About the Randi} Connectivity, Modify Randi} Connectivity and Sum-connectivity Indices of Titania Nanotubes TiO2(m,n)

The Randi} Connectivity Index R(G) is one of the oldest connectivity index, introduced by Randi} in 1975. Another connectivity indices is the Sum-Connectivity Index X(G) introduced in 2008 by Zhou and Trinajsti}. Recently in 2011, a modification of the Randi} Connectivity Index of a graph G was introduced by Dvorak et al. In this paper, we compute these connectivity topological indices for a family of molecular graphs known as titania nanotubes TiO2(m,n).


Introduction
A graph is a collection of points and lines connecting a subset of them. The points and lines in a graph are respectively called vertices and edges of the graph. An edge in E(G) with end vertices u and v is denoted by uv. Two vertices u and v are said to be adjacent if there is an edge between them. In chemical graph theory, the vertices of molecular graph G correspond to the atoms and its edges correspond to the chemical bonds. We denoted the order and size and degree of a vertex/atom v of a molecular graph G by |V(G)|, |E(G)| and dv, respectively. The set of all vertices adjacent to a vertex v in V(G) is said to be the neighborhood of v, denoted as N(v). The number of vertices in N(v) is said to be the degree of v. The minimum and maximum vertex degrees in a graph G denoted by δ(G) and Δ(G), respectively and are defined as min{dv | v∈V(G)} and max{dv | v∈V(G)}, respectively. Our notation is standard and mainly taken from standard books of chemical graph theory. [1][2][3] We have many connectivity topological indices, for an arbitrary graph with connected structure in chemical graph theory. The oldest of them is Randi} Connectivity Index which has shown to reflect molecular branching, introduced by Milan Randi} in 1975, 4 and defined as (1) where, d u and d v are the degrees of the vertices u and v, respectively.
Another connectivity indices is the Sum-Connectivity Index that was introduced by Zhou and Trinajsti} in 2008. 5,6 The sum-connectivity index X(G) is defined as the sum over all edges of the graph of the terms d u + d v ) -2/2 and is equal to (2) Recently in 2011, Dvorak et al. introduced a modification of the Randi} Connectivity Index of G and is defined as (3) that is more tractable from computational point of view. It is much easier to compute Modify Randi} index R'(G) than Randi} index R'(G) (see 7 for more details). Some basic properties of these indices can be found in the recent letters. For more study, see reference. [8][9][10][11][12][13] In this paper, we investigate the topological Connectivity indices, and compute some formulas for the Randi}, Sum-Connectivity and Modify Randi} indices of a family of molecular graphs that called titania nanotubes TiO 2 (m,n) for positive integers n, m (see Figure 1).

Main results and Discussion
In this section, we compute the Randi}, Sum-connectivity and Modify Randi} Indices for the titania nanotubes TiO 2 (m,n) (∀ m,n∈» »). Titania nanotubes were systematically synthesized during the last 10-15 years using different methods and carefully studied as prospective technological materials. Since the growth mechanism for TiO 2 Nanotubes is still not well defined, their comprehensive theoretical studies attract enhanced attention. The TiO 2 sheets with a thickness of a few atomic layers were found to be remarkably stable. 14-17 Molecular graphs titania TiO 2 (m,n) is a family of nanotubes, such that the structure of this family of nanotubes consist of the cycles with length four C 4 and eight C 8 . Several topological indices of titania nanotubes (TiO 2 ) have been studied in the literature. [18][19][20] Let us denote the number of Octagons or cycles C 8 in the first row and column of the 2-Dimensional lattice of TiO 2 nanotubes (Figure 1) by m and n, respectively.
Before we prove the main results, let us introduce some definitions. Definition 1. Consider the graph G = (V, E), then we divide the vertex set V(G) and edge set E(G) of G into several partitions based on the degrees of vertices/atoms in G as follows. 9 (7) Where d u (1 ≤ d v ≤ n -1) be the degrees of v∈V(G) and δ and Δ are the minimum and maximum, respectively.
In particular, let G = (V, E) be a connected molecular graph or nanotubes, then we can divide the vertex set and edge set of G in following partitions: (8) Since the degree of an atom (or vertex) of the molecular graph is equal to 1, 2,…, 5 and the hydrogen atoms (with degree 1) in G are often omitted.
In particular, let TiO 2 (m,n) be the titania nanotubes (∀m,n∈» ») with 6n(m+1) vertices and 10mn+8n edges, then from its structure, the vertex and edge partitions of  (TiO 2 (m,n)) and edge set E (TiO 2 (m,n)) and their order and size are as follow. 17 (9) and (10) By above mentioned formulas, one can see that (11) Now, we have the following computations of the Randi}, Sum-connectivity and Modify Randi} Indices for the titania nanotubes TiO 2 (m,n) ∀ m,n ∈ » ».
Thus the Randi} connectivity index of TiO 2 (m,n) nanotubes is equal to (12) Also, (14) Hence the Sum-Connectivity index of TiO 2 (m,n) nanotubes is Now, by using Definition 1, we see that there are two modify edges partitions E 4 + and E 5 + for the titania nanotubes TiO 2 (m,n) (∀m,n∈» ») as: Therefore the Modify Randi} index of TiO 2 (m,n) is equal to: Here, we complete the proof of main theorem of this article and all main results are computed.

Discussion
Now we study the change of the values of Randi}, Sum-connectivity and Modify Randi} Indices of TiO 2 (m,n) nanotubes when the parameters m and n are slightly changed. The graphs of these nanotubes corresponding to some small values of m and n are shown in Figure 2. Similarly, the values of the studied topological indices corresponding to small change in the values of m and n is summarized in Table 1.

Conclusion
In this paper, we considered an infinite class of the titania nanotubes TiO 2 (m,n), that were systematically