THE VERTEX SZEGED INDEX OF TITANIA CARBON NANOTUBES TiO2 (m,n)
HTML Full TextTHE VERTEX SZEGED INDEX OF TITANIA CARBON NANOTUBES TiO2 (m,n)
Mohammad Reza Farahani1*, 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.
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 nu(e|G) is the number of vertices of G lying closer to u than v and nv(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:
SZv(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 nu(e|TiO2(m,n)) and nv(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 nu(e|TiO2(m,n)) as the number of vertices in the left component of TiO2(m,n)-C(e) and alternatively nv(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
nu1(e1|TiO2(m,n))=2(m+1)
and
nv1(e2|TiO2(m,n))=6mn+4m+6n+4-2(m+1)=6mn+2m+6n+2.
For the edge e2=u2v2:
nu2(e2|TiO2(m,n))=3×2(m+1)+1×2(m+1)=8(m+1)
and
nv2(e2|TiO2(m,n))=6mn+4m+6n+4-8(m+1)=6mn-4m+6n-4.
For the edge en+1=u n+1v n+1:
nu n+1(e n+1|TiO2(m,n))=(3n+1)×2(m+1)
and
nv 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 ei=u iv i:
nui(e i|TiO2(m,n))=2(m+1)×(3(i-1)+1)
and
nvi(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
nu1(f1|TiO2(m,n))=m+1
and
nv1(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=u2v2:
nu2(f2|TiO2(m,n))=(m+1)+3×2(m+1)=7(m+1)
and
nv2(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:
nu(n+1) (f(n+1)|TiO2(m,n))=(m+1)+3n×2(m+1)=(6n+1)(m+1)
and
nv(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 fj=u jv j:
nuj(f j|TjO2(m,n))=3(j-1)×2(m+1)+(m+1)=(m+1)(6j-5)
and
nvj(fj|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
nu1(g1|TiO2(m,n))=2(m+1)+(m+1)
and
nv1(g1|TiO2(m,n))=(6n+4)(m+1)-3(m+1)=(6n+1)(m+1)
For g2=u2v2:
nu2(g2|TiO2(m,n))=3×2(m+1)+2(m+1)+(m+1)=9(m+1)
and
nv2(g2|TiO2(m,n))=(6n+4)(m+1)-9(m+1)=(6n-5)(m+1)
For gn+1=un+1 vn+1:
nun+1 (gn+1|TiO2(m,n))=3n×2(m+1)+2(m+1)+(m+1)=(6n+3)(m+1)
and
nvn+1 (gn+1|TiO2(m,n))=(6n+4)(m+1)- (6n+3)(m+1)=(m+1).
And these imply that ∀j=1,2,…,n+1
nuj(g j|TjO2(m,n))=3j×2(m+1)+3(m+1)=(m+1)(6j+3)
and
nvj(gj|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
nu1(h1|TiO2(m,n))=2×2(m+1)
and nu2(l2|TiO2(m,n))=3×2(m+1)
nv1(h1|TiO2(m,n))=(6n+4)(m+1)- 2(m+1)=(6n+2)(m+1)
and nv2(l2|TiO2(m,n))=(6n+1)(m+1)
And by a simple induction on ∀i=1,2,…,n; for the edges hi=u iv i and li=a ib i we have
nui(hi|TiO2(m,n))=3(i-1)×2(m+1)+2×2(m+1)=2(m+1)(3i-1)
and
nv i (hi|TiO2(m,n))=(6n+4)(m+1)- 2(m+1)(3i-1)
=(m+1)(6n-6i+6)
=6(m+1)(n+1-i).
nai(li|TiO2(m,n))=3(i-1)×2(m+1)+3×2(m+1)=6i(m+1)
and
nbi (li|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(ei)|=|C(hi)|=|C(li)|=2(m+1)
And
|C(fi)|=|C(gi)|=2m+1.
Here by above mentions results of nu(e|TiO2(m,n)) and nv(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).
Szv(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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Article Information
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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