e3|TiO2(m,n))=|E(TiO2(m,n))|-(3|C1|+3|F1|) =10mn+6m+8n+4-(12m+9)=10mn+8n-6m-5.
  • For C4: mu(e4|TiO2(m,n))=3|C1|+4|F1|=14m+11 and mv(e4|TiO2(m,n))=|E(TiO2(m,n))| -(4|C1|+4|F1|) =10mn+6m+8n+4-(16m+12)=10mn+8n-10m-8.
  • For C(2h-1):

     mu(e(2h-1)|TiO2(m,n))=(2h-2)|C1|+(3h-3)|F1|=(2h-2)(2m+1)+(3h-3) (2m+2)=(10m+8)(h-1)

      and mv(e(2h-1)|TiO2(m,n))=|E(TiO2(m,n))|-((2h-1)|C1|+(3h-3)|F1|)=10mn+6m+8n+4-(10m+8)(h-1)-(2m+1)

  • For C(2h):

    mu(e(2h)|TiO2(m,n))=(2h-1)|C1|+(3h-2)|F1|=(2h-1)(2m+1)+(3h-2) (2m+2)=10hm+8h-6m-5

and mv(e(2h)| TiO2(m,n))=|E(TiO2(m,n))|-(2h|C1|+(3h-2)|F1|)=10m(n-h)+10m+8(n-h)+8.

  • For C2n+2:

    mu(e2n+2|TiO2(m,n))=(2n+1)|C1|+(3n+1)|F1|=(2n+1)(2m+1)+(3n+1) (2m+2)=10nm+8n+4m+3

    and mv(e2n+2|TiO2(m,n))=0.

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

In case the orthogonal cuts Fj ­(j=1,…,3n+1) , see Fig. 2:

  • For F1: mu(e1|TiO2(m,n))=2m+1=|Ci| and mv(e1|TiO2(m,n))=|E(TiO2(m,n))|-(|C1|+|F1|)= 10mn+ 6m+8n+4-(4m+3)=10mn+8n+2m+1.
  • For F2: mu(e2|TiO2(m,n))=2|C1|+|F1|=6m+4 and mv(e2|TiO2(m,n))=|E(TiO2(m,n))|-(2|C1|+2|F1|) =10mn+6m+8n+4-(8m+6)=10mn+8n-2m-2.
  • For F3: mu(e3|TiO2(m,n))=2|C1|+2|F1|=8m+6 and mv(e3|TiO2(m,n))=|E(TiO2(m,n))|-(2|C1|+3|F1|)

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

  • For F4: mu(e4|TiO2(m,n))=3|C1|+3|F1|=12m+9 and mv(e4|TiO2(m,n))=|E(TiO2(m,n))|-(3|C1|+4|F1|)

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

  • For F5: mu(e5|TiO2(m,n))=4|C1|+4|F1|=16m+12 and mv(e5|TiO2(m,n))=|E(TiO2(m,n))|-(4|C1|+5|F1|) =10mn+6m+8n+4-(18m+14)=10mn+8n-12m-10.
  • For F6: mu(e6|TiO2(m,n))=4|C1|+5|F1|=18m+14 and mv(e6|TiO2(m,n))=|E(TiO2(m,n))|-(4|C1|+6|F1|) =10mn+6m+8n+4-(20m+16)=10mn+8n-14m-12.
  • For F7: mu(e7|TiO2(m,n))=5|C1|+6|F1|=22m+17 and mv(e7|TiO2(m,n))=|E(TiO2(m,n))|-(5|C1|+7|F1|) =10mn+6m+8n+4-(24m+19).
  • For F8: mu(e8|TiO2(m,n))=6|C1|+7|F1|=26m+20 and mv(e8|TiO2(m,n))=|E(TiO2(m,n))|-(6|C1|+8|F1|) =10mn+6m+8n+4-(28m+22).
  • For F3h+1 (h=0,…,n):

        mu(F3h+1|TiO2(m,n))=(2h+1)|C1|+(3h)|F1|=(2h+1)(2m+1)+(3h) (2m+2)=10hm+2m+8h+1.

        mv(F3h+1|TiO2(m,n))=|E(TiO2(m,n))|-(10hm+2m+8h+1)=(10m+8)(n-h)+4m+3.

  • For F3h-1 (h=1,…,n):

         mu(F3h-1|TiO2(m,n))=(2h)|C1|+(3h-2)|F1|=(2h)(2m+1)+(3h-2) (2m+2)

                   =(10m+8)h-2|F1|=10hm-4m+ 8h-4.

        mv(F3h-1|TiO2(m,n))=(10mn+6m+8n+4)-(10hm-4m+8h-4)=(10m+8)(n-h)+10m+8.

  • For F3h (h=1,…,n):

        mu(F3h|TiO2(m,n))=mu(F3h-1|TiO2(m,n))+ |F1|=2h|C1|+(3h-1)|F1|

       =(10m+8)h-|F1|=(10m+8)h-2m-2.

       mv(F3h|TiO2(m,n))=mv(F3h-1|TiO2(m,n))-|F1|=(10m+8)(n-h)+8m+6.

Here, we can compute the edge Szeged index of the Titania Nanotubes TiO2(m,n) (∀m,n>1) as:

 


CONCLUSION: In this paper, we computed the closed formulas of the edge-PI index of Titania Nanotubes TiO2. Nano-structured TiO2 has been widely used in various applications such as biosensors, solar cells and biomaterials. Synthesis of nano-structured Titanium dioxide (TiO2) 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 et al., proposed a topological index named the edge-PI index (shortly PIe) as:

where mu(e) is the number of edges of G whose distance to the vertex u is smaller than the distance to the vertex v, analogously mv(e) denotes the number of edges of 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.

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    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 TiO2(m,n). Int J Pharm Sci Res 2018; 9(3): 1274-80.doi: 10.13040/IJPSR.0975-8232.9(3).1274-80.

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Article Information

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1274-1280

635

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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

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