**1. Introduction**

Mathematical chemistry becomes an interesting branch of science in which we talk about and foresee the concoction structure by utilizing numerical apparatuses and does not really allude to the quantum mechanics. As a branch of numerical science where we apply devices of graph hypothesis, chemical graph theory was introduced and extensively studied to show the compound wonder scientifically. This is more imperative to state that the hydrogen particles are regularly overlooked in any sub-atomic graph. Topological indices are really a numeric measures related to the constitution synthetic material implying for relationship of concoction structure with numerous physio-substance features, compound responsiveness or biological activity. Motivated by the wide applications of topological indices, the topological indices of graphs are studied extensively [1–3].

A nano structure is a question of middle size among both molecular and microscopic structures. Such a material is determined through designing at atomic scale, which is something that has a physical measurement littler than one hundred nanometers, running from bunches of particles to many dimensional layers. Carbon Nanotubes (CNTs) with allotropes of carbon whose shapes are usually hollow and round possess some kinds of nanostructure.

For a graph *G*, the degree of a vertex *w* is the cardinality of edges incident to *w* and denoted by *dgr*(*s*). A molecular graph is a basic limited graph in which vertices mean the atoms and edges indicate the compound bonds in fundamental substance structures.

For a graph *G*, a topological index *T p*(*G*) is a value which can be obtained by a computing method from *G*. Moreover, if graphs *G* and *F* are isomorphic, then the result *T p*(*F*) = *T p*(*G*) holds. Wiener [4] initially figured out an idea for a topological index in the early years, and at that time, he took a shot at breaking point of paraffin. He defined this record to be the way number. Afterwards, such a concept was renamed the Wiener index. As we know, the Wiener record is the first posed index and it is one of the most attractive indices, from not only a hypothetical perspective but applications, and characterized as the total of separations among vertices in *G*, see for subtle elements [5].

The first Zagreb index, a very old topological index, was initiated in 1972 [6] and later many variations of Zagreb index were proposed, e.g., Shirdel et al. [7] in 2013 described a novel index under the name of "hyper-Zagreb index" and it was defined to be

$$HM(G) = \sum\_{sr \in E(G)} \left[ dgr(s) + dgr(r) \right]^2 \tag{1}$$

in [8], two new versions of Zagreb indices were put forward, which are the first multiple Zagreb index *PM*1(*G*) and the second multiple Zagreb index *PM*2(*G*). More precisely, they are formulated as follows.

$$PM\_1(G) = \prod\_{sr \in E(G)} [dgr(s) + dgr(r)]\tag{2}$$

$$PM\_2(G) = \prod\_{sr \in E(G)} [dgr(s) \times dgr(r)]\tag{3}$$

Some properties of the indices *PM*1(*G*), *PM*2(*G*) of specific chemical structures were investigated in [9].

To investigate more interesting properties of *PM*1(*G*), *PM*2(*G*) of a graph *G*, the first Zagreb Polynomial *<sup>M</sup>*1(*<sup>G</sup>*, *x*) and the second Zagreb Polynomial *<sup>M</sup>*2(*<sup>G</sup>*, *x*) are proposed [10,11] and put forward as

$$M\_1(G, \mathbf{x}) = \sum\_{sr \in E(G)} \mathbf{x}^{[dgr(s) + dgr(r)]} \tag{4}$$

$$M\_2(G, \mathbf{x}) = \sum\_{sr \in E(G)} \mathbf{x}^{[dgr(s) \times dgr(r)]} \tag{5}$$
