Degree Based Multiplicative Connectivity Indices of Nanostructures

Published online: March 6, 2019 The Multiplicative topological indices of Phenylenic, Naphatalenic, Anthracene and Tetracenic Nanotubes are calculated. The indices like Multiplicative Zagreb, Multiplicative Hyper-Zagreb, Multiplicative Sum connectivity, Multiplicative product connectivity, General multiplicative Zagreb, Multiplicative ABC and Multiplicative GA indices are expressed as a closed formula for the known values of s, t. The proposed formulae will be very useful for the study of nanostructure in the field of nanotechnology.


Introduction
A molecular graph is defined as a simple graph in which the vertices represent the atoms and the edges represent the bonds between the atoms. Graph theory is a developing branch of Mathematics. From the Mathematical modeling of the chemical compound, we can apply the graph theory concepts to the graphs of chemical structure. The application of graph theory is a powerful tool for studying in QSPR and QSAR. Topological indices are invariant values given by the structure of chemical compounds, which correlates with their physic-chemical properties. Phenylenic nanotubes have cycles of length 4, 6 and 8. The cycles are arranged in the alternating manner. The structure of nanotubes is either a cylinder or in the form of a torus. Naphatalenic nanotubes have cycles of length 4, 6 and 8. The first row of the Naphatalenic nanotubes contains only C 4 and C 6 . Similarly, the second row contains the sequences of cycles of length 6 and 8. So we can say that it is a lattice containing the cycles of length 4, 6 and 8, and it is a plane. The entire structure can cover either a cylinder or torus. Anthracene is denoted by the formula C 14 H 10 , which is a solid polycyclic aromatic hydrocarbon. It consists of 3 benzene rings which are fused. It is also known as cool tar. It is used in the production of dyes like red alizarin. Tetracenic is also a polycyclic aromatic hydrocarbon. It appeared to be orange powder light in color. The chemical graphs of Phenylenic, Naphatalenic, Anthracene and Tetracenic Nanotubes are shown in the Figures 1, 2, 3 Kulli (2016b) Many literatures (Veylaki et. al. 2015), (Wei Gao 2017), (Farahani 2015), (Kulli 2018), (Imran Nadeem et. al. 2016), (Liu et. al. 2016), (Wei Geo et. al. 2017) are available in the study of topological indices based on additive and multiplicative indices. Recently these topological invariants of nanotubes correlates perfectly with the properties of nanotubes (Doslic et. al. 2011), (Vukicevic et. al. 2011), (Diudea 2010), (Diudea 2006). Therefore, the studies of four different types of nanotubes are selected and their topological invariants are computed and explained with their structures to the field of nanotechnology.

Results for V-Phenylenic nanotubes (V P )
This type of nanotubes is defined as VPHX [s, t]. Here s-number of joining hexagons in row first and t -number of another hexagons in column first. These nanotubes are defined in figure 1. [ , ] VPHX s t is the V-Phenylenic nanotubes, the vertex cardinality is | ( ) | 6 .

Results for V-Anthracene Nanotubes (V A )
This type is defined as Anthracene [ , ].
s t Here s-number of joining hexagons in row first and t -number of another hexagons in column first. These nanotubes are defined in figure 3. Lemma 2.5. Let V-Anthracene nanotubes, the vertex cardinality is Lemma 2.6. Let V-Anthracene nanotubes, the edge cardinality is

Results for V-Tetracenic (V T )
This type is defined Tetracenic [ , ].
s t Here s-number of joining hexagons in row first and t -number of another hexagons in column first. These nanotubes are defined in figure 4.

Conclusion
In this article, we have studied the Multiplicative indices of some nanostructures. The analytical expression for the topological invariants is presented by using edge set dividing method. Also, we expressed the generalized form for the computational formulas. These proposed results will be useful for the study of nanostructures. The results obtained in this study have a wide application prospect in nanoscience, biology, pharmacy and other fields.