Nano properties analysis via fourth multiplicative ABC indicator calculating

Wei Gao; Weifan Wang; Darko Dimitrov; Yiqiao Wang

doi:10.1016/j.arabjc.2017.12.024

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11 (

); 793-801

doi:

10.1016/j.arabjc.2017.12.024

Nano properties analysis via fourth multiplicative ABC indicator calculating

Wei Gao^{^a,⁎}, Weifan Wang^{^b}, Darko Dimitrov^{^c}, Yiqiao Wang^{^d}

a School of Information Science and Technology, Yunnan Normal University, Kunming 650500, China

b Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

c Hochschule für Technik und Wirtschaft Berlin, Wilhelminenhofstraße 75A, D-12459 Berlin, Germany

d School of Management, Beijing University of Chinese Medicine, Beijing 100029, China

⁎Corresponding author. gaowei@ynnu.edu.cn (Wei Gao)

Received: 2017-8-30, Accepted: 2017-12-21, Published: 2018-09-01

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Peer review under responsibility of King Saud University.

Abstract

In the field of nanoscience, there are a large number of new nanomaterials produced in the laboratory every year. These nanomaterials need to go through a lot of tests in engineering application before, to determine their physical, chemical and nanomedicine properties. The work demands a large number of experimental personnel, equipment and reagents, and it’s time-consuming as well. In the theoretical nanoscience, the various features can be obtained through the computation of topological index on nano molecular structures. In this paper, we study the fourth multiplicative atom-bond connectivity indices of some special molecular structures which commonly appeared in the compound of nanomaterials, and their specific expressions are given. The results yielded in this work will be a guidance for the practical nanoscience applications.

Keywords

Theoretical nanoscience

Molecular structure

Fourth multiplicative atom-bond connectivity index

Dendrimer

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1 Introduction

With the rapid development of experimental methods, there are a large number of new nanomaterials from the laboratory emerging every year. These nanomaterials need to be based on a variety of tests and make sure that they are useful before entering the market, which brings new challenges to nanomaterial testing and regulation. Every year it requires a lot of manpower, material and financial resources to carry out nano related properties test. With the development of theoretical nanoscience, the researchers found that there is a close relationship between the chemical and biological characteristics and the molecular structure of nanomaterial itself. Hence, it emerges as a new branch of theoretical chemistry, and it relies on the calculation of chemistry index to determine the characteristics of nanomaterials. The mathematical model can be stated as follows: each atom is regarded as vertices, and chemical bond between atoms can be considered an edge between the vertices, and then chemical molecular structure of drug can be expressed by a molecular graph. By defining the topological index on the molecular graph, and corresponding chemical properties of nanomaterials can be studied by the index calculation. The advantage of this approach to understand the nature of the nanomaterial is that it does not require laboratory equipment and reagents, thus it saves the cost and receives the favor of underdeveloped areas.

Throughout our paper, let G =  $(V (G), E (G))$ be a molecular graph with vertex set $V (G)$ and edge set $E (G)$ , and then a topological index is regarded as a function f: G $\to R^{+}$ . In the past 40 years, several degree-based or distance-based indices like harmonic index, Wiener index, Gutman index, PI index, Randic index, and sum connectivity index et al. are defined and used in the nanoscience engineering applications. Some important contributions on distance-based and degree-based indices of special molecular structures can be referred to Harishchandra and Ramane (2016), and Gao et al. (2016a,b,c,d,e, 2017a,b,c,d).

Estrada et al. (1998) defined the atom-bond connectivity index (shortly, ABC index) which was sated as ${ABC}_{1} (G) = \sum_{uv \in E (G)} \sqrt{\frac{d (u) + d (v) - 2}{d (u) d (v)}} .$

Very recently, Kulli (2016) introduced the first multiplicative atom-bond connectivity index as follows: ${ABC}_{1} II (G) = \prod_{uv \in E (G)} \sqrt{\frac{d (u) + d (v) - 2}{d (u) d (v)}} .$

As a kind of important topological index, many articles contributed to various kinds of atom-bond connectivity indices. The fourth atom-bond connectivity index of circumcoronene series of benzenoid is calculated by Farahani (2013). Goubko et al. (2015) showed a counterexample which contradicts the main conclusion of previous work. The atom bond connectivity index of an infinite class of nanostar dendrimers and several graphs are computed by Ahmadi and Sadeghimehr (2010), Sukor et al. (2017), Basheer et al. (2017), Rahman et al. (2017). The atom bond connectivity index of two families of nanostar dendrimers is obtained by Husin et al. (2013). The atom–bond connectivity index of quasi-tree graphs is learned by Dehghan-Zadeh and Ashrafi (2014). Dehghan-Zadeh et al. (2014) raised the upper bound of atom–bond connectivity index for the class of tetracyclic graphs. Farahani (2013) determined the fourth atom-bond connectivity index of V-phenylenic nanotori and nanotubes. Dimitrov (2013) presented an efficient calculating technique for trees with smallest atom-bond connectivity index. Ashrafi et al. (2015) deduced the maximum value of atom-bond connectivity index of cactus graphs with given order of molecular graph. Das et al. (2012) derived the upper bounds and Nordhaus–Gaddum type conclusions for the atom bond connectivity index.

Moreover, Kulli (2017) defined the fourth multiplicative atom-bond connectivity index which was formulated as follows: ${ABC}_{4} II (G) = \prod_{uv \in E (G)} \sqrt{\frac{S (u) + S (v) - 2}{S (u) S (v)}},$ where $S (v) = \sum_{uv \in E (G)} d (u)$ .

Although there have been several recent advances in developing topological indices for various structures of the molecular nanomaterials, the study of biomedicine properties of more nano structures has been largely limited. Furthermore, as widespread and critical structures in nanomaterials, the applications of dendrimers, $C_{5} C_{7}$ net and carbon nanocones are frequently appeared in nanoscience engineering. It inspires us to study the fourth multiplicative atom-bond connectivity indices of these common structures in nanomaterials.

The main contribution in this work is reflected in the following aspects: (1) the fourth multiplicative atom-bond connectivity index of dendrimers are determined; (2) the expressions of the fourth multiplicative atom-bond connectivity index of $C_{5} C_{7}$ net are raised; (3) the formulas of the fourth multiplicative atom-bond connectivity index of carbon nanocones are also obtained.

2 Main results and proofs

In this section, we present the main results and their detailed proofs. The main technique to determine the conclusions is edge dividing which divides the edge set into several subsets according to the value of $S (u)$ and $S (v)$ .

2.1

2.1 The fourth multiplicative atom-bond connectivity index of dendrimers

In this subsection, we determine the fourth multiplicative atom-bond connectivity index of dendrimers which is widely appeared in the nano structures. More nanoscience engineering applications on different kinds of dendrimers can be referred in Cevik et al. (2016), Heredero-Bermejo et al. (2016), Worley et al. (2016), Vacas-Cordoba et al. (2016), Rivero-Buceta et al. (2015), Ozcan and Sezginturk (2015), Heredero-Bermejo et al. (2015), Gothwal et al. (2015), and Sepulveda-Crespo et al. (2015), Halin et al. (2017), Hassan et al. (2017), Ismail and Hanafiah, (2017).

First, we discuss the ${ABC}_{4} II$ of dendrimers denoted by ${NS}_{2} [n]$ . Using the results raised in Alikhani et al. (2014), the number of vertices and edges in ${NS}_{2} [n]$ are $16 \times 2^{n} - 4$ and $18 \times 2^{n} - 5$ , respectively. The structure of ${NS}_{2} [3]$ can be shown in the Fig. 1.

Theorem 1

Let $n \in N$ , then the fourth multiplicative atom-bond connectivity index for dendrimer ${NS}_{2} [n]$ is given by ${ABC}_{4} II ({NS}_{2} [n]) = {(\sqrt{\frac{8}{25}})}^{2^{n + 1} + 2} {(\sqrt{\frac{3}{10}})}^{6 \times 2^{n}} {(\sqrt{\frac{5}{18}})}^{6 \times 2^{n} - 12} {(\sqrt{\frac{3}{8}})}^{2 \times 2^{n}} {(\sqrt{\frac{7}{20}})}^{2 \times 2^{n}} \sqrt{\frac{12}{49}} {(\sqrt{\frac{2}{7}})}^{4} .$

Proof

The dendrimer ${NS}_{2} [n]$ has edge dividing with the form presented in Table 1. Thus, according to the definition of the fourth multiplicative atom-bond connectivity index, we infer ${ABC}_{4} II ({NS}_{2} [n]) = \prod_{uv \in E ({NS}_{2} [n])} \sqrt{\frac{S (u) + S (v) - 2}{S (u) S (v)}} = {(\sqrt{\frac{5 + 5 - 2}{5 \times 5}})}^{2^{n + 1} + 2} {(\sqrt{\frac{5 + 6 - 2}{5 \times 6}})}^{6 \times 2^{n}} {(\sqrt{\frac{6 + 6 - 2}{6 \times 6}})}^{6 \times 2^{n} - 12} {(\sqrt{\frac{4 + 4 - 2}{4 \times 4}})}^{2 \times 2^{n}} {(\sqrt{\frac{5 + 4 - 2}{5 \times 4}})}^{2 \times 2^{n}} \times \sqrt{\frac{7 + 7 - 2}{7 \times 7}} {(\sqrt{\frac{5 + 7 - 2}{5 \times 7}})}^{4} .$

Hence, the desired result is obtained after simplification. □

Table 1 Edge dividing of

{NS}_{2} [n]

based on

S (u)

and

S (v)

$(S (u), S (v))$ where $uv \in E ({NS}_{2} [n])$	Number of edges
$(4, 4)$	$2 \times 2^{n}$
$(5, 4)$	$2 \times 2^{n}$
$(5, 5)$	$2 \times 2^{n} + 2$
$(5, 6)$	$6 \times 2^{n}$
$(7, 7)$	$1$
$(5, 7)$	$4$
$(6, 6)$	$6 \times 2^{n} - 12$

Next, we consider another type of dendrimers ${NS}_{3} [n]$ with $n ⩾ 1$ . By Alikhani et al. (2014), Samad et al. 2017, Halim et al. (2017), Aziz and Hanafiah, (2017), ${NS}_{3} [n]$ contains $18 \times 2^{n} - 12$ vertices and $21 \times 2^{n} - 15$ edges. The molecular strcture of ${NS}_{3} [n]$ can refer to Fig. 2.

Theorem 2

For $n ⩾ 1$ , then the fourth multiplicative atom-bond connectivity index of dendrimer ${NS}_{3} [n]$ is ${ABC}_{4} II ({NS}_{3} [n]) = {(\sqrt{\frac{3}{8}})}^{3 \times 2^{n}} {(\sqrt{\frac{7}{20}})}^{3 \times 2^{n}} {(\sqrt{\frac{2}{7}})}^{3 \times 2^{n}} {(\sqrt{\frac{11}{42}})}^{9 \times 2^{n} - 12} {(\sqrt{\frac{12}{49}})}^{3 (2^{n} - 1)} .$

Proof

For ${NS}_{3} [n]$ , we have edges of the form $(4, 4)$ , $(4, 5)$ , $(5, 7)$ , $(6, 7)$ and $(7, 7)$ . In light of Table 2 and the definition of the fourth multiplicative atom-bond connectivity index, we deduce ${ABC}_{4} II ({NS}_{3} [n]) = \prod_{uv \in E ({NS}_{3} [n])} \sqrt{\frac{S (u) + S (v) - 2}{S (u) S (v)}} = {(\sqrt{\frac{4 + 4 - 2}{4 \times 4}})}^{3 \times 2^{n}} {(\sqrt{\frac{5 + 4 - 2}{5 \times 4}})}^{3 \times 2^{n}} {(\sqrt{\frac{5 + 7 - 2}{5 \times 7}})}^{3 \times 2^{n}} {(\sqrt{\frac{6 + 7 - 2}{6 \times 7}})}^{9 \times 2^{n} - 12} {(\sqrt{\frac{7 + 7 - 2}{7 \times 7}})}^{3 (2^{n} - 1)} .$

Thus, the expected result is yielded after simplification. □

Table 2 Edge dividing for

{NS}_{3} [n]

based on

S (u)

and

S (v)

$(S (u), S (v))$ where $uv \in E ({NS}_{3} [n])$	Number of edges
$(4, 4)$	$3 \times 2^{n}$
$(4, 5)$	$3 \times 2^{n}$
$(5, 7)$	$3 \times 2^{n}$
$(6, 7)$	$9 \times 2^{n} - 12$
$(7, 7)$	$3 \times 2^{n} - 3$

Next, we discuss the polyphenylene dendrimers. Fig. 3 presents the structure of polyphenylene dendrimers $D_{4} [n]$ of generations $G_{2}$ with $2$ growth stages.

Theorem 3

Let $n ⩾ 1$ , then the fourth multiplicative atom-bond connectivity index of polyphenylene dendrimers is ${ABC}_{4} II (D_{4} [n]) = \frac{12}{49} {(\sqrt{\frac{7}{20}})}^{24 \cdot 2^{n} - 16} {(\sqrt{\frac{3}{8}})}^{24 \cdot 2^{n} - 16} {(\frac{4}{9})}^{8 \cdot 2^{n} - 8} {(\frac{2 \sqrt{2}}{5})}^{8 \cdot 2^{n} - 8} \times {(\sqrt{\frac{5}{24}})}^{8 \cdot 2^{n} - 8} {(\frac{1}{2})}^{8 \cdot 2^{n} - 8} {(\sqrt{\frac{2}{7}})}^{40 \cdot 2^{n} - 32} {(\sqrt{\frac{13}{56}})}^{8 \cdot 2^{n} - 8} {(\frac{\sqrt{2}}{3})}^{12 \cdot 2^{n} - 12} .$

Proof

The $D_{4} [n]$ has edges with the form $(4, 4), (4, 5), (5, 5), (5, 7), (6, 8), (7, 7), (7, 8), (7, 9), (8, 9)$ and $(9, 9)$ . In view of Table 3, we get ${ABC}_{4} II (D_{4} [n]) = \prod_{uv \in E (D_{4} [n])} \sqrt{\frac{S (u) + S (v) - 2}{S (u) S (v)}} = {(\sqrt{\frac{4 + 5 - 2}{4 \times 5}})}^{24 \cdot 2^{n} - 16} {(\sqrt{\frac{4 + 4 - 2}{4 \times 4}})}^{24 \cdot 2^{n} - 16} {(\sqrt{\frac{9 + 9 - 2}{9 \times 9}})}^{8 \cdot 2^{n} - 8} {(\sqrt{\frac{7 + 7 - 2}{7 \times 7}})}^{2} {(\sqrt{\frac{5 + 5 - 2}{5 \times 5}})}^{8 \cdot 2^{n} - 8} \times {(\sqrt{\frac{8 + 9 - 2}{8 \times 9}})}^{8 \cdot 2^{n} - 8} {(\sqrt{\frac{6 + 8 - 2}{6 \times 8}})}^{8 \cdot 2^{n} - 8} {(\sqrt{\frac{5 + 7 - 2}{5 \times 7}})}^{40 \cdot 2^{n} - 32} {(\sqrt{\frac{7 + 8 - 2}{7 \times 8}})}^{8 \cdot 2^{n} - 8} {(\sqrt{\frac{7 + 9 - 2}{7 \times 9}})}^{12 \cdot 2^{n} - 12} .$

After an easy simplification, we get the result. □

The polyphenylene dendrimers D 4 [ n ] .

Table 3 Edge dividing for

D_{4} [n]

based on

S (u)

and

S (v)

$(S (u), S (v))$ where $uv \in E (D_{4} [n])$	Number of edges
$(4, 4)$	$24 \times 2^{n} - 16$
$(4, 5)$	$24 \times 2^{n} - 16$
$(5, 5)$	$8 \times 2^{n} - 8$
$(5, 7)$	$40 \times 2^{n} - 32$
$(6, 8)$	$8 \times 2^{n} - 8$
$(7, 7)$	2
$(7, 8)$	$8 \times 2^{n} - 8$
$(7, 9)$	$12 \times 2^{n} - 12$
$(8, 9)$	$8 \times 2^{n} - 8$
$(9, 9)$	$8 \times 2^{n} - 8$

At the last of this subsection, we compute the fourth multiplicative atom-bond connectivity index of polyphenylene dendrimer which is denoted by $D_{2} [n]$ . Fig. 4 shows the graph of polyphenylene dendrimer $D_{2} [n]$ of the generations $G_{3}$ with $3$ growth stages, and Table 4 presents its edge set dividing.

Theorem 4

Let $n ⩾ 1$ , then the fourth multiplicative atom-bond connectivity index of polyphenylene dendrimers $D_{2} [n]$ is ${ABC}_{4} II (D_{2} [n]) = \frac{2 \sqrt{3}}{7} {(\sqrt{\frac{11}{42}})}^{12} {(\sqrt{\frac{13}{56}})}^{8 \cdot 2^{n} - 8} {(\frac{1}{2})}^{8 \cdot 2^{n} - 8} {(\sqrt{\frac{2}{7}})}^{16 (2^{n} - 1) + 16} \times {(\sqrt{\frac{7}{20}})}^{32 \cdot 2^{n} - 32} {(\frac{\sqrt{6}}{4})}^{32 \cdot 2^{n} - 32} {(\sqrt{\frac{5}{24}})}^{8 \cdot 2^{n} - 8} {(\frac{4}{9})}^{8 \cdot 2^{n} - 8} {(\frac{\sqrt{2}}{3})}^{12 \cdot 2^{n} - 12} {(\frac{2 \sqrt{2}}{5})}^{16 (2^{n} - 1) - 16} .$

Proof

In terms of Table 4, we know that edge set of $D_{2} [n]$ can be divided into several subsets with the form $(4, 4), (4, 5), (5, 5), (5, 7), (6, 7), (6, 8), (7, 7), (7, 8), (7, 9), (8, 9)$ and $(9, 9)$ . Hence, by means of the definition of the fourth multiplicative atom-bond connectivity index, we yield ${ABC}_{4} II (D_{2} [n]) = \prod_{uv \in E (D_{2} [n])} \sqrt{\frac{S (u) + S (v) - 2}{S (u) S (v)}} = \sqrt{\frac{7 + 7 - 2}{7 \times 7}} {(\sqrt{\frac{7 + 6 - 2}{7 \times 6}})}^{12} {(\sqrt{\frac{7 + 8 - 2}{7 \times 8}})}^{8 \cdot 2^{n} - 8} {(\sqrt{\frac{6 + 8 - 2}{6 \times 8}})}^{8 \cdot 2^{n} - 8} {(\sqrt{\frac{7 + 5 - 2}{7 \times 5}})}^{16 (2^{n} - 1) + 16} \times {(\sqrt{\frac{5 + 4 - 2}{5 \times 4}})}^{32 \cdot 2^{n} - 32} {(\sqrt{\frac{4 + 4 - 2}{4 \times 4}})}^{32 \cdot 2^{n} - 32} {(\sqrt{\frac{8 + 9 - 2}{8 \times 9}})}^{8 \cdot 2^{n} - 8} {(\sqrt{\frac{9 + 9 - 2}{9 \times 9}})}^{8 \cdot 2^{n} - 8} \times {(\sqrt{\frac{9 + 7 - 2}{9 \times 7}})}^{12 \cdot 2^{n} - 12} {(\sqrt{\frac{5 + 5 - 2}{5 \times 5}})}^{16 (2^{n} - 1) - 16} .$

After simplification, we get the expression presented in Theorem 4. □

The polyphenylene dendrimers D 2 [ n ] .

Table 4 Edge dividing for

D_{2} [n]

based on

S (u)

and

S (v)

$(S (u), S (v))$ where $uv \in E (D_{2} [n])$	Number of edges
$(4, 4)$	$32 \cdot 2^{n} - 32$
$(4, 5)$	$32 \cdot 2^{n} - 32$
$(5, 5)$	$16 (2^{n} - 1) - 16$
$(5, 7)$	$16 (2^{n} - 1) + 16$
$(6, 7)$	$12$
$(6, 8)$	$8 \cdot 2^{n} - 8$
$(7, 7)$	$1$
$(7, 8)$	$8 \cdot 2^{n} - 8$
$(7, 9)$	$12 \cdot 2^{n} - 12$
$(8, 9)$	$8 \cdot 2^{n} - 8$
$(9.9)$	$8 \cdot 2^{n} - 8$

2.2

2.2 The fourth multiplicative atom-bond connectivity index of ${SiO}_{4}$ layer structure

A silicate (SiO₄) molecule consists of one silicon ion and four oxygen ions, as presented in Fig. 5.

Relied on their polymerization, silicates form different networks but the networks are usually motivated by the ${SiO}_{4}$ layer structure. Four silicate molecules join together to form a sort of octagon, as shown in Fig. 6.

Finally, these octagons join together with other octagons to form the ${SiO}_{4}$ layer structure. We define the rows as the number of lines of vertical octagons and columns as the number of lines of horizontal octagons. The number of rows and columns are denoted by p and q respectively. The concept is extended to any general number of rows and columns. To give a better idea, Fig. 7 gives the ${SiO}_{4}$ network with $p = 4$ and $q = 5$ .

Theorem 5

The fourth multiplicative atom-bond connectivity index of ${SiO}_{4}$ layer structure is given by ${ABC}_{4} II ({SiO}_{4}) = {(\sqrt{\frac{1}{3}})}^{2 q + 2 p + 4} {(\frac{1}{2})}^{2 q + 2 p + 4 pq} .$

Proof

By analyzing the molecular structure of ${SiO}_{4}$ layer structure, we get its edge dividing as follows:

Therefore, in view of the definition of the fourth multiplicative atom-bond connectivity index and Table 5, we derive ${ABC}_{4} II ({SiO}_{4}) = \prod_{uv \in E ({SiO}_{4})} \sqrt{\frac{S (u) + S (v) - 2}{S (u) S (v)}} = {(\sqrt{\frac{4 + 6 - 2}{4 \times 6}})}^{2 q + 2 p + 4} {(\sqrt{\frac{8 + 6 - 2}{8 \times 6}})}^{2 q + 2 p + 4 pq} .$

The Theorem 5 is proved. □

The molecular structure of SiO 4 ( 4 , 5 ) .

Table 5 Edge dividing for

{SiO}_{4}

layer structure based on

S (u)

and

S (v)

$(S (u), S (v))$ where $uv \in E ({SiO}_{4})$	Number of edges
$(4, 6)$	$2 p + 2 q + 4$
$(6, 8)$	$2 p + 2 q + 4 pq$

2.3

2.3 The fourth multiplicative atom-bond connectivity index of $C_{5} C_{7}$ networks

In recent years, the nano structures are widely used in nanomedicine, see Elahian et al. (2017), Sharma et al. (2014), Mohanpuria et al. (2008), Dar et al. (2013), Turner et al. (2008), Anil Kumar et al. (2007), Ramamurthy et al. (2013), Tran and Webster (2011), Bhainsa and D'Souza (2006), Singh et al. (2016), Griffiths et al. (2015), and Jokerst et al. (2011), Khan et al. (2017), Zaidi et al. (2017), Rahman et al. (2017). A $C_{5} C_{7}$ net is a trivalent decoration which is constructed by alternating $C_{5}$ and $C_{7}$ . It can cover either a cylinder or a torus. In this part, we discuss the three kinds of $C_{5} C_{7}$ net: ${HC}_{5} C_{7}$ , ${VC}_{5} C_{7}$ and ${VAC}_{5} C_{7}$ . These nano structures are defined by Diudea and can be widely used in nanomedicine.

Theorem 6

The fourth multiplicative atom-bond connectivity index of ${VC}_{5} C_{7}$ is ${ABC}_{4} II ({VC}_{5} C_{7}) = {(\frac{2 \sqrt{2}}{5})}^{p} {(\sqrt{\frac{11}{42}})}^{10 p} {(\frac{\sqrt{2}}{3})}^{24 pq - 14 p} {(\frac{4}{9})}^{24 pq - 14 p} .$

Proof

By analyzing the molecular structure of ${VC}_{5} C_{7}$ , we yield its edge dividing as follows:

Hence, in terms of the definition of the fourth multiplicative atom-bond connectivity index and Table 6, we get ${ABC}_{4} II ({VC}_{5} C_{7}) = \prod_{uv \in E ({VC}_{5} C_{7})} \sqrt{\frac{S (u) + S (v) - 2}{S (u) S (v)}} = {(\sqrt{\frac{5 + 5 - 2}{5 \times 5}})}^{p} {(\sqrt{\frac{6 + 7 - 2}{6 \times 7}})}^{10 p} {(\sqrt{\frac{7 + 9 - 2}{7 \times 9}})}^{24 pq - 14 p} {(\sqrt{\frac{9 + 9 - 2}{9 \times 9}})}^{24 pq - 14 p}$

Therefore, we get the desired conclusion. □

Theorem 7

The fourth multiplicative atom-bond connectivity index of ${HC}_{5} C_{7}$ is ${ABC}_{4} II ({HC}_{5} C_{7}) = {(\frac{1}{2})}^{8 p} {(\sqrt{\frac{7}{32}})}^{24 pq - 10 p} {(\sqrt{\frac{5}{24}})}^{24 pq - 10 p} {(\frac{4}{9})}^{24 pq - 10 p} .$

Proof

By analyzing the molecular structure of ${HC}_{5} C_{7}$ , we yield its edge dividing as follows:

Therefore, in view of Table 7 and the definition of the fourth multiplicative atom-bond connectivity index, we derive ${ABC}_{4} II ({HC}_{5} C_{7}) = \prod_{uv \in E ({HC}_{5} C_{7})} \sqrt{\frac{S (u) + S (v) - 2}{S (u) S (v)}} = {(\sqrt{\frac{6 + 8 - 2}{6 \times 8}})}^{8 p} {(\sqrt{\frac{8 + 8 - 2}{8 \times 8}})}^{24 pq - 10 p} {(\sqrt{\frac{8 + 9 - 2}{8 \times 9}})}^{24 pq - 10 p} {(\sqrt{\frac{9 + 9 - 2}{9 \times 9}})}^{24 pq - 10 p}$

The conclusion is proved. □

Theorem 8

The fourth multiplicative atom-bond connectivity index of ${VAC}_{5} C_{7}$ is ${ABC}_{4} II ({VAC}_{5} C_{7}) = {(\frac{2 \sqrt{2}}{5})}^{2 p} {(\sqrt{\frac{11}{40}})}^{8 p} {(\frac{1}{2})}^{8 p} {(\sqrt{\frac{7}{32}})}^{24 pq - 13 p + 3} {(\sqrt{\frac{5}{24}})}^{24 pq - 13 p + 3} {(\frac{4}{9})}^{24 pq - 13 p + 3} .$

Proof

By analyzing the molecular structure of ${VAC}_{5} C_{7}$ , we yield its edge dividing as follows:

In light of the definition of the fourth multiplicative atom-bond connectivity index and Table 8, we derive ${ABC}_{4} II ({VAC}_{5} C_{7}) = \prod_{uv \in E ({VAC}_{5} C_{7})} \sqrt{\frac{S (u) + S (v) - 2}{S (u) S (v)}} = {(\sqrt{\frac{5 + 5 - 2}{5 \times 5}})}^{2 p} {(\sqrt{\frac{5 + 8 - 2}{5 \times 8}})}^{8 p} {(\sqrt{\frac{6 + 8 - 2}{6 \times 8}})}^{8 p} {(\sqrt{\frac{8 + 8 - 2}{8 \times 8}})}^{24 pq - 13 p + 3} \times {(\sqrt{\frac{8 + 9 - 2}{8 \times 9}})}^{24 pq - 13 p + 3} {(\sqrt{\frac{9 + 9 - 2}{9 \times 9}})}^{24 pq - 13 p + 3} .$

Thus, we complete the proving of Theorem 8. □

Table 6 Edge dividing for

{VC}_{5} C_{7}

based on

S (u)

and

S (v)

$(S (u), S (v))$ where $uv \in E ({VC}_{5} C_{7})$	Number of edges
$(5, 5)$	$p$
$(6, 7)$	$10 p$
$(7, 9)$	$24 pq - 14 p$
$(9, 9)$	$24 pq - 14 p$

Table 7 Edge dividing for

{HC}_{5} C_{7}

based on

S (u)

and

S (v)

$(S (u), S (v))$ where $uv \in E ({HC}_{5} C_{7})$	Number of edges
$(6, 8)$	$8 p$
$(8, 8)$	$24 pq - 10 p$
$(8, 9)$	$24 pq - 10 p$
$(9, 9)$	$24 pq - 10 p$

Table 8 Edge dividing for

{VAC}_{5} C_{7}

based on

S (u)

and

S (v)

$(S (u), S (v))$ where $uv \in E ({VAC}_{5} C_{7})$	Number of edges
$(5, 5)$	$2 p$
$(5, 8)$	$8 p$
$(6, 8)$	$8 p$
$(8, 8)$	$24 pq - 13 p + 3$
$(8, 9)$	$24 pq - 13 p + 3$
$(9, 9)$	$24 pq - 13 p + 3$

2.4

2.4 The fourth multiplicative atom-bond connectivity index of carbon nanocones $C_{n} [k]$

As the last conclusion part of our paper, we determine the fourth multiplicative atom-bond connectivity index of carbon nanocones $C_{n} [k]$ (it contain $n {(k + 1)}^{2}$ vertices and $\frac{n}{2} (k + 1) (3 k + 2)$ edges) which is widely used in the nano engineering. As an example, the Fig. 8 presents the molecular structure for carbon nanocones $C_{n} [k]$ with $n = 4$ and $k = 2$ .

Theorem 9

The fourth multiplicative atom-bond connectivity index of $C_{n} [k]$ is ${ABC}_{4} II (C_{n} [k]) = {(\frac{2 \sqrt{2}}{5})}^{n} {(\sqrt{\frac{2}{7}})}^{2 n} {(\sqrt{\frac{11}{42}})}^{2 (k - 1) n} {(\frac{\sqrt{2}}{3})}^{nk} {(\frac{4}{9})}^{\frac{nk}{2} (3 k - 1)} .$

Proof

By analyzing the molecular structure of $C_{n} [k]$ , we yield its edge dividing as follows:

According to the definition of the fourth multiplicative atom-bond connectivity index and Table 9, we have ${ABC}_{4} II (C_{n} [k]) = \prod_{uv \in E (C_{n} [k])} \sqrt{\frac{S (u) + S (v) - 2}{S (u) S (v)}} = {(\sqrt{\frac{5 + 5 - 2}{5 \times 5}})}^{n} {(\sqrt{\frac{5 + 7 - 2}{5 \times 7}})}^{2 n} {(\sqrt{\frac{6 + 7 - 2}{6 \times 7}})}^{2 (k - 1) n} {(\sqrt{\frac{7 + 9 - 2}{7 \times 9}})}^{nk} {(\sqrt{\frac{9 + 9 - 2}{9 \times 9}})}^{\frac{nk}{2} (3 k - 1)} .$

Hence, the last conclusion is proved. □

Table 9 Edge dividing for

C_{n} [k]

based on

S (u)

and

S (v)

$(S (u), S (v))$ where $uv \in E (C_{n} [k])$	Number of edges
$(5, 5)$	$n$
$(5, 7)$	$2 n$
$(6, 7)$	$2 (k - 1) n$
$(7, 9)$	$nk$
$(9, 9)$	$\frac{nk}{2} (3 k - 1)$

3 Conclusion

This paper proposed the fourth multiplicative atom-bond connectivity index of dendrimers, $C_{5} C_{7}$ net and carbon nanocones by analyzing certain molecular structural, degree computation and mathematical derivation. The conclusion also demonstrates the wide and promising application prospects in nanoscience engineering.

Conflict of interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgements

We thank the reviewers for their constructive comments in improving the quality of this paper. This work was supported in part by the National Natural Science Foundation of China (11761083, 11771402, 11671053).

References

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Ismail H., Hanafiah M.M., . Management of end-of-life electrical and electronic products: the challenges and the potential solutions for management enhancement in developing countries context. Acta Sci. Malay.. 2017;1(2):05-08.
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Khan I.U., Sajid S., Javed A., Sajid S., Shah S.U., Khan S.N., Ullah K., . Comparative diagnosis of typhoid fever by polymerase chain reaction and widal test in southern districts (Bannu, Lakki Marwat And D.I.Khan) of Khyber Pakhtunkhwa, Pakistan. Acta Sci. Malay.. 2017;1(2):12-15.
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Kulli V.R., . Two new multiplicative atom bond connectivity indices. Ann. Pure Appl. Math.. 2017;13(1):1-7.
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Introduction

Main results and proofs

The fourth multiplicative atom-bond connectivity index of dendrimers
The fourth multiplicative atom-bond connectivity index of SiO 4 layer structure
The fourth multiplicative atom-bond connectivity index of C 5 C 7 networks
The fourth multiplicative atom-bond connectivity index of carbon nanocones C n [ k ]

Conclusion

Conflict of interests

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[1] Ahmadi M.B., Sadeghimehr M., . Atom bond connectivity index of an infinite class NS₁[n] of dendrimer nanostars. Optoelectron. Adv. Mater. – Rapid Commun.. 2010;4(7):1040-1042.
[Google Scholar]

[2] Alikhani S., Hasni R., Arif N.E., . On the atom-bond connectivity index of some families of dendrimers. J. Comput. Theor. Nanosci.. 2014;11(8):1802-1805.
[Google Scholar]

[3] Anil Kumar S., Abyaneh M.K., Gosavi S.W., Kulkarni S.K., Pasricha R., Ahmad A., . Nitrate reductase-mediated synthesis of silver nanoparticles from AgNO3. Biotechnol. Lett.. 2007;29:439-445.
[Google Scholar]

[4] Ashrafi A.R., Dehghan-Zadeh T., Habibi N., . Extremal atom-bond connectivity index of cactus graphs. Commun. Korean Math. Soc.. 2015;30(3):283-295.
[Google Scholar]

[5] Aziz N.I.H.A., Hanafiah M.M., . The potential of palm oil mill effluent (POME) as a renewable energy source. Acta Sci. Malay.. 2017;1(2):09-11.
[Google Scholar]

[6] Basheer A.O., Hanafiah M.M., Abdulhasan M.J., . A study on water quality from langat river. Selangor. Acta Sci. Malay.. 2017;1(2):01-04.
[Google Scholar]

[7] Bhainsa K.C., D'Souza S.F., . Extracellular biosynthesis of silver nanoparticles using the fungus Aspergillus fumigatus. Colloids Surf. B. 2006;47:160-164.
[Google Scholar]

[8] Cevik E., Bahar O., Senel M., Abasiyanik M.F., . Construction of novel electrochemical immunosensor for detection of prostate specific antigen using ferrocene-PAMAM dendrimers. Biosens. Bioelectron.. 2016;86:1074-1079.
[Google Scholar]

[9] Dar M.A., Ingle A., Rai M., . Enhanced antimicrobial activity of silver nanoparticles synthesized by cryphonectria sp evaluated singly and in combination with antibiotics. Nanomedicine. 2013;9:105-110.
[Google Scholar]

[10] Das K.C., Gutman I., Furtula B., . On atom–bond connectivity index. Filomat. 2012;26(4):733-738.
[Google Scholar]

[11] Dehghan-Zadeh T., Ashrafi A.R., . Atom–bond connectivity index of quasi-tree graphs. Rendiconti del Circolo Matematico di Palermo. 2014;63:347-354.
[Google Scholar]

[12] Dehghan-Zadeh T., Ashrafi A.R., Habibi N., . Maximum values of atom–bond connectivity index in the class of tetracyclic graphs. J. Appl. Math. Comput.. 2014;46:285-303.
[Google Scholar]

[13] Dimitrov D., . Efficient computation of trees with minimal atom-bond connectivity index. Appl. Math. Comput.. 2013;224:663-670.
[Google Scholar]

[14] Elahian F., Reiisi S., Shahidi A., Mirzaei S.A., . High-throughput bioaccumulation, biotransformation, and production of silver and selenium nanoparticles using genetically engineered pichia pastoris. Nanomedicine: nanotechnology. Biology Med.. 2017;13:853-861.
[Google Scholar]

[15] Estrada E., Torres L., Rodriguez L., Gutman I., . An atombond connectivity index: modelling the enthalpy of formation of alkanes. Indian J. Chem. A. 1998;37(10):849-855.
[Google Scholar]

[16] Farahani M.R., . Computing atom-bond connectivity (ABC₄) index for circumcoronene series of benzenoid. J. Adv. Chem.. 2013;2(1):68-72.
[Google Scholar]

[17] Farahani M.R., . Computing fourth atom-bond connectivity index of V-phenylenic nanotubes and nanotori. Acta Chim. Slov.. 2013;60(2):429-432.
[Google Scholar]

[18] Gao W., Baig A.Q., Ali H., Sajjad W., Farahani M.R., . Margin based ontology sparse vector learning algorithm and applied in biology science. Saudi J. Biol. Sci.. 2017;24(1):132-138.
[Google Scholar]

[19] Gao W., Farahani M.R., Shi L., . The forgotten topological index of some drug structures. Acta Med. Mediterr.. 2016;32:579-585.
[Google Scholar]

[20] Gao W., Siddiqui M.K., . Molecular descriptors of nanotube, oxide, silicate, and triangulene networks. J. Chem. 2017 Article ID 6540754, 10 pages
[CrossRef] [Google Scholar]

[21] Gao W., Siddiqui M.K., Imran M., Jamil M.K., Farahani M.R., . Forgotten topological index of chemical structure in drugs. Saudi Pharm. J.. 2016;24:258-264.
[Google Scholar]

[22] Gao W., Wang W.F., . The eccentric connectivity polynomial of two classes of nanotubes. Chaos Solitons Fract.. 2016;89:290-294.
[Google Scholar]

[23] Gao W., Wang W.F., . The fifth geometric arithmetic index of bridge graph and carbon nanocones. J. Differ. Eq. Appl. 2017
[CrossRef] [Google Scholar]

[24] Gao W., Wang W.F., Farahani M.R., . Topological indices study of molecular structure in anticancer drugs. J. Chem. 2016 Article ID 3216327, 8 pages
[CrossRef] [Google Scholar]

[25] Gao W., Wang W.F., Jamil M.K., Farahani M.R., . Electron energy studying of molecular structures via forgotten topological index computation. J. Chem. 2016 Article ID 1053183, 7 pages
[CrossRef] [Google Scholar]

[26] Gao W., Yan L., Shi L., . Generalized Zagreb index of polyomino chains and nanotubes. Optoelectron. Adv. Mater. – Rapid Commun.. 2017;11(1–2):119-124.
[Google Scholar]

[27] Gothwal A., Kesharwani P., Gupta U., Khan I., Amin M.C.I.M., Banerjee S., Iyer A.K., . Dendrimers as an effective nanocarrier in cardiovascular disease. Curr. Pharm. Des.. 2015;21(30):4519-4526.
[Google Scholar]

[28] Goubko M., Magnant C., Nowbandegani P.S., Gutman I., . ABC index of trees with fixed number of leaves. MATCH Commun. Math. Comput. Chem.. 2015;74:697-702.
[Google Scholar]

[29] Griffiths P.C., Cattoz B., Ibrahim M.S., Anuonye J.C., . Probing the interaction of nanoparticles with mucin for drug delivery applications using dynamic light scattering. Eur. J. Pharm. Biopharm.. 2015;97(A):218-222.
[Google Scholar]

[30] Halim H., Abdullah R., Nor M.J.M., Aziz H.A., Rahman N.A., . Comparison between measured traffic noise in klang valley, malaysia and existing prediction models. Eng. Heritage J.. 2017;1(2):10-14.
[Google Scholar]

[31] Halin N.I.A., Huyop F., Hamid T.H.T.A., Halim K.B.A., Hamid A.A.A., . In silico binding interactions of dehalogenase (Dehe) with various haloalkanoic acids. Sci. Heritage J.. 2017;1(1):04-06.
[Google Scholar]

[32] Harishchandra S., Ramane R.B., . Note on forgotten topological index of chemical structure in drugs. Appl. Math. Nonlinear Sci.. 2016;1(2):369-374.
[Google Scholar]

[33] Hassan S.R., Zaman N.Q., Dahlan I., . Influence of seed loads on start up of modified anaerobic hybrid baffled (MAHB) reactor treating recycled paper wastewater. Eng. Heritage J.. 2017;1(2):05-09.
[Google Scholar]

[34] Heredero-Bermejo I., Copa-Patino J.L., Soliveri J., Fuentes-Paniagua E., de la Mata F.J., Gomez R., Perez-Serrano J., . Evaluation of the activity of new cationic carbosilane dendrimers on trophozoites and cysts of acanthamoeba polyphaga. Parasitol. Res.. 2015;114(2):473-486.
[Google Scholar]

[35] Heredero-Bermejo I., Sanchez-Nieves J., Soliveri J., Gomez R., de la Mata F.J., Copa-Patino J.L., Perez-Serrano J., . In vitro anti-Acanthamoeba synergistic effect of chlorhexidine and cationic carbosilanedendrimers against both trophozoite and cyst forms. Int. J. Pharm.. 2016;509(1–2):1-7.
[Google Scholar]

[36] Husin N.M., Hasni R., Arif N.E., . Atom-bond connectivity and geometric arithmetic indices of dendrimer nanostars. Aust. J. Basic Appl. Sci.. 2013;7(9):10-14.
[Google Scholar]

[37] Ismail H., Hanafiah M.M., . Management of end-of-life electrical and electronic products: the challenges and the potential solutions for management enhancement in developing countries context. Acta Sci. Malay.. 2017;1(2):05-08.
[Google Scholar]

[38] Jokerst J.V., Lobovkina T., Zare R.N., Gambhir S.S., . Nanoparticle PEGylation for imaging and therapy. Nanomedicine (Lond). 2011;6(4):715-728.
[Google Scholar]

[39] Khan I.U., Sajid S., Javed A., Sajid S., Shah S.U., Khan S.N., Ullah K., . Comparative diagnosis of typhoid fever by polymerase chain reaction and widal test in southern districts (Bannu, Lakki Marwat And D.I.Khan) of Khyber Pakhtunkhwa, Pakistan. Acta Sci. Malay.. 2017;1(2):12-15.
[Google Scholar]

[40] Kulli V.R., . Multiplicative connectivity indices of certain nanotubes. Ann. Pure Appl. Math.. 2016;12(2):169-176.
[Google Scholar]

[41] Kulli V.R., . Two new multiplicative atom bond connectivity indices. Ann. Pure Appl. Math.. 2017;13(1):1-7.
[Google Scholar]

[42] Mohanpuria P., Rana N.K., Yadav S.K., . Biosynthesis of nanoparticles: technological concepts and future applications. J. Nanopart. Res.. 2008;10:507-517.
[Google Scholar]

[43] Ozcan H.M., Sezginturk M.K., . Detection of parathyroid hormone using an electrochemical impedance biosensor based on PAMAM dendrimers. Biotechnol. Prog.. 2015;31(3):815-822.
[Google Scholar]

[44] Rahman N.A., Halim H., Gotoh H., Harada E., . Validation of microscopic dynamics of grouping pedestrians behavior: from observation to modeling and simulation. Eng. Heritage J.. 2017;1(2):15-18.
[Google Scholar]

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[Google Scholar]

[46] Ramamurthy C., Sampath K.S., Arunkumar P., Kumar M.S., Sujatha V., Premkumar K., . Green synthesis and characterization of selenium nanoparticles and its augmented cytotoxicity with doxorubicin on cancer cells. Bioprocess Biosyst. Eng.. 2013;36:1131-1139.
[Google Scholar]

[47] Rivero-Buceta E., Doyaguez E.G., Colomer I., Quesada E., Mathys L., Noppen S., Liekens S., Camarasa M.J., Perez-Perez M.J., Balzarini J., San-Felix A., . Tryptophan dendrimers that inhibit HIV replication, prevent virus entry and bind to the HIV envelope glycoproteins gp120 and gp41. Eur. J. Med. Chem.. 2015;106:34-43.
[Google Scholar]

[48] Samad N.S.A., Amid A., Jimat D.N., Shukor N.A.A., . Isolation and identification of halophilic bacteria producing halotolerant protease. Sci. Heritage J.. 2017;1(1):07-09.
[Google Scholar]

[49] Sepulveda-Crespo D., Sanchez-Rodriguez J., Serramia M.J., Gomez R., De La Mata F.J., Jimenez J.L., Munoz-Fernandez M.A., . Triple combination of carbosilane dendrimers, tenofovir and maraviroc as potential microbicide to prevent HIV-1 sexual transmission. Nanomedicine. 2015;10(6):899-914.
[Google Scholar]

[50] Sharma G., Sharma A.R., Bhavesh R., Park J., Ganbold B., Nam J.S., . Biomolecule-mediated synthesis of selenium nanoparticles using dried Vitis vinifera (raisin) extract. Molecules. 2014;19:2761-2770.
[Google Scholar]

[51] Singh P., Kim Y.J., Zhang D., Yang D.C., . Biological synthesis of nanoparticles from plants and microorganisms. Trends Biotechnol.. 2016;34:588-599.
[Google Scholar]

[52] Sukor N.S.A., Jarani N., Fisal S.F.M., . Analysis of passengers’ access and egress characteristics to the train station. Eng. Heritage J.. 2017;1(2):01-04.
[Google Scholar]

[53] Tran T.A., Webster T.J., . Selenium nanoparticles inhibit staphylococcus aureus growth. Nanomedicine. 2011;6:1553-1558.
[Google Scholar]

[54] Turner M., Golovko V.B., Vaughan O.P., Abdulkin P., Berenguer-Murcia A., Tikhov M.S., . Selective oxidation with dioxygen by gold nanoparticle catalysts derived from 55-atom clusters. Nature. 2008;454:981-983.
[Google Scholar]

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Nano properties analysis via fourth multiplicative ABC indicator calculating

Abstract

Keywords

Theoretical nanoscience

Molecular structure

Fourth multiplicative atom-bond connectivity index

Dendrimer

1 Introduction

2 Main results and proofs

2.1 The fourth multiplicative atom-bond connectivity index of dendrimers

2.2 The fourth multiplicative atom-bond connectivity index of ${SiO}_{4}$ layer structure

2.3 The fourth multiplicative atom-bond connectivity index of $C_{5} C_{7}$ networks

2.4 The fourth multiplicative atom-bond connectivity index of carbon nanocones $C_{n} [k]$

3 Conclusion

Conflict of interests

Acknowledgements

References

Suggested read for related articles:

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Nano properties analysis via fourth multiplicative ABC indicator calculating

Abstract

Keywords

Theoretical nanoscience

Molecular structure

Fourth multiplicative atom-bond connectivity index

Dendrimer

1 Introduction

2 Main results and proofs

2.1 The fourth multiplicative atom-bond connectivity index of dendrimers

2.2 The fourth multiplicative atom-bond connectivity index of SiO 4 layer structure

2.3 The fourth multiplicative atom-bond connectivity index of C 5 C 7 networks

2.4 The fourth multiplicative atom-bond connectivity index of carbon nanocones C n [ k ]

3 Conclusion

Conflict of interests

Acknowledgements

References

Suggested read for related articles:

2.2 The fourth multiplicative atom-bond connectivity index of ${SiO}_{4}$ layer structure

2.3 The fourth multiplicative atom-bond connectivity index of $C_{5} C_{7}$ networks

2.4 The fourth multiplicative atom-bond connectivity index of carbon nanocones $C_{n} [k]$