Computing the theta polynomial Θ(G,x) and the theta index Θ(G) of titania nanotubes TiO₂(m,n)

Let G = (V ,E) be a simple connected molecular graph in chemical graph theory, where the vertex/ atom set and edge/bond set of G denoted by V (G) and E(G), respectively and its vertices correspond to the atoms and the edges correspond to the bonds. Two counting polynomials the "Omega Ω(G,x) and Theta Θ(G,x)" polynomials of a molecular graph G were defined by Diudea as Θ(G,x) =Σ_e∈E(G) x_n(e) and Θ(G,x) = Σ_e∈E(G) n(e)x^n(e) where n(e) denotes the number of edges co-distant with the edge e. From definition of these counting polynomials, we can obtain the Theta polynomial by inserting the coefficient n(e) in the Omega polynomial. Then the Theta index will be the first derivative of the Theta polynomial Θ(G,x) evaluated at x = 1. The goal of this paper is to compute the Theta polynomial Θ(G,x) and the Theta index Θ(G) of an infinite family of the Titania Nanotubes TiO₂(m,n) for the first time.