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Introduction and Motivations
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 et al. Because of its wide engineering applications, many works contributed to determining the indices of special molecular graphs (See Yan et al., [24], Gao and Shi [7], Gao and Wang [8], [9] and [10] and Jamil et al. [13] for more details).
In our article, we only consider simple (molecular) graphs which are finite, loopless, and without multiple edges. Let G = (V(G),E(G)) be a graph in which the vertex set and edge set are expressed as V(G) and E(G), respectively. Readers can refer Bondy and Mutry [3] for any notations and terminologies used but not clearly explained in our paper.
The first Zagreb index can be regarded as one of the oldest graph invariants which was defined in 1972 by Gutman and Trinajsti [11] as
where d(v) is the degree vertex v in G. Another alternative formulation for M1(G) is denoted as Σe=uv∈E(G)(d(u) + d(v)). And, the second Zagreb index was later introduced as
where α ∈ R. Obviously, $\begin{array}{}
\displaystyle
M_1^0(G) = |V(G)|,M_1^1(G) = 2|E(G)|
\end{array}$ and $\begin{array}{}
\displaystyle
M_1^2 \equiv {M_1}(G)
\end{array}$. Furthermore, we have
So the output of ReZG1(G) is always n, simply the vertices in the graph G. Note that their identities for ReZG1(G) in our paper is calculated in terms of the other graph parameters like p,q.
As degree-based topological indices, the redefined version of Zagreb indices of a graph G introduced by Ranjini et al., [19], and Usha et al., [20].
The Redefined first Zagreb index of a molecular graph G is defined by
The Redefined third Zagreb index was also independently defined by Mansour and Song [15]. Moreover, the generalized version was presented in [23].
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 redefined Zagreb indices of nano structures has been largely limited. Furthermore, nanotube, nanostar, polyomino chain and benzenoid series are critical and widespread molecular structures which have been widely applied in medical science, chemical engineering and pharmaceutical fields (see Mirzargar [18], Ashrafi and Karbasioun [1], Manuel et al., [16], Baca et al., [4], DeBorde et al., [5], Matsuno et al., [17], Vilela et al., [22], Velichko, Nosich [21], Haslam and Raeymaekers [12], Ashrafi and Karbasioun [2]). Also, these structures are the basic and primal structures of other more complicated chemical molecular structures and nano materials. Based on these grounds, we have attracted tremendous academic and industrial interests in determining the redefined Zagreb indices of special family of nanotube and nanostar from a computation point of view. In addition, the redefined Zagreb indices of polyomino chain and benzenoid series are considered.
The main contribution of our paper is two-folded. First, we focus on four classes of nanotubes: VC5C7[p,q], HC5C7[p,q], polyhex zigzag TUZC6 and polyhex armchair TUAC6, and theredefined Zagreb indices of these four classes of nanotubes are determined. Second, we compute the redefined Zagreb indices of dendrimer nanostar D3[n]. As supplement, we calculate the redefined Zagreb indices of some special families of polyomino chains and benzenoid series.
Main Results and Proofs
redefined Zagreb indices of nanotubes
The purpose of this part is to yield the redefined Zagreb indices of certain special classes nanotubes. Our work in this part can be divided into two parts: (1) VC5C7[p,q] and HC5C7[p,q] nanotubes; (2) zigzag TUZC6 and armchair TUAC6.
Nanotubes Covered by C5 and C7
In this subsection, we discuss VC5C7[p,q] and HC5C7[p,q] nanotubes which consisting of cycles C5 and C7 (or it is a trivalent decoration constructed by C5 and C7 in turn, and thus called C5C7-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 VC5C7[p,q] and HC5C7[p,q]. 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 p,q ∈ ℕ, there exist 16p edges and 6p vertices in each period of VC5C7[p,q] which are adjacent at the end of the molecular structure. By simple computation, we check that |V(VC5C7[p,q])| = 16pq + 6p and |E(VC5C7[p,q])| = 24pq + 6p since there are 6p vertices with d(v) = 2 and other 16pq vertices with d(v) = 3.
Furthermore, there are 8p vertices and 12p edges in any periods of HC5C7[p,q]. We get |V(HC5C7[p,q])| = 8pq + 5p and |E(HC5C7[p,q])| = 12pq + 5p since there are 5p vertices adjacent at the end of structure, and exists q repetition and 5p addition edges.
Let δ and Δ be the minimum and maximum degree of graph G, respectively. In the whole following context, for any graph G, its vertex set V(G) and edge set E(G) are divided into several partitions:
for any i, 2δ(G) ≤ i 2Δ(G), let Ei = {e = uv ∈ E(G)|d(u) + d(v) = i};
for any j, (δ)2 ≤ j (Δ)2, let $\begin{array}{}
\displaystyle
E_j^* = \{ e = uv \in E(G)|d(u)d(v) = j\}
\end{array}$;
for any k, δ ≤ Δ k ≤ Δ, let Vk = {v ∈ V(G)|d(v) = k}.
Therefore, by omitting the single carbon atoms and the hydrogen, we infer two partitions V2 = {v ∈ V(G)|d(v) = 2} and V3 = {v ∈ V(G)|d(v) = 3} for VC5C7[p,q] and HC5C7[p,q]. Moreover, the edge set of VC5C7[p,q] and HC5C7[p,q] can be divided into the following three edge sets.
Proof. First, considering nanotubes VC5C7[p,q] for arbitrary p,q ∈ ℕ. By analyzing its structure, we have |V2| = 6p, |V3| = 16pq, $\begin{array}{}
\displaystyle
\left| {{E_5}\left| = \right|E_6^*} \right| = 12p
\end{array}$ and $\begin{array}{}
\displaystyle
|{E_6}| = |E_9^*| = 24pq - 6p
\end{array}$. In terms of the definitions of redefined Zagreb indices, we infer
Second, we consider nanotube HC5C7[p,q] for arbitrary p,q ∈ ℕ. According to its chemical structure, we verify |V2| = 5p, |V3| = 8pq, $\begin{array}{}
\displaystyle
|{E_4}| = |E_4^*| = p,|{E_5}| = |E_6^*| = 8p
\end{array}$, and $\begin{array}{}
\displaystyle
|{E_6}| = |E_9^*| = 12pq - 4p
\end{array}$. Therefore, by means of the definitions of redefined Zagreb indices, we infer
We study the redefined Zagreb indices of polyhex nanotubes: zigzag TUZC6 and armchair TUAC6 in this subsection. We use parameter m ∈ ℕ to denote the number of hexagons in the 1-st row of the TUZC6 and TUAC6. Analogously, the positive integer n is used to express the number of hexagons in the 1-st column of the 2D-lattice of TUZC6 and TUAC6. In view of structure analysis, we conclude |V(TUZC6)| = |V(TUAC6)| = 2m(n + 1) and |E(TUZC6)| = |E(TUAC6)| = 3mn + 2m.
Clearly, the degree of vertex in polyhex nanotubes can’t exceed three. For nanotubes TUZC6[m,n] with any m,n ∈ ℕ, we infer |V2| = 2m, |V3| = 2mn, $\begin{array}{}
\displaystyle
|{E_5}| = |E_6^*| = 4m
\end{array}$ and $\begin{array}{}
\displaystyle
|{E_6}| = |E_9^*| = 3mn - 2m
\end{array}$. Moreover, for nanotube TUAC6[m,n] with any m,n ∈ ℕ, we get |V2| = 2m, |V3| = 2mn, $\begin{array}{}
\displaystyle
|{E_4}| = |E_4^*| = m,|{E_5}| = |E_6^*| = 2m
\end{array}$ and $\begin{array}{}
\displaystyle
|{E_6}| = |E_9^*| = 3mn - m
\end{array}$. Therefore, the results stated as follows are obtained by means of above discussions and the definitions of redefined Zagreb indices.
Dendrimer is a basic structure in nanomaterials. In this section, for any n ∈ ℕ, D3[n] is denoted as the n-th growth of dendrimer nanostar. We aim to determine the redefined Zagreb indices of dendrimer nanostar D3[n] (its structure can be referred to Figure 1 for more details).
Fig. 1
The structure of 2-dimensional of dendrimer nanostar D3[n].
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 C6 or chemically benzene, and D3[n] is constituted by adding 3 · 2n leafs in the n-th growth of D3[n − 1]. Therefore, there are in all 3 · 2n+1 − 3 leafs (C6) in the dendrimer D3[n]. The main contribution in this section can be stated as follows.
Proof. Let Vi[n] be the number of vertices with degree i(i ∈ {1,2,3,4}) in D3[n]. In terms of hierarchy structural of D3[n], we deduce V1[n + 1] = 2V1[n] = 3 · 2n+1, V2[n + 1] = V2[n] + 12 · 2n+1 and V3[n + 1] = V3[n] + 6 · 2n+1 + V1[n]. Hence, by means of the induction on n with V1[0] = 3, V2[0] = 12 and V3[0] = 7, we get V2[n + 1] = 12(2n+2 − 1) and V3[n + 1] = 15 · 2n+1 − 8.
Set $\begin{array}{}
\displaystyle
E_3^* = \{ e = uv \in E({D_3}[n])|d(u) = 3,d(v) = 1\}
\end{array}$. We infer
Therefore, the expected results are obtained by the definition of the first, the second and the third redefined Zagreb indices.
More Findings
Redefined Zagreb indices of polymino 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 C4. 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 ci as the center of the i-th square) of its adjacent regular forms a path c1c2 ··· cn. Let 𝒫n be the set of polyomino chains with n squares. We have |E(PCn)| = 3n + 1 for each PCn ∈ 𝒫n. PCn is called a linear chain expressed as LCn if the subgraph of PCn induced by V3 has exactly n − 2 squares. Moreover, PCn is called a zig-zag chain denoted as ZCn if the subgraph of PCn induced by V>2 (all the vertices with degree larger than two) is a path has exactly n − 1 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 l(S) 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 S1,S2,···,Sm with m ≥ 1, and we denote l(Si) = li for i ∈ {1,2,···,m} with property that $\begin{array}{}
\displaystyle
{\rm{\Sigma }}_{i = 1}^m{l_i} = n + m - 1
\end{array}$. For arbitrary segment S in a polyomino chain, we have 2 ≤ l(S) ≤ n. Specially, we get m = 1 and l1 = n for a linear chain LCn, and m = n − 1 and li = 2 for a zig-zag chain ZCn.
The theorems presented in the below reveal clearly how the redefined Zagreb indices of certain families of polyomino chain are expressed.
Theorem 4
Let LCn, ZCn be the polyomino chains presented above. Then, we get
Proof. The results are obvious for n = 1, and we only focus on n ≥ 2 in the following discussion. It is not hard to check that |E(LCn)| = |E(ZCn)| = 3n + 1.
For the polyomino chain LCn, we obtain $\begin{array}{}
\displaystyle
|{E_4}| = |E_4^*| = 2,|{E_5}| = |E_6^*| = 4
\end{array}$ and $\begin{array}{}
\displaystyle
|{E_6}| = |E_9^*| = 3n - 5
\end{array}$. By the definitions of redefined Zagreb indices, we have
The expected results are got from the fact m = n − 1 for ZCn.
Theorem 5
Let$\begin{array}{}
\displaystyle
PC_n^1(n \ge 3)
\end{array}$be a polyomino chain with n squares and two segments which l1 = 2 and l2 = n − 1. Then, we have
Proof. For n = 3, it is trivial. For n ≥ 4, we obtain $\begin{array}{}
\displaystyle
|{E_4}| = |E_4^*| = 2,|{E_5}| = |E_6^*| = 5,|E_8^*| = 1,|{E_7}| = |E_{12}^*| = 3,|E_9^*| = 3n - 10
\end{array}$ and $\begin{array}{}
\displaystyle
|{E_6}| = |E_9^*| + |E_8^*| = 3n - 9
\end{array}$. Therefore, by means of simply calculation, we obtain the desired results.
Theorem 6
Let$\begin{array}{}
\displaystyle
PC_n^2
\end{array}$be a polyomino chain with n squares and m segments S1,S2,···,Sm (m ≥ 3) such that l1 = lm = 2 and l2,··· ,lm−1 ≥ 3. Then
Proof. For this chemical structure, we get $\begin{array}{}
\displaystyle
|{E_4}| = |E_4^*| = 2,|{E_5}| = |E_6^*| = 2m,|E_8^*| = 2,|{E_7}| = |E_{12}^*| = 4m - 6,|E_9^*| = 3n - 6m + 3
\end{array}$ and $\begin{array}{}
\displaystyle
|{E_6}| = |E_9^*| + |E_8^*| = 3n - 6m + 5
\end{array}$. Therefore, in view of the definitions of redefined Zagreb indices, we obtain the desired results.
The last two results obtained using similarly tricks.
Theorem 7
Let$\begin{array}{}
\displaystyle
PC_n^3(n \ge 4)
\end{array}$be a polyomino chain with n squares and m segments S1, S2,··· ,Sm (m ≥ 3) such that l1 = 2, l2,··· ,lm ≥ 3 or lm = 2, l1,l2 ··· ,lm−1 ≥ 3. Then
Let$\begin{array}{}
\displaystyle
PC_n^4
\end{array}$be a polyomino chain with n squares and m segments S1,S2,··· ,Sm (m ≥ 3) such that li ≥ 3 (i = {1,··· ,m}). Then
Redefined Zagreb indices of two classes of benzenoid series
The last part of our paper is to determine the redefined Zagreb indices of two classes of benzenoid series. First, we consider circumcoronene series of benzenoid Hk. When k = 1,2,3, the structures are presented in Figure 2.
Fig. 2
The structure of Hk when k = 1,2,3.
Thus, this family of circumcoronene homologous series of benzenoid is consisted several copy of benzene C6 on circumference, the more details for this structure can refer to Figure 3.
Fig. 3
The circumcoronene series of benzenoid Hk.
Clearly, its vertex set can be divided into two part: V2 and V3 such that |V2| = 6k and |V3| = 6k(k − 1). Moreover, by simply calculation, we have $\begin{array}{}
\displaystyle
|{E_4}| = |E_4^*| = 6,|{E_6}| = |E_9^*| = 9{k^2} - 15k + 6
\end{array}$ and $\begin{array}{}
\displaystyle
|{E_5}| = |E_6^*| = 12(k - 1)
\end{array}$. Hence, we immediately deduce the following conclusion.
Next, we consider capra-designed planar benzenoid series Cak(C6) (the structure can refer to Farahani and Vlad [6] for more details). By means of intermediate results presented in Farahani and Vlad [6], we present the redefined Zagreb indices of Cak(C6) which are stated as follows.
The purpose of this paper is to discuss the redefined Zagreb indices of several nano structures, and these molecular graphs we considered here are fundamentally and commonly used in chemical and nano engineering. Specifically, the main contributions in this report can be concluded into two aspects: first, we compute the redefined Zagreb indices of four classes of nanotubes; then, the redefined Zagreb indices of dendrimer nanostars D3[n] are calculated. As supplement, we also discuss some families of polyomino chains and benzenoid series. As redefined 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, chemical and nanosciences.
