ABAIDUR Rehman Virk

(Department of Mathematics， University of Sialkot， Sialkot, Pakistan 22161)

Abstract: Titania is one of the most comprehensively studied nanostructures due to the widespread applications in the production of catalytic, gas sensing, and corrosion-resistant materials. Zagreb indices are the most important topological indices, so we computed first and second generalized multiplicative Zagreb indices for the three and six layered single-walled Titania nanotubes. We also recovered first and second multiplicative Zagreb indices from the generalized first and second multiplicative Zagreb indices.

Key words: TiO2；Nanotube；topological index；multiplicative Zagreb index

Titania, TiO2, attracts considerable technological interest due to its unique properties in biology, optics, electronics, and photo-chemistry[1]. Recent experimental studies show that titania nanotubes (NTs) improve TiO2bulk properties for photocatalysis, hydrogen-sensing, and photo-voltaic applications[2]. Titanium nanotubes have been observed in two types of morphologies: single-walled titanium (SW TiO2) nanotubes and multi-walled (MW TiO2) nanotubes[3]. Here, we are interested only in single-walled TiO2nanotubes because we consider their chemical graphs to work on molecular descriptors. Titania nanotubes are formed by rolling up the stoichiometric two-periodic (2D) sheets cut from the energetically stable anatase surface.

In applied mathematics, chemical reaction network theory was initiated in 1960s and got huge attraction of researcher because this theory is used to model the chemical system phenomena′s. This theory is applicable in theoretical and bio-chemistry.

Another interesting field of research is cheminformatics. In this study, topological indices together with quantitative structure-activity (QSAR) and Structure-property (QSPR) relationships guess different properties of chemical structures.

The area of research in chemistry in which mathematics is used to deal with the problems of chemistry is named as Mathematical Chemistry. For example, graph theory is a mathematical tool which is used to model the chemical structure and with the help of graph theoretical technics, one can obtain information about different chemical structures by using symmetry present in that structure. This particular branch of Mathematical Chemistry is known as Chemical Graph Theory[4].

The union of dots (vertices) and lines (edges) is called a graph and is denoted byG.The graphGis said to be connected if all of its vertices have connection between them. By degree of a vertexv, we mean the number of vertices at distance one fromvand is represented bydv.

Now, we give definitions of multiplicative Zagreb indices. Throughout this paperGdenotes the connected graph without loops and multiple edges.

The first and second generalized multiplication Zagreb indices are defined as：

and

respectively.

The first and second multiplication Zagreb indices are defined as：

and

respectively.

In this paper, we computed multiplicative three and six layered single-walled Titania Nanotubes.

1 Multiplicative Zagreb Indices of Three Layered Single-Walled Titania Nanotubes

Three layered single-walled Titania Nanotube is denoted byTNT3[m,n] and the molecular graph is given in Figure 1.

Figure 1 Graph of TNT3[m,n]

The Table 1 contains the edge partition ofTNT3[m,n] based on the degree of end vertices.

Table 1 Edge Partition of TNT3[m,n]

Theorem1LetGbe the graph of three layered single-walled Titania Nanotube, then we have：

ProofUsing edge partition from Table 1, we have following computations for the first generalized multiplication Zagreb index：

=(6α)|ij∈E1(G)|×(7α)|ij∈E2(G)|×(8α)|ij∈E3(G)|×(9α)|ij∈E4(G)|

=(6α)4m×(7α)4m×(8α)4m×(9α)2m(6n-5)

=216αm×38αm(3n-2)×74αm.

Theorem2LetGbe the graph of three layered single-walled Titania Nanotube, then we have：

ProofUsing edge partition from Table 1, we have following computations for the second generalized multiplication Zagreb index：

=(8α)|ij∈E1(G)|×(12α)|ij∈E2(G)|×(12α)|ij∈E3(G)|×(18α)|ij∈E4(G)|

=(8α)4m×(12α)4m×(12α)4m×(18α)2m(6n-5)

=26αm(2n-3)×312αm(2n-1).

Theorem3LetGbe the graph of three layered single-walled Titania Nanotube, then we have：

MZ1(G)=216m×38m(3n-2)×74m.

ProofThe result can be obtained immediately from Theorem 1.

Theorem4LetGbe the graph of three layered single-walled Titania Nanotube, then we have：

MZ2(G)=26m(2n-3)×312m(2n-1).

ProofThe result can be obtained immediately from Theorem 2.

2 Multiplicative Zagreb Indices of Six Layered Single-Walled Titania Nanotubes

LetTNT6[m,n] be the six layered single-walled Titania Nanotube as shown in Figure 2.

Figure 2 Graph of Six-Layered Single-Walled Titania Nanotube

The edge partition ofTNT6[m,n] based on the degree of end vertices is given in Table 2.

Table 2 Edge Partition of TNT6[m,n]

Theorem5LetGbe the graph of six layered single-walled Titania Nanotube, then we have：

ProofUsing edge partition from Table 2, we have following computations for the first generalized multiplication Zagreb index：

=(4α)|ij∈E1(G)|×(5α)|ij∈E2(G)|×(6α)|ij∈E3(G)|×(7α)|ij∈E4(G)|×(7α)|ij∈E5(G)|× (8α)|ij∈E6(G)|

=(4α)2m×(5α)2m×(6α)6m×(7α)8mn×(7α)2m×(8α)2m(6n-5)

=24αm(9n-5)×36αm×52αm×72αm(4n+1).

Theorem6LetGbe the graph of six layered single-walled Titania Nanotube, then we have：

ProofUsing edge partition from Table 2, we have following computations for the second generalized multiplication Zagreb index：

=(4α)|ij∈E1(G)|×(6α)|ij∈E2(G)|×(8α)|ij∈E3(G)|×(10α)|ij∈E4(G)|×(12α)|ij∈E5(G)|×(15α)|ij∈E6(G)|

=(4α)2m×(6α)2m×(8α)6m×(10α)8mn×(12α)2m×(15α)2m(6n-5)

=22αm(4n+13)×32αm(6n-3)×510αm(2n-1).

Theorem7LetGbe the graph of six layered single-walled Titania Nanotube, then we have：

MZ1(G)=24m(9n-5)×36m×52m×72m(4n+1)．

ProofThis result can be obtained immediately from Theorem 5.

Theorem8LetGbe the graph of six layered single-walled Titania Nanotube, then we have：

MZ2(G)=22m(4n+13)×32m(6n-3)×510m(2n-1).

ProofThis result can be obtained immediately form Theorem 6.

3 Conclusions

In this paper, we computed generalized version of first and second multiplicative Zagreb indices for two important classes of Nanotubes. From the computed results, we recover first and second multiplicative Zagreb indices. One can also recover some other versions of multiplicative indices from our results, for example, multiplicative first and second Harmonic indices and multiplicative sum and product connectivity indices can also be obtained from our results.