On Multiple Zagreb Indices of TiO2 Nanotubes

The first and second Zagreb indices were first introduced by I. Gutman and N. Trinajstic in 1972. It is reported that these indices are useful in the study of anti-inflammatory activities of certain chemical instances, and in elsewhere. Recently, the first and second multiple Zagreb indices of a graph G were introduced by Ghorbani and Azimi in 2012. In this paper, we calculate the Zagreb indices and the multiplicative versions of the Zagreb indices of an infinite class of titania nanotubes TiO2[m,n].


Introduction
Mathematical chemistry is a branch of theoretical chemistry in which we discuss and predict the chemical structure by using mathematical tools. Chemical graph theory is a branch of mathematical chemistry in which we apply tools from graph theory to model the chemical phenomenon mathematically. This theory plays a prominent role in the fields of chemical sciences.
A molecular graph is a simple graph in which the vertices denote atoms and the edges represent chemical bonds between these atoms. The hydrogen atoms are often omitted in a molecular graph. Let G be a molecular graph with vertex set V(G) = {v 1 , v 2 , …, v n } and edge set E(G). We denote the order and size of G by |V|(G)| and |E(G)|, respectively. 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. The set of all vertices adjacent to a vertex u is said to be the neighbourhood of u, denoted as N(u). The number of vertices in N(u) is said to be the degree of u, denoted by d(u). The maximum and minimum vertex degrees in a graph G, respectively denoted by Δ(G) and δ(G), are defined as max {d(u)|u∈V(G)} and min {d(u)| u∈V(G)}, respectively. A (v 1 , v n ) -path on n vertices is denoted by P n and is defined as a graph with vertex set {v 1 :1≤ i ≤n} and edge set v i v i+1 , for 1 ≤ i ≤ n-1. The length of a path P n is the number of edges in it, that is, n-1.
A topological index is a molecular graph invariant which correlates the physico-chemical properties of a molecular graph with a number. 1 The first topological index was introduced by a chemist Harold Wiener in 1947 to calculate the boiling points of paraffins. 2 This numerical representation of a molecular graph has shown to be very useful quantity to use in the quantitative structure-property relationship. 3 It also has many applications in communication, facility location, cryptography, etc., that are effectively modeled by a connected graph G with some restrictions. 4 This index was originally defined for trees to correlate certain physico-chemical properties of alkanes, alcohols, amines and their compounds. Hosoya 5 defined the notion of Wiener index for any graph G as (1) A pair of molecular descriptors known as the first Zagreb index M 1 (G) and the second Zagreb index M 2 (G) were first appeared in the topological formula for total π-energy of conjugated molecules that was derived in 1972 by Gutman and Trinajstic. 6 Afterwards, these indices have been used as branching indices 7 . Later on, the Zagreb indices found applications in QSPR and QSAR studies. 8,9 Many topological indices have been studied in the literature for different chemical structures. [10][11][12][13] The first Zagreb index M 1 (G) and the second Zagreb index M 2 (G) of a molecular graph G are respectively defined as Ali Malik and Imran: On Multiple Zagreb Indices of TiO 2 Nanotubes ...
The new multiplicative versions of M 1 (G) and M 2 (G) indices, denoted by PM 1 (G) and PM 2 (G) (respectively), were first defined by Ghorbani and Azimi. 14 These indices are defined as follows.
(4) (5) In this paper, we study the Zagreb and multiplicative versions of Zagreb indices of titania TiO 2 nanotubes. As a well-known semiconductor with numerous technological applications, Titania nanotubes are comprehensively studied in materials science. 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. 15 Recently, the multiple Zagreb indices of circumcoronene homologous series of benzenoids were studied by Farahani. 16 The Zagreb indices of some other nanotubes and benzenoid graphs can also be found in the literature. 17,18

Main Results
The graph of the titania nanotube TiO 2 [m,n] is presented in Figure 1, where m denotes the number of octa-gons in a column and n denotes the number of octagons in a row of the Titania nanotube.
In the following, we perform some necessary calculations for computing the Zagreb indices and multiplicative versions of Zagreb indices defined in the previous section.
Let us define the partitions for the vertex set and edge set of the titania nanotube TiO 2 , for δ(G) ≤ k ≤ Δ(G), 2δ(G) ≤ i ≤ 2Δ(G), and δ(G) 2 ≤ j ≤ Δ (G) 2 , then we have In the molecular graph of TiO 2 nanotubes, we can see that 2 ≤ d(v) ≤ 5. So, we have the vertex partitions as follows.  Since for every vertex v∈ V(G), d(v) belongs to exactly one class V k for 2 ≤ k ≤ 5 and for every edge uv∈-E(G), d(u) + d(v) (resp. d(u)d(v)) belongs to exactly one class E i (resp. E j * ) for 2δ(G) ≤ i ≤ 2Δ (G), and δ(G) 2 ≤ i ≤ Δ(G) 2 . So, the vertex partitions V k and the edge partitions E i and E j * are collectively exhaustive, that is (17) When m = 1, the number of vertices of degree 2 in TiO 2 [1,n] are 2n + n + 1 × (2n) + n. Thus for TiO 2 [m,n], we have |V 2 | = 2n + n + m(2n) + n = 2mn + 4n. Similarly, the cardinalities of all the vertex and edge partitions can be obtained, which are presented in Table 1.

Conclusion
The Zagreb indices were first appeared in the topological formula for the total π-energy of conjugated molecules that has been derived in 1972 by Gutman and Trinajstic 3 . These indices have also been used as branching indices and have found applications in QSPR and QSAR studies. In this paper, we studied the Zagreb indices and the newly defined multiplicative versions of the Zagreb indices of an infinite class of Titania nanotubes TiO 2 [m,n]. Closed form formulas have been derived for the above mentioned indices.