Vertex PI _v Topological Index of Titania Carbon Nanotubes TiO ₂ (m,n)

Abstract

A topological index of a graph G is a numeric quantity related to G which is invariant under automorphisms of G. The Padmakar-Ivan (PI) index of a graph G is defined as PI(G) = Σe=uv∈E(G) [nu + nv], where nu is the number of edges of G lying closer to v than u, analogously nv. In this paper, we compute the vertex PI index of Titania carbon Nanotubes TiO2[m,n].

Computing Sanskruti Index of Titania Nanotubes

“…T iO 2 sheets with a thickness of a few atomic layers were found to be remarkably stable [12]. Recently Malik and Imran [13] and some others researchers computed multiple Zagreb index of T iO 2 nanotubes, readers can see the paper series [14,15,16,17,18,19,20,21,22,23,24,25,26,27]. The graph of the Titania nanotube T iO 2 [m, n] is presented in Fig.…”

Untitled

“…The graph of the Titania Nanotubes TiO 2 (m,n) is presented in Fig.1, where m denotes the number of octagons in a column and n denotes the number of octagons in a row of the Titania Nanotubes. We encourage the reader to consult papers [19][20][21][22][23][24][25][26][27][28][29][30] , for further study and more information of Titania Nanotubes TiO 2 . …”

Physical-chemical properties studying of molecular structures via topological index calculating

“…Since D 1 [n] is symmetrical, we can mark the vertices some representative symbols which are showed in Figure 1. Next, we only present the detailed proof of GA 4 index, and other parts of result can be obtained in the similar way.…”

“…By the analysis of molecular structure, the edge set of D 1 [n] can be divided into six subsets which are described as follows: Thus, by the definition of the fourth GA index, we infer …”

“…There were many contributions to report these distance-based or degree-based indices of special molecular structures (See Farahani et al [1], Jamil et al [2], Gao and Wang [3], Gao et al [4][5][6][7] and Gao and Wang [8,9] for more details). The notations and terminologies that were used (but undefined in our paper) can be found in Bondy and Murty [10].…”