The Multiplicative Zagreb Indices of Nanostructures and Chains ()
Received 11 December 2015; accepted 5 April 2016; published 8 April 2016

1. Introduction
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., [1] and [2] , Gao and Shi [3] and [4] , Xi and Gao [5] , Gao and Wang [6] - [8] , Gao and Farahani [9] , and Gao et al., [10] 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 [11] 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 [12] 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 [13] , and Ghorbani and Azimi [14] 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., [15] , Xu et al., [16] , and Farahani [17] and [18] .
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.
(or
):
;
(or
):
;
(or
),
and
.
Now, we state the main results in this subsection.
Theorem 1.
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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
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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
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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.
Theorem 2.
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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.
Theorem 3.
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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
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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
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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
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By the same fashion, we yield
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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
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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
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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
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Corollary 2. Let
be a polyomino chain with n squares and m segments
(
) such that
(
). Then
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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 [19] and [20] 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.
Acknowledgements
We thank all the reviewers for their constructive comments in improving the quality of this paper. Research is supported partially by NSFC (No. 11401519).