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In theoretical chemistry, the researchers use graph models to express the structure of molecular, and the Zagreb indices and multiplicative Zagreb indices defined on molecular graph G are applied to measure the chemical characteristics of compounds and drugs. In this paper, we present the exact expressions of multiplicative Zagreb indices for certain important chemical structures like nanotube, nanostar and polyomino chain.

For the past 40 years, chemical graph theory, as an important branch of both computational chemistry and graph theory, has attracted much attention and the results obtained in this field have been applied in many chemical and pharmaceutical engineering applications. In these frameworks, the molecular is represented as a graph in which each atom is expressed as a vertex and covalent bounds between atoms are represented as edges between vertices. Topological indices were introduced to determine the chemical and pharmaceutical properties. Such indices can be regarded as score functions which map each molecular graph to a non-negative real number. There were many famous degree-based or distance-based indices such as Wiener index, PI index, Zagreb index, atom-bond connectivity index, Szeged index and eccentric connectivity index. Because of its wide engineering applications, many works contributed to determining the indices of special molecular graphs (see Yan et al., [

In our article, we only consider simple (molecular) graphs which are finite, loopless, and without multiple edges. Let

The first Zagreb index could be regarded as one of the oldest graph invariants which was defined in 1972 by Gutman and Trinajsti [

where

As degree-based topological indices, the multiplicative version of these Zagreb indices of a graph G is introduced by Gutman [

Here

There have been many advances in Wiener index, Szeged index, PI index, and other degree-based or distance- based indices of molecular graphs, while the study of the first and second multiplicative Zagreb index of special chemical structures has been largely limited. Furthermore, nanotube, nanostar and polyomino chain are critical and widespread molecular structures which have been widely applied in medical science, chemical engineering and pharmaceutical fields. Also, these structures are the basic and primal structures of other more complicated chemical molecular structures. Based on these grounds, we have attracted tremendous academic and industrial interests in determining the multiplicative Zagreb indices of special family of nanotube, nanostar and polyomino chain from a computation point of view.

The contribution of our paper is three-folded. First, we focus on four classes of nanotubes:

The purpose of this section is to yield the multiplicative Zagreb indices of certain special classes nanotubes. Our work in this part can be divided into two parts: 1)

In this subsection, we discuss

The parameter p is denoted as the number of pentagons in the 1-st row of

Furthermore, there are 8p vertices and 12p edges in any periods of

Let

for any i,

for any j,

for any k,

Therefore, by omitting the single carbon atoms and the hydrogen, we infer two partitions

Now, we state the main results in this subsection.

Theorem 1.

Proof. First, considering nanotubes

Second, we consider nanotube

We study the multiplicative version of polyhex nanotubes: zigzag

Clearly, the degree of vertex in polyhex nanotubes can’t exceed three. For nanotubes

Theorem 2.

Dendrimer is a basic structure in nanomaterials. In this section, for any

This class of dendrimer nanostar has a core presented in

Theorem 3.

Proof. Let

Set

Therefore, the expected results are obtained by the definition of the first and the second multiplicative Zagreb index.

From the perspective of mathematical, a polyomino system can be considered as a finite 2-connected plane graph in which each interior cell is surrounded by a

The branched or angularly connected squares in a polyomino chain are called a kink, and a maximal linear chain in a polyomino chain including the kinks and terminal squares at its end is called a segment represented by S. We use

The theorems presented in the below reveal clearly how the multiplicative Zagreb indices of certain families of polyomino chain are expressed.

Theorem 4. Let

Proof. The results are obvious for

For the polyomino chain

By the same fashion, we yield

The expected results are got from the fact

Theorem 5. Let

Proof. For

Theorem 6. Let

Proof. For this chemical structure, we get

The last two results obtained using similarly tricks.

Corollary 1. Let

Corollary 2. Let

The purpose of this paper is to discuss the multiplicative Zagreb indices of several chemical structures, and these molecular graphs we consider here are fundamentally and commonly used in chemical engineering. Spe- cifically, the contributions in this report can be concluded into three aspects: first, we compute the multiplicative Zagreb indices of four classes of nanotubes; then, the multiplicative Zagreb indices of dendrimer nanostars

A closely related concept of the Zagreb index is the Estrada index (see Shang [

We thank all the reviewers for their constructive comments in improving the quality of this paper. Research is supported partially by NSFC (No. 11401519).

Wei Gao,Mohammad Reza Farahani,M. R. Rajesh Kanna, (2016) The Multiplicative Zagreb Indices of Nanostructures and Chains. Open Journal of Discrete Mathematics,06,82-88. doi: 10.4236/ojdm.2016.62008