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Introduction
Let G be a connected graph with vertex and edge sets V(G) and E(G), respectively. As usual, the distance between the vertices u and v of G is the number of edges in a minimal path connecting them, denoted as d(u,v). Define NG(u) to be the set of all vertices adjacent to u. The diameter is the greatest distance between two vertices of G, denoted as diam(G).
Let e=uv be an edge of the graph G. The number of vertices of G whose distance to the vertex u is smaller than the distance to the vertex v is denoted by nu(e). Analogously, nv(e) is the number of vertices of G whose distance to the vertex v is smaller than the distance to the vertex u.
Suppose X denoted the class of all graphs. A map Top from X into real numbers is called a topological index if G ≌ H implies that Top(G) = Top(H). Obviously, the maps Topvand Tope defined as the number of edges and vertices respectively, are topological indices.
The Wiener index was the first reported topological index based on graph distances [1]. This index is defined as the sum of all distances between vertices of the graph under consideration. The Wiener index is applicable to acyclic graphs only. For cyclic compounds a novel molecular graph based descriptor, referred to as the Szeged index [2]. This is considered as the modification of Wiener index to cyclic graph. The Szeged index is defined as
Note that vertices equidistant to u and v are not counted.
For acyclic graphs the Szeged and Wiener indices coincide. As a consequence, Padmaker and Ivan introduced another index called Padmakar-Ivan index (PIv) [3, 4]. PI of a graph G is defined as
Note that vertices equidistant to u and v are not counted. Many methods for the calculation of these indices of some systems are considered in [5–11].
As a well-known semiconductor with numerous technological applications, Titania nanotubes are comprehensively studied in material sciences. 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 TiO2nanotubes is still not well defined, their comprehensive theoretical studies attract enhanced attention. The TiO2 sheets with a thickness of a few atomic layers were found to be remarkably stable.
In this paper, we compute the vertex PI index of the Titania nanotubes. For further results we refer [12–14].
Main Results
The 2-dimensional graph of the Titania nanotube, TiO2[m,n], is shown in Figure 1, where m and n denotes the number of octagons in a column and the number of octagons in a row of the Titania nanotube. This graph has 2(3n+2)(m+1) vertices and 10mn + 6m + 8n + 4 edges.
By using the orthogonal cuts and cut method of the Titania nanotubes, we can determine all edge cuts of the Titania nanotubes. 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. For further research and study of the cut method and orthogonal cuts in some classes of chemical graphs see [8, 9, 11, 15].
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.
Theorem 1
Let TiO2[m,n] be the Titania Nanotubes, where m,n∈N. Then vertex PI index of TiO2[m,n] is:
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
On the other hands, by according to Figure 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):