Received 11 December 2015; accepted 5 April 2016; published 8 April 2016
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.,  and  , Gao and Shi  and  , Xi and Gao  , Gao and Wang  -  , Gao and Farahani  , and Gao et al.,  for more details).
In our article, we only consider simple (molecular) graphs which are finite, loopless, and without multiple edges. Let be a graph in which the vertex set and edge set are expressed as and, respectively. Here, each edge can be regarded as the subset of with exactly two elements, and edge set consists of all such edges. Readers can refer Bondy and Mutry  for any notations and terminologies used but not clearly explained in our paper.
The first Zagreb index could be regarded as one of the oldest graph invariants which was defined in 1972 by Gutman and Trinajsti  as
where is the degree vertex v in G. Another alternative formulation for is denoted as
. And, the second Zagreb index was later introduced as
As degree-based topological indices, the multiplicative version of these Zagreb indices of a graph G is introduced by Gutman  , and Ghorbani and Azimi  as:
Here is the first multiplicative Zagreb index and is the second multiplicative Zagreb index. Several conclusions on these two classes of multiplicative Zagreb indices can be refered to Eliasi et al.,  , Xu et al.,  , and Farahani  and  .
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:, , polyhex zigzag and polyhex armchair, and the multiplicative Zagreb indices of these four classes of nanotubes are determined. Second, we compute the multiplicative Zagreb indices of dendrimer nanostar. At last, we calculate the multiplicative Zagreb indices of some special families of polyomino chains.
2. Multiplicative Zagreb Indices 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) and nanotubes; 2) zigzag and armchair.
2.1. Nanotubes Covered by C5 and C7
In this subsection, we discuss and nanotubes which consisting of cycles and (or it is a trivalent decoration constructed by and in turn, and thus called -net). It can cover either a cylinder or a torus.
The parameter p is denoted as the number of pentagons in the 1-st row of and. The vertices and edges in first four rows are repeated alternatively. In these nanotubes, and we set q as the number of such repetitions. For arbitrary, there exist 16p edges and 6p vertices in each period of which are adjacent at the end of the molecular structure. By simple computation, we check that and since there are 6p vertices with and other vertices with.
Furthermore, there are 8p vertices and 12p edges in any periods of. We get and since there are 5p vertices adjacent at the end of structure, and exists q repetition and 5p addition edges.
Let and be the minimum and maximum degree of graph G, respectively. In the whole following context, for any graph G, its vertex set and edge set are divided into several partitions:
for any i, , let;
for any j, , let;
for any k, , let.
Therefore, by omitting the single carbon atoms and the hydrogen, we infer two partitions and for and. Moreover, the edge set of and can be divided into the following three edge sets.
Now, we state the main results in this subsection.
Proof. First, considering nanotubes for arbitrary. By analyzing its structure, we have, , and. In terms of the definitions of multiplicative version of these Zagreb indices, we infer
Second, we consider nanotube for arbitrary. According to its chemical structure, we verify, , , , and. Therefore, by means of the definitions of multiplicative version of these Zagreb indices, we get
2.2. Two Classes of Polyhex Nanotubes
We study the multiplicative version of polyhex nanotubes: zigzag and armchair in this sub- section. We use parameter to denote the number of hexagons in the 1-st row of the and. Analogously, the positive integer n is used to express the number of hexagons in the 1-st column of the 2D-lattice of and. In view of structure analysis, we conclude and.
Clearly, the degree of vertex in polyhex nanotubes can’t exceed three. For nanotubes with any, we infer, , and. Moreover, for nanotube with any, we get, , , and. Therefore, the results stated as follows are obtained by means of above discussions and the definitions of multiplicative Zagreb indices.
3. Multiplicative Zagreb Indices of Dendrimer Nanostars
Dendrimer is a basic structure in nanomaterials. In this section, for any, is denoted as the n-th growth of dendrimer nanostar. We aim to determine multiplicative Zagreb indices of dendrimer nanostar (its structure can be referred to Figure 1 for more details).
This class of dendrimer nanostar has a core presented in Figure 1 and we call an element as a leaf. It is not difficult to check that a leaf is actually consisted of or chemically benzene, and is constituted by adding leafs in the n-th growth of. Therefore, there are in all leafs () in the dendrimer. The main contribution in this section can be stated as follows.
Figure 1. The structure of 2-dimensional of dendrimer nanostar.
Proof. Let be the number of vertices with degree i () in. In terms of hierarchy structural of, we deduce, and . Hence, by means of the induction on n with, and, we get and.
Set. We infer
Therefore, the expected results are obtained by the definition of the first and the second multiplicative Zagreb index.
4. Multiplicative Zagreb Indices of Polyomino Chains
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. In other words, it can be regarded as an edge-connected union of cells in the planar square lattice. For instance, polyomino chain is a special polyomino system in which the joining of the centers (denoted as the center of the i-th square) of its adjacent regular forms a path. Let be the set of polyomino chains with n squares. We have for each. is called a linear chain expressed as if the subgraph of induced by has exactly squares. Moreover, is called a zig-zag chain denoted as if the subgraph of induced by (all the vertices with degree larger than two) is a path has exactly edges.
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 to denote the length of S which is determined by the number of squares in S. Assume a polyomino chain consists of a sequence of segments with, and we denote for with property that. For arbitrary segment S in a polyomino chain, we have. Specially, we get and for a linear chain, and and for a zig-zag chain.
The theorems presented in the below reveal clearly how the multiplicative Zagreb indices of certain families of polyomino chain are expressed.
Theorem 4. Let, be the polyomino chains presented above. Then, we get
Proof. The results are obvious for, and we only focus on in the following discussion. It is not hard to check that.
For the polyomino chain, we obtain, and. By the definitions of multiplicative Zagreb indices, we have
By the same fashion, we yield
The expected results are got from the fact for.
Theorem 5. Let () be a polyomino chain with n squares and two segments which and. Then, we have
Proof. For, it is trivial. For, we obtain, , , , and. Therefore, by means of simply calculation, we obtain the desired results.
Theorem 6. Let be a polyomino chain with n squares and m segments () such that and. Then
Proof. For this chemical structure, we get, , , , and. Therefore, in view of the definitions of multiplicative Zagreb indices, we obtain the desired results.
The last two results obtained using similarly tricks.
Corollary 1. Let () be a polyomino chain with n squares and m segments () such that, or,. Then
Corollary 2. Let be a polyomino chain with n squares and m segments () such that (). Then
5. Conclusions and Further Work
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 are calculated; at last, we also discuss some families of polyomino chains. As multiplicative Zagreb indices can been used in QSPR/QSAR study and play a crucial role in analyzing both the boiling point and melting point for medicinal drugs and chemical compounds, the results obtained in our paper illustrate the promising prospects of application for medical, pharmacal, biological and chemical sciences.
A closely related concept of the Zagreb index is the Estrada index (see Shang  and  for more details) and the techniques used in our paper can be potentially applicable to the Estrada indices. The Estrada index of special chemical graph structures can be considered in the further works.
We thank all the reviewers for their constructive comments in improving the quality of this paper. Research is supported partially by NSFC (No. 11401519).
 Yan, L., Li, J.S. and Gao, W. (2014) Vertex PI Index and Szeged Index of Certain Special Molecular Graphs. The Open Biotechnology Journal, 8, 19-22.
 Gao, W. and Wang, W.F. (2014) Second Atom-Bond Connectivity Index of Special Chemical Molecular Structures. Journal of Chemistry, 2014, Article ID: 906254.
 Gao, W. and Wang, W.F. (2015) The Vertex Version of Weighted Wiener Number for Bicyclic Molecular Structures. Computational and Mathematical Methods in Medicine, 2015, Article ID: 418106.
 Gao, Y., Gao, W. and Liang, L. (2014) Revised Szeged Index and Revised Edge Szeged Index of Certain Special Molecular Graphs. International Journal of Applied Physics and Mathematics, 4, 417-425.
 Bondy, J.A. and Mutry, U.S.R. (2008) Graph Theory. Spring, Berlin.
 Gutman, I. and Trinajsti, N. (1972) Graph Theory and Molecular Orbitals. III. Total φ-Electron Energy of Alternant Hydrocarbons. Chemical Physics Letters, 17, 535-538.
 Xu, K. and Das, K.Ch. (2012) Trees, Unicyclic, and Bicyclic Graphs Extremal with Respect to Multiplicative Sum Zagreb Index. MATCH Communications in Mathematical and in Computer Chemistry, 68, 257-272.
 Shang, Y.L. (2015) Laplacian Estrada and Normalized Laplacian Estrada Indices of Evolving Graphs. PLoS ONE, 10, E0123426.
 Shang, Y.L. (2015) The Estrada Index of Evolving Graphs. Applied Mathematics and Computation, 250, 415-423.