Calculating of degree-based topological indices of nanostructures

Abstract A larger amount of studies reveal that there is strong inherent connection between the chemical characteristics of nanostructures and their molecular structures. Degree-based topological indices introduced on these chemical molecular structures can help material scientists better understand its chemical and biological features, thus they can make up for the lack of chemical experiments. In this paper, by means of edge dividing trick, we present several degree-based indices of special widely employed nanostructures: nanotubes, polyphenylene dendrimers, H-Naphtalenic nanotubes NPHX[m, n], nanotubes and PAMAM dendrimers.


Introduction
As the development of nanotechnology, more and more of nanomaterials are emerging every year. Thus, identification of the chemical properties of these nanomaterials has become more and more cumbersome. Fortunately, previous studies have shown that chemical characteristics of nanomaterials and their molecular structures are closely related. By defining the chemical topological indices to study indicators of these nanostructures can help researchers to determine their chemical properties, which make up the chemical experiments defects.
Specifically, the nanostructure is modelled as a graph, where each vertex represents an atom and each edge denotes a chemical bond between two atoms. Let G be a (molecular) graph with vertex set V(G) and edge set E(G). A topological index can be regarded as a real-valued function f: G→ ℝ + which maps each nanostructure to a real number. As numerical descriptors of the molecular structure yielded from the corresponding nanostructures, topological indices have been proofed several applications in nanoengineering, for example, QSPR/ QSAR study. In the past years, harmonic index, Wiener index, sum connectivity index were introduced to measure certain structural features of nanomolecules. There were several papers contributing to determine these topological indices of special molecular graph in chemical engineering (See Hosamani (2016), Gao & Wang, 2014, 2015, 2016, 2017, , and Gao, Farahani, & Shi, 2016;Gao, Siddiqui, Imran, Jamil, & Farahani, 2016;Gao, Wang, & Farahani, 2016) for more detail). The notations and terminologies used but not clearly defined in our article can be referred in book (Bondy & Mutry, 2008) written by Bondy and Mutry. Bollobas and Erdos (1998) defined the general Randic index which was stated as follows: where k is a real number and d(u) denotes the degree of vertex u in molecular graph G. Liu and Gutman (2007) determined the estimating for general Randic index and its special cases. Throughout, we always assume that k is a real number.

ARTICLE HISTORY
Received 19 april 2017 accepted 21 July 2017  introduced the general harmonic index as: If we take k = 1 in formula (3), then it becomes a normal harmonic index which was described by: Eliasi and Iranmanesh (2011) reported the ordinary geometric-arithmetic index (or, called general geometric-arithmetic index) as the extension of geometric-arithmetic index which was stated as follows: Clearly, GA (geometric-arithmetic) index is a special case of ordinary geometric-arithmetic index when k = 1. Azari and Iranmanesh (2011) proposed the generalized Zagreb index of molecular graph G expressed by: where t 1 and t 2 are arbitrary non-negative integers.
Several polynomials related to degree-based indices are also introduced. For instance, the first and the second Zagreb polynomials are expressed by: and respectively.
Moreover, the third Zagreb index and third Zagreb polynomial are denoted as: and The multiplicative version of first and second Zagreb indices were introduced by Gutman (2011) and Ghorbani and Azimi (2012) as follows: .
Several conclusions on PM 1 (G) and PM 2 (G)can be referred to Eliasi, Iranmanesh, and Gutma (2012) and Xu and Das (2012). Furthermore, Ranjini, Lokesha, and Usha (2013) re-defined the Zagreb indices, i.e., the redefined first, second and third Zagreb indices of a (molecular) graph G were manifested as follows: and respectively.
Although there have been several advances in distance-based indices of molecular graphs, the study of degree-based indices for special nanomolecular structures are still largely limited. In addition, as widespread and critical nanostructures, SC 5 C 7 [p, q] nanotubes, polyphenylene dendrimers, H-Naphtalenic nanotubes NPHX [m, n], TUC 4 [m, n] nanotubes and PAMAM dendrimers are widely used in medical science and material field. For these reasons, we give the exact expressions of above-mentioned degree-based indices for these nanostructures.
The rest of the context is arranged as follows: first, we present the degree-based indices of SC 5 C 7 p, q nanotubes; then, the nanostructure of polyphenylene dendrimers are considered; third, we focus on the H-Naphtalenic nanotubes NPHX [m, n]; the degreebased indices computation of TUC 4 [m, n] nanotubes are presented in Section 5; at last, we consider three kinds of PAMAM dendrimers: PD 1 [n], PD 2 [n] and DS 1 [n].

Degree-based indices of SC 5 C 7 [p, q] nanotubes
The purpose of this section is to manifest several degreebased indices of SC 5 C 7 [p, q] nanotubes. Actually, this nanotube is a kind of C 5 C 7 -net which is obtained by alternating C 5 and C 7 . This classical tiling of C 5 and C 7 can either cover a cylinder or a torus. A period of SC 5 C 7 [p, q] (here p is the number of heptagons in each row and q is the number of periods in whole lattice) is consisted of three rows (see Figure 1 for more details on i-th period). Clearly, there are 8p vertices in one period of the lattice, and thus | | | V SC 5 C 7 p, q | | | = 8pq. Using the similar fashion, there are 12p edges in one period and exists 2p extra edges joined to the end of this nanostructure. Therefore, we have The main technique in this paper to obtain the desired conclusion is edge dividing approach. Throughout this paper, we use the following notations for edge dividing.
Let δ(G) and Δ(G) be the minimum and maximum degree of G. We divide edge set E(G) and vertex set V(G) into several partitions: for any Now, we state the main conclusion in this section.
Theorem 1: Then, the result follows from the definitions of these degree-based indices. ✷

Remark 1:
From what we have deduced in Theorem 1, we yield that

Degree-based indices of polyphenylene dendrimers
The aim of this section is to show the degree-based indices of polyphenylene dendrimers D 4 [n] and D 2 [n], where n ∊ ℕ. These two molecular structures are widely appeared in the nanomaterials. The kernel structure of D 4 [n] and D 2 [n] can be referred to Figure 2. Additionally, the following Figure 3 present the D 2 [n] with three growth stages.
The main results in this section are manifested as follows: H SC 5 C 7 p, q = 4pq − 1 10 p, M 3 SC 5 C 7 p, q = 6p.

Degree-based indices of H-Naphtalenic nanotubes
In this part, we consider the degree-based indices of H-Naphtalenic nanotubes NPHX [m, n] (here m is denoted as the number of pairs of hexagons in first row and n is represented as the number of alternative hexagons in a column) which is a trivalent decoration with sequence of C 6 , C 6 , C 4 , C 6 , C 6 , C 4 , … in the first row and a sequence of C 6 , C 8 , C 6 , C 8 , … in the other rows. That is to say, this nanolattice can be regarded as a plane tiling of C 4 , C 6 and C 8 . Thus, such type of tiling can either cover a cylinder or a torus (see Figure 4 as

Proof:
By observation of TUC 4 [m, n] nanotubes, we ensure that its edge set can be divided into three partitions: