Fifth Geometric-Arithmetic Index of Polyhex Zigzag TUZC6[m,n] Nanotube and Nanotori

Among topological descriptors, connectivity indices are very important and they have a prominent role in chemistry. The geometric-arithmetic index (GA) index is a topological index was defined as  in which degree of a vertex v denoted by dv. A new version of GA index was defined by A. Graovac et al recently, and is equale to  where  In this paper, we compute this new topological connectivity index for polyhex zigzag TUZC6[m,n] Nanotube and Nanotori.


INTRODUCTION
Let G be a simple connected graph in chemical graph theory. In mathematices chemistry, the vertices and edges of a graph also correspond to the atoms and bonds of the molecular graph, respectively. If e is an edge/bond of G, connecting the vertices/atoms u and v, then we write e=uv and say "u and v are adjacent".
A simple graph is an unweighted, undirected graph without loops or multiple edges. And also a connected graph is a graph such that there is a path between all pairs of vertices. Clearly, a molecular graph is a simple connected graph.
In mathematices chemistry, there are many topological indices and structure descriptors. A topological index is a numeric quantity from the structural graph of a molecule and is invariant on the automorphism of the graph. Nowadays thousands and thousands topological indices are defined for different goals, such as stability of alkanes, the strain energy of cycloalkanes, prediction of boiling point and etc. [1][2][3].
One of the best known and widely used is the connectivity index introduced in 1975 by Milan Randić [3], who has shown this index to reflect molecular branching and was defined as in which degree of a vertex v denoted by dv.
One of the important topological connectivity index is the geometric-arithmetic index (GA) considered by Vukičević and Furtula [4] as Recently, the fifth geometric-arithmetic topological indices was defined by A. Graovac et al [5] as In Refs [6][7][8][9][10][11][12][13][14][15][16][17] some connectivity and geometric-arithmetic topological indices of some nanotubes and nanotorus are computed. The goal of this paper is to study the fifth geometric-arithmetic index and investigate this new index in one of the famous nano structure of polyhex zigzag nanotubes TUZC6 (see Figure 1).

Main Results and Discussions
Consider the molecular graph polyhex zigzag nanotube TUZC6 and let we denote the number of hexagons in the first row/column of the 2D-lattice of TUZC6 (Figures 2 and 3) by m and n, respectively. For other related research and historical details, see the paper series [19][20][21][22][23][24][25][26] and the general representation of this nano structure is shown in Figure 1 and Figure  2. The fifth geometric-arithmetic index of polyhex Zigzag TUZC6[m,n] Planar, Nanotube and Nanotori are given in the following theorem.