On Eccentric Connectivity Index of TiO 2 Nanotubes

The eccentric connectivity index (ECI) is a distance based molecular structure descriptor that was recently used for mathematical modeling of biological activities of diverse nature. The ECI has been shown to give a high degree of predictability compare to Wiener index with regard to diuretic activity and anti-inflammatory activity. The prediction accuracy rate of ECI is better than the Zagreb indices in case of anticonvulsant activity. Titania nanotubular materials are of high interest metal oxide substances due to their widespread technological applications. The numerous studies on the use of this material also require theoretical studies on the other properties of such materials. Recently, the Zagreb indices were studied of an infinite class of titania (TiO2) nanotubes [32]. In this paper, we study the eccentric connectivity index of these nanotubes.


Introduction
Cheminformatics is a new subject which is a combination of chemistry, mathematics and information science.It studies quantitative structure activity relationships (QSAR) and structure property relationships (QSPR) that are used to predict the biological activities and properties of chemical compounds.In the QSAR/QSPR study, physicochemical properties and topological indices are used to predict biological activity of the chemical compounds.
A topological index is a numerical descriptor of the molecular structure based on certain topological features of the corresponding molecular graph.2][3] Topological indices are also used as a measure of structural similarity or diversity and thus they may give a measure of the diversity of chemical databases.There are two major classes of topological indices such as distance based topological indices and degree based topological indices.Among these classes, distance based topological indices are of great importance and play a vital role in chemical graph theory and particularly in chemistry.
A graph G with vertex set V(G) and edge set E(G) is connected if there exists a path between any pair of vertices in G.The degree of a vertex u ∈ V is the number of edges incident to u and denoted by deg(u).For two vertices u, v of a graph G their distance d (u, v) is defined as the length of any shortest path connecting u and v in G.For a given vertex u of G its eccentricity ε(u) is the largest distance between u and any vertex v of G.
Sharma et al. 9 introduced a distance based topological index, the eccentric connectivity index (ECI) of G, defined as (1)   It is reported in [4][5][6][7][8] that ECI provides excellent correlations with regard to both physical and biological properties.The eccentric connectivity index is successfully used for mathematical models of biological activities of diverse nature.0][11] The prediction accuracy rate of ECI is better than the Wiener index with regard to diuretic activity 12 and anti-inflammatory activity. 13Compare to Zagreb indices, the ECI has been shown to give a high degree of predictability in case of anticonvulsant activity. 14][22] The titanium nanotubular materials, called titania by a generic name, are of high interest metal oxide substances due to their widespread applications in production of catalytic, gas-sensing and corrosionresistance materials. 23s a well-known semiconductor with numerous technological applications, Titania (TiO 2 ) nanotubes are comprehensively studied in materials science. 24The TiO 2 nanotubes were systematically synthesized using different methods 25 and carefully studied as prospective technological materials.][31] Recently, M. A. Malik et al. 32 studied the Zagreb indices of an infinite class of TiO 2 nanotubes.In this paper, we study eccentric connectivity index of these nanotubes.

Main Results
The molecular graph of titania nanotubes TiO 2 [m,n] is presented in Figure 1, where m denotes the number of octagons in a row and n denotes the number of octagons in a column of the titania nanotube.In the molecular graph, G, of TiO 2 nanotubes, we can see that 2≤deg(v)≤ 5. So, we have the vertex partitions as follows. (2) The cardinalities of all vertex partitions are presented in Table 1.

Table 1:
The vertex partitions of the TiO 2 nanotubes along with their cardinalities.

Vertex partition Cardinality
In the following, we compute the exact formulas for eccentric connectivity index of TiO 2 [m,n] nanotubes.

Case 1. When p = 2n
In this case the eccentricity of the vertices u ij , v ij is 3p + 2n + 1 where i = 1,2n + 2. The eccentricity of each vertex in the remaining 2n rows is 4p.Hence Case 2. when and p ≠ 2n In this case the eccentricity of the vertices u ij , v ij is same as the eccentricity of vertices u (2n+3-i)j , v (2n+3-i)j .where i = 1,2, •••, 2n -p + 1.The eccentricity of these vertices in i th row is given by (7)   The eccentricity of vertices u ij , v ij in remaining 2p -2n rows is 4p.

Conclusion
The eccentric connectivity index provides excellent prediction accuracy rate compare to other indices in certain biological activities of diverse nature such as diuretic activity, anticonvulsant activity and anti-inflammatory activity.In this sense, this index is very useful in QSPR/QSAR studies.In this paper, we study eccentric connectivity index of an infinite class of TiO 2 nanotubes.By using this index, we can find mathematical models of certain biological activities for this material.With the help of these models, we can predict about certain biological activities for this material.

Theorem 2 . 1 4 ) 2 . 2
Let TiO 2 [m,n] be the graph of titania nanotube, then for we have (3) Proof.Consider G = TiO 2 [m,n].When , the eccentricity of every vertex in every row is 2m.From Table 1, we have (Theorem Let TiO 2 [m,n] be the graph of titania nanotube, where m = 2p then for p even we have (5) Proof.Consider G = TiO 2 [m,n].With respect to the eccentricity of vertices, we have the following cases. Hence

( 14 ) 2 . 3 Case 1 .
Theorem Let TiO 2 [m,n] be the graph of titania nanotube, where m = 2p then for p odd we have Proof.Consider G = TiO 2 [m,n].With respect to the eccentricity of vertices, we have the following cases.When p = 2n -1

Case 4 .
When n ≥ n -1 and n is even.