Topological aspects of 2D structures of trans-



Pd
(


NH


2


)
S




 lattice and a metal-organic superlattice

Mehran Azeem; Adnan Aslam; Zahid Iqbal; Muhammad Ahsan Binyamin; Wei Gao

doi:10.1016/j.arabjc.2020.102963

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Original article

2020

:14;

202103

doi:

10.1016/j.arabjc.2020.102963

Topological aspects of 2D structures of trans- $Pd ({NH}_{2}) S$ lattice and a metal-organic superlattice

Mehran Azeem^{^a}, Adnan Aslam^{^a}, Zahid Iqbal^{^b}, Muhammad Ahsan Binyamin^{^c}, Wei Gao^{^d}

a Department of Natural Sciences and Humanities, University of Engineering and Technology, Lahore (RCET), Pakistan

b School of Natural Sciences, National University of Sciences and Technology, Sector H-12, Islamabad, Pakistan

c Department of Mathematics, Govt College University, Faislabad, Pakistan

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

Received: 2020-4-24, Accepted: 2020-12-20,

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

Abstract

Abstract

In quantitative structure property (QSPR) and quantitative structure activity (QSAR) relationships studies, topological indices are utilized to associate the biological activity of underline structures with their physical properties like strain energy, distortion, melting point, boiling point, and stability, etc. In these studies, the degree based topological indices have obtained center position between various kind of descriptors, due to the simplicity of finding and the rapidity with which these estimations can be accomplished. In this paper, we compute the general Randic, first and second Zagreb, Atom Bond Connectivity(ABC) and Geometric Arithmetic(GA) indices of the $2 D$ structures of trans- $Pd - ({NH}_{2}) S$ lattice and a metal–organic superlattice. Furthermore, we present the precise formulas of the fifth version of Geometric Arithmetic index and the fourth version of Atom-Bond Connectivity index of these lattices.

Keywords

Trans-pd-(NH2) lattice

Metal-organic superlattice

Degree-based topological descriptors

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1 Introduction

Two-dimensional $(2 D)$ covalent-organic frameworks and $2 D$ materials have earned remarkable interest after the invention of graphene (Novoselov et al., 2004). Like graphene, many other $2 D$ materials such as metal carbides (Lukatskaya et al., 2013), $2 D$ boron nitride nanomesh (Corso et al., 2004), silicene (Vogt et al., 2012), graphene oxide (Xie et al., 2013), MoS₂ (Mak et al., 2010), and porous graphene (Jungthawan et al., 2013) have been explored theoretically or successfully to be manufactured experimentally for their applications in nanotechnology and nanodevices (Radisavljevic et al., 2011; Papasimakis et al., 2013). Furthermore, there is significant progress also in the synthesis of different types of metal–organic frameworks and in the research of their applications in various fields (Kambe et al., 2013; Aslam et al., 2018; Gao et al., 2019 ). In order to meet the heavy request of materials for the forthcoming nanotechnology (Lukatskaya et al., 2013; Clough et al., 2014) and other $2 D$ materials, the analysis of their different properties are urgently required.

Throughout the paper, all the graphs we consider are finite, simple and connected molecular graphs. Let $G$ be such a graph with edge set $E (G)$ and vertex set $V (G)$ . Vertices of $G$ are related with atoms, and the edges related to the chemical bonds among atoms. The vertices k and l are adjacent vertices if and only if they are end vertices of a common edge $f \in E (G)$ , and we write it as $f = kl$ . The set of neighbor vertices of a vertex k is represented by $N_{k}$ , i.e., $N_{k} = {l \in V (G) : kl \in E (G)}$ . The degree of a vertex $k \in V (G)$ is symbolized by $η_{k}$ , that represents the number of vertices incident to k. Let $S_{k}$ stands for the sum of the degrees of all neighboring vertices of vertex k. Let $E_{(i, j)}$ denotes the set of all edges having end vertices with degrees i and j.

Graph theory has become a significant and needed part of the predictive toxicology and drug discovery, as it played a major role in the assessment of (QSPR) and (QSAR) studies. That is, the various features of molecules are based on their structures and therefore, quantitative structure–activity, quantitative structure–property and quantitative structure toxicity relationships (QSAR/QSPR/QSTR) investigation have turn out to be an effective area of study in the valuation of biological, physicochemical properties and pharmacological activities of materials and chemical compounds. These studies have been broadly used to pharmacodynamics, pharmacokinetics, toxicology, chemometrics, and so on (Liu and Long, 2009). Topological descriptors have made sure their vital importance, due to their easy formation and the speed with which these assessments can be executed. There are various graph-associated numerical descriptors, which have verified their value in different areas. Thereby, the process of finding the topological descriptors has become a fascinating and appealing direction of research. Some fruitful kinds of graph-related descriptors are counting-related, distance-based, and degree-based. Among these, degree-based have the noticeable status and can play the leading role in the characterization of the chemical compounds and takes into account their properties like molecular weight, density, boiling and melting points, etc. The mathematical equations of degree-based topological descriptors, which we discussed in this article are depicted in Table 1.

Table 1 Degree-based topological descriptors.

Topological indices	Mathematical from
General Randi $\overset{́}{c}$ index (Li and Gutman, 2006)	$R_{α} (G) = \sum_{xy \in E (G)} {(η_{x} η_{y})}^{α}$
First Zagreb (Gutman and Trinajstić, 1972)	$M_{1} (G) = \sum_{xy \in E (G)} [η_{x} + η_{y}]$
Second Zagreb (Gutman and Trinajstić, 1972)	$M_{2} (G) = \sum_{xy \in E (G)} [η_{x} η_{y}]$
Sum-connectivity index (Zhou and Trinajstić, 2010)	$SCI (G) = \sum_{xy \in E (G)} \frac{1}{\sqrt{η_{x} + η_{y}}}$
Atom-bond connectivity index (Estrada et al., 1998)	$ABC (G) = \sum_{xy \in E (G)} \sqrt{\frac{η_{x} + η_{y} - 2}{η_{x} η_{y}}}$
Geometric-arithmetic (Vukičević and Furtula, 2009)	$GA (G) = \sum_{xy \in E (G)} \frac{2 \sqrt{η_{x} + η_{y}}}{η_{x} + η_{y}}$
Fourth version of atom-bond connectivity index(Ghorbani and Hosseinzadeh, 2010)	${ABC}_{4} (G) = \sum_{xy \in E (G)} \sqrt{\frac{S_{x} + S_{y} - 2}{S_{x} S_{y}}}$
Fifth version of geometric-arithmetic index (Graovac et al., 2011)	${GA}_{5} (G) = \sum_{xy \in E (G)} \frac{2 \sqrt{S_{x} S_{y}}}{S_{x} + S_{y}}$

For the detailed discussions of topological indices, we referred the interested reader to (Aslam et al., 2017; Bashir et al., 2017; Gao et al., 2019; Gao et al., 2019; Iqbal et al., 2019; Iqbal et al., 2019; Iqbal et al., 2020; Yang et al., 2019; Zhao et al., 2019; Zheng et al., 2019 ).

2 Topological aspects of $2 D$ trans- $Pd - ({NH}_{2}) S$ lattice

Topological insulators stand for the type of materials that have applications in spintronics and quantum computing. Qionghua Zhou et al. (Zhou et al., 2015) proposed a $2 D$ structure of trans- $Pd - ({NH}_{2}) S$ lattice, where the ${NH}_{2}$ and S groups are coordinated to the central ${Pd}^{2 +}$ ion in a trans-like fashion with preserved space inversion symmetry. This structure may be readily fabricated and may serve as topological insulator. The molecular graph of trans- $Pd - ({NH}_{2}) S$ lattice is shown in Fig. 1. Note that each unit cell of trans- $Pd - ({NH}_{2}) S$ lattice is a kagome lattice. Let us denote $G_{1} (c, d)$ , the molecular graph of trans- $Pd - ({NH}_{2}) S$ lattice with c unit cells in each row and d unit cells in each column. Fig. 1 is the molecular graph of $G_{1} (3, 3)$ . A simple calculation shows that $G_{1} (c, d)$ has $58 cd - 2 c - 2 d$ vertices and $168 cd - 84 c + 84 - 84 d$ edges.

Theorem 1

The Randic connectivity index of graph $G_{1} (c, d)$ is $R_{α} (G_{1} (c, d)) = \{\begin{matrix} 1284 cd - 682 c + 682 - 682 d, & if α = 1, \\ (12 cd - 2 c - 2 d + 2) (\sqrt{6}) \\ + (36 cd - 18 c - 18 d + 18) (\sqrt{3}) \\ + 288 cd - 136 c - 136 d + 136 \\ + (12 cd - 10 c - 10 d + 10) (\sqrt{8} + 2 \sqrt{3}), & if α = \frac{1}{2} \\ \frac{326 cd - 153 c + 153 - 153 d}{12}, & if α = - 1, \\ 32 cd - 14 c - 14 d + 14 + (12 cd - 2 c - 2 d + 2) (\frac{1}{\sqrt{6}}) \\ + (36 cd - 18 c - 18 d + 18) (\frac{1}{\sqrt{3}}) \\ + (12 cd - 10 c - 10 d + 10) (\frac{1}{2 \sqrt{2}} + \frac{1}{2 \sqrt{3}}) & if α = \frac{- 1}{2} \end{matrix}$

Proof

By using the values of Table 2 and the mathematical equation of Randic connectivity index presented in Table 1, the value of this index can be determined as follows: For $α = 1$ $\begin{matrix} R_{1} (G_{1} (c, d)) = & \sum_{xy \in E_{(1, 3)} (G_{1} (c, d))} η_{x} η_{y} + \sum_{xy \in E_{(2, 2)} (G_{1} (c, d))} η_{x} η_{y} + \sum_{xy \in E_{(2, 3)} (G_{1} (c, d))} η_{x} η_{y} \\ + & \sum_{xy \in E_{(2, 4)} (G_{1} (c, d))} η_{x} η_{y} + \sum_{xy \in E_{(3, 3)} (G_{1} (c, d))} η_{x} η_{y} + \sum_{xy \in E_{(3, 4)} (G_{1} (c, d))} η_{x} η_{y} \end{matrix}$ $\begin{matrix} = & (36 cd - 18 c - 18 d + 18) (3) + (4 c + 4 d - 4) (4) \\ + & (12 cd - 10 c - 10 d + 10) (8) + (96 cd - 48 c - 48 d + 48) (9) \\ + & (12 cd - 10 c - 10 d + 10) (12) + (12 cd - 2 c - 2 d + 2) (6) \\ = & 1284 cd - 682 c + 682 - 682 d . \end{matrix}$ For $α = \frac{1}{2}$ $\begin{matrix} R_{\frac{1}{2}} (G_{1} (c, d)) = & \sum_{xy \in E_{(1, 3)} (G_{1} (c, d))} \sqrt{η_{x} η_{y}} + \sum_{xy \in E_{(2, 2)} (G_{1} (c, d))} \sqrt{η_{x} η_{y}} + \sum_{xy \in E_{(2, 3)} (G_{1} (c, d))} \sqrt{η_{x} η_{y}} \\ + & \sum_{xy \in E_{(2, 4)} (G_{1} (c, d))} \sqrt{η_{x} η_{y}} + \sum_{xy \in E_{(3, 3)} (G_{1} (c, d))} \sqrt{η_{x} η_{y}} + \sum_{xy \in E_{(3, 4)} (G_{1} (c, d))} \sqrt{η_{x} η_{y}} \end{matrix}$ $\begin{matrix} = & (36 cd - 18 c - 18 d + 18) (\sqrt{3}) + (4 c + 4 d - 4) (2) \\ + & (12 cd - 10 c - 10 d + 10) (\sqrt{8}) + (96 cd - 48 c - 48 d + 48) (3) \\ + & (12 cd - 10 c - 10 d + 10) (2 \sqrt{3}) + (12 cd - 2 c - 2 d + 2) (\sqrt{6}) \\ = & (12 cd - 2 c - 2 d + 2) (\sqrt{6}) + (36 cd - 18 c - 18 d + 18) (\sqrt{3}) \\ + & 288 cd - 136 c - 136 d + 136 + (12 cd - 10 c - 10 d + 10) (\sqrt{8} + 2 \sqrt{3}) . \end{matrix}$ For $α = - 1$ $\begin{matrix} R_{- 1} (G_{1} (c, d)) = & \sum_{xy \in E_{(1, 3)} (G_{1} (c, d))} \frac{1}{η_{x} η_{y}} + \sum_{xy \in E_{(2, 2)} (G_{1} (c, d))} \frac{1}{η_{x} η_{y}} + \sum_{xy \in E_{(2, 3)} (G_{1} (c, d))} \frac{1}{η_{x} η_{y}} \\ + & \sum_{xy \in E_{(2, 4)} (G_{1} (c, d))} \frac{1}{η_{x} η_{y}} + \sum_{xy \in E_{(3, 3)} (G_{1} (c, d))} \frac{1}{η_{x} η_{y}} + \sum_{xy \in E_{(3, 4)} (G_{1} (c, d))} \frac{1}{η_{x} η_{y}} \\ = & \frac{36 cd - 18 c - 18 d + 18}{3} + \frac{12 cd - 2 c - 2 d + 2}{6} \\ + & \frac{12 cd - 10 c - 10 d + 10}{8} + \frac{4 c + 4 d - 4}{4} \\ + & \frac{96 cd - 48 c - 48 d + 48}{9} + \frac{12 cd - 10 c - 10 d + 10}{12} \\ = & \frac{326 cd - 153 c + 153 - 153 d}{12} \end{matrix}$ when $α = - \frac{1}{2}$ $\begin{matrix} R_{- \frac{1}{2}} (G_{1} (c, d)) = \sum_{xy \in E_{(1, 3)} (G_{1} (c, d))} \frac{1}{\sqrt{η_{x} η_{y}}} + \sum_{xy \in E_{(2, 2)} (G_{1} (c, d))} \frac{1}{\sqrt{η_{x} η_{y}}} + \sum_{xy \in E_{(2, 3)} (G_{1} (c, d))} \frac{1}{\sqrt{η_{x} η_{y}}} \\ + \sum_{xy \in E_{(2, 4)} (G_{1} (c, d))} \frac{1}{\sqrt{η_{x} η_{y}}} + \sum_{xy \in E_{(3, 3)} (G_{1} (c, d))} \frac{1}{\sqrt{η_{x} η_{y}}} + \sum_{xy \in E_{(3, 4)} (G_{1} (c, d))} \frac{1}{\sqrt{η_{x} η_{y}}} \end{matrix}$ $\begin{matrix} = (\frac{36 cd - 18 c - 18 d + 18}{\sqrt{3}}) + (\frac{4 c + 4 d - 4}{2}) \\ + (\frac{12 cd - 2 c - 2 d + 2}{\sqrt{6}}) + (\frac{12 cd - 10 c - 10 d + 10}{2 \sqrt{2}}) \\ + (\frac{96 cd - 48 c - 48 d + 48}{3}) + (\frac{12 cd - 10 c - 10 d + 10}{2 \sqrt{3}}) . \end{matrix}$ After simplifications, we have the required result.

Theorem 2

The sum conductivity index of the graph $G_{1} (c, d)$ is $\begin{matrix} S (G_{1} (c, d)) = & 18 cd - 7 c - 7 d + 7 + (12 cd - 2 c - 2 d + 2) (\frac{1}{\sqrt{5}}) \\ + & (108 cd - 58 c - 58 d + 58) (\frac{1}{\sqrt{6}}) + (12 cd - 10 c - 10 d + 10) (\frac{1}{\sqrt{7}}) . \end{matrix}$

Proof

In the light of Table 2 and the definition of sum conductivity index given in Table 1, this index can be computed as follows: $\begin{matrix} S (G_{1} (c, d)) = & \sum_{xy \in E_{(1, 3)} (G_{1} (c, d))} \frac{1}{\sqrt{η_{x} + η_{y}}} + \sum_{xy \in E_{(2, 2)} (G_{1} (c, d))} \frac{1}{\sqrt{η_{x} + η_{y}}} + \sum_{xy \in E_{(2, 3)} (G_{1} (c, d))} \frac{1}{\sqrt{η_{x} + η_{y}}} \\ + & \sum_{xy \in E_{(2, 4)} (G_{1} (c, d))} \frac{1}{\sqrt{η_{x} + η_{y}}} + \sum_{xy \in E_{(3, 3)} (G_{1} (c, d))} \frac{1}{\sqrt{η_{x} + η_{y}}} + \sum_{xy \in E_{(3, 4)} (G_{1} (c, d))} \frac{1}{\sqrt{η_{x} + η_{y}}} \\ = & (4 c + 4 d - 4) (\frac{1}{2}) + (12 cd - 2 c - 2 d + 2) (\frac{1}{\sqrt{5}}) \\ + & (36 cd - 18 c - 18 d + 18) (\frac{1}{2}) + (96 cd - 48 c - 48 d + 48) (\frac{1}{\sqrt{6}}) \\ + & (12 cd - 10 c - 10 d + 10) (\frac{1}{\sqrt{6}}) + (12 cd - 10 c - 10 d + 10) (\frac{1}{\sqrt{7}}) \\ = & 18 cd - 7 c - 7 d + 7 + (12 cd - 2 c - 2 d + 2) (\frac{1}{\sqrt{5}}) \\ + & (108 cd - 58 c - 58 d + 58) (\frac{1}{\sqrt{6}}) + (12 cd - 10 c - 10 d + 10) (\frac{1}{\sqrt{7}}) . \end{matrix}$

Theorem 3

The ABC index of graph $G_{1} (c, d)$ is $\begin{matrix} ABC (G_{1} (c, d)) = & (24 cd - 8 c - 8 d + 8) (\frac{1}{\sqrt{2}}) + (36 cd - 18 c - 18 d + 18) (\sqrt{\frac{2}{3}}) \\ + & 32 (2 cd - c + 1 - d) + (12 cd - 10 c - 10 d + 10) (\frac{\sqrt{15} + 3 \sqrt{2}}{6}) . \end{matrix}$

Proof

By using the values of Table 2 and the definition of ABC index manifested in Table 1, this index can be determined as follows: $\begin{matrix} = & (36 cd - 18 c - 18 d + 18) (\sqrt{\frac{2}{3}}) + (4 c + 4 d - 4) (\frac{1}{\sqrt{2}}) \\ + & (12 cd - 2 c - 2 d + 2) (\frac{1}{\sqrt{2}}) + (96 cd - 48 c - 48 d + 48) (\frac{2}{3}) \\ + & (12 cd - 10 c - 10 d + 10) (\frac{1}{\sqrt{2}}) + (12 cd - 10 c - 10 d + 10) (\sqrt{\frac{5}{12}}) \\ = & (24 cd - 8 c - 8 d + 8) (\frac{1}{\sqrt{2}}) + (36 cd - 18 c - 18 d + 18) (\sqrt{\frac{2}{3}}) \\ + & 32 (2 cd - c + 1 - d) + (12 cd - 10 c - 10 d + 10) (\frac{\sqrt{15} + 3 \sqrt{2}}{6}) . \end{matrix}$

Theorem 4

The GA index of graph $G_{1} (c, d)$ is $\begin{matrix} GA (G_{1} (c, d)) = & 96 cd - 44 c - 44 d + 44 + (12 cd - 2 c - 2 d + 2) (\frac{2 \sqrt{6}}{5}) \\ + & (36 cd - 18 c - 18 d + 18) (\frac{\sqrt{3}}{2}) + \frac{2 \sqrt{2} (12 cd - 10 c - 10 d + 10)}{3} \\ + & \frac{4 \sqrt{3} (12 cd - 10 c - 10 d + 10)}{7} . \end{matrix}$

Proof

From what we have computed in Table 2 and the definition of GA index given in Table 1, the value of this index can be computed as follows: $\begin{matrix} GA (G_{1} (c, d)) = & \sum_{xy \in E_{(1, 3)} (G_{1} (c, d))} \frac{2 \sqrt{η_{x} η_{y}}}{η_{x} + η_{y}} + \sum_{xy \in E_{(2, 2)} (G_{1} (c, d))} \frac{2 \sqrt{η_{x} η_{y}}}{η_{x} + η_{y}} \\ + & \sum_{xy \in E_{(2, 3)} (G_{1} (c, d))} \frac{2 \sqrt{η_{x} η_{y}}}{η_{x} + η_{y}} + \sum_{xy \in E_{(2, 4)} (G_{1} (c, d))} \frac{2 \sqrt{η_{x} η_{y}}}{η_{x} + η_{y}} \\ + & \sum_{xy \in E_{(3, 3)} (G_{1} (c, d))} \frac{2 \sqrt{η_{x} η_{y}}}{η_{x} + η_{y}} + \sum_{xy \in E_{(3, 4)} (G_{1} (c, d))} \frac{2 \sqrt{η_{x} η_{y}}}{η_{x} + η_{y}} \end{matrix}$ $\begin{matrix} = & (36 cd - 18 c - 18 d + 18) (\frac{\sqrt{3}}{2}) + (4 c + 4 d - 4) (1) \\ + & (12 cd - 2 c - 2 d + 2) (\frac{2 \sqrt{6}}{5}) + (96 cd - 48 c - 48 d + 48) (1) \\ + & (12 cd - 10 c - 10 d + 10) (\frac{2 \sqrt{2}}{3}) + (12 cd - 10 c - 10 d + 10) (\frac{4 \sqrt{3}}{7}) \\ = & 96 cd - 44 c - 44 d + 44 + (12 cd - 2 c - 2 d + 2) (\frac{2 \sqrt{6}}{5}) \\ + & (36 cd - 18 c - 18 d + 18) (\frac{\sqrt{3}}{2}) + (12 cd - 10 c - 10 d + 10) (\frac{2 \sqrt{2}}{3} + \frac{4 \sqrt{3}}{7}) . \end{matrix}$

Molecular Graph of trans- Pd - ( NH 2 ) S lattice.

Table 2 The edge partition of graph

G_{1} (c, d)

based on degrees of end vertices of each edge.

$(η_{x}, η_{y})$ where $xy \in E (G_{1} (c, d))$	Number of edges
$(1, 3)$	$36 cd - 18 c - 18 d + 18$
$(2, 2)$	$4 c + 4 d - 4$
$(2, 3)$	$12 cd - 2 c - 2 d + 2$
$(2, 4)$	$12 cd - 10 c - 10 d + 10$
$(3, 3)$	$96 cd - 48 c - 48 d + 48$
$(3, 4)$	$12 cd - 10 c - 10 d + 10$

In the next results, we find the exact values of Zagreb indices.

Theorem 5

The first Zagreb index of graph $G_{1} (c, d)$ is $M_{1} (G_{1} (c, d)) = 936 cd - 484 c - 484 d + 484 .$

Proof

The edge partitioning given in Table 2 and the definition of first Zagreb index depicted in Table 1, this index can be computed in the following way: $\begin{matrix} M_{1} (G_{1} (c, d)) = \sum_{xy \in E_{(1, 3)} (G_{1} (c, d))} (η_{x} + η_{y}) + \sum_{xy \in E_{(2, 2)} (G_{1} (c, d))} (η_{x} + η_{y}) \\ + \sum_{xy \in E_{(2, 3)} (G_{1} (c, d))} (η_{x} + η_{y}) + \sum_{xy \in E_{(2, 4)} (G_{1} (c, d))} (η_{x} + η_{y}) \\ + \sum_{xy \in E_{(3, 3)} (G_{1} (c, d))} (η_{x} + η_{y}) + \sum_{xy \in E_{(3, 4)} (G_{1} (c, d))} (η_{x} + η_{y}) . \end{matrix}$ $\begin{matrix} = (4 c + 4 d - 4) (4) + (12 cd - 2 c - 2 d + 2) (5) + (36 cd - 18 c - 18 d + 18) (4) \\ + & (96 cd - 48 c - 48 d + 48) (6) + (12 cd - 10 c - 10 d + 10) (13) \\ = 936 cd - 484 c - 484 d + 484 . \end{matrix}$

Theorem 6

The second Zagreb index of graph $G_{1} (c, d)$ is $M_{2} (G_{1} (c, d)) = 1284 cd - 682 c + 682 - 682 d .$

Proof

From what we have provided in Table 2 and the definition of second Zagreb index illustrated in Table 1, this index can be computed as follows: $\begin{matrix} M_{2} (G_{1} (c, d)) = \sum_{xy \in E_{(1, 3)} (G_{1} (c, d))} η_{x} η_{y} + \sum_{xy \in E_{(2, 2)} (G_{1} (c, d))} η_{x} η_{y} + \sum_{xy \in E_{(2, 3)} (G_{1} (c, d))} η_{x} η_{y} \\ + \sum_{xy \in E_{(2, 4)} (G_{1} (c, d))} η_{x} η_{y} + \sum_{xy \in E_{(3, 3)} (G_{1} (c, d))} η_{x} η_{y} + \sum_{xy \in E_{(3, 4)} (G_{1} (c, d))} η_{x} η_{y} \end{matrix}$ $\begin{matrix} = (4 c + 4 d - 4) (4) + (12 cd - 2 c - 2 d + 2) (6) + (36 cd - 18 c - 18 d + 18) (3) \\ + (96 cd - 48 c - 48 d + 48) (9) + (12 cd - 10 c - 10 d + 10) (20) \\ = 1284 cd - 682 c + 682 - 682 d . \end{matrix}$

Theorem 7

The ${ABC}_{4}$ index of graph $G_{1} (c, d)$ is $\begin{matrix} {ABC}_{4} (G_{1} (c, d)) = & (4 c + 4 d - 4) (\frac{1}{\sqrt{10}} + \frac{2 \sqrt{5}}{2} + \sqrt{\frac{11}{40}} + \frac{\sqrt{13}}{3 \sqrt{6}} + \frac{\sqrt{7}}{3 \sqrt{2}}) \\ + & (12 cd - 10 c - 10 d + 10) (\frac{3}{\sqrt{14}} + \frac{1}{\sqrt{5}} + \frac{\sqrt{3}}{2 \sqrt{2}}) \\ + & (24 cd - 16 c - 16 d + 16) (\sqrt{\frac{31}{56}} + \frac{\sqrt{5}}{2 \sqrt{6}}) \\ + & (24 cd - 12 c - 12 d + 12) (\frac{2 \sqrt{2}}{\sqrt{21}} + \frac{4}{9}) \\ + & (36 cd - 18 c - 18 d + 18) (\frac{\sqrt{2}}{3}) . \end{matrix}$

Proof

The edge partition of graph $G_{1} (c, d)$ based on degree sum of neighbor vertices of end vertices of each edge is given in Table 3. By using Table 3 and the definition of ${ABC}_{4}$ index index which is formulated in Table 1, we compute ${ABC}_{4} (G_{1} (c, d))$ in the following way: $\begin{matrix} {ABC}_{4} (G_{1} (c, d)) = (4 c + 4 d - 4) (\frac{1}{\sqrt{10}}) + (4 c + 4 d - 4) (\frac{2 \sqrt{5}}{2}) + (4 c + 4 d - 4) (\sqrt{\frac{11}{40}}) \\ + (4 c + 4 d - 4) (\frac{\sqrt{13}}{3 \sqrt{6}}) + (4 c + 4 d - 4) (\frac{\sqrt{7}}{3 \sqrt{2}}) \\ + (12 cd - 10 c - 10 d + 10) (\frac{1}{\sqrt{5}}) + (12 cd - 10 c - 10 d + 10) (\frac{\sqrt{3}}{2 \sqrt{2}}) \\ + (24 cd - 16 c - 16 d + 16) (\sqrt{\frac{31}{56}}) + (24 cd - 16 c - 16 d + 16) (\frac{\sqrt{5}}{2 \sqrt{6}}) \\ + (24 cd - 12 c - 12 d + 12) (\frac{2 \sqrt{2}}{\sqrt{21}}) + (36 cd - 18 c - 18 d + 18) (\frac{\sqrt{2}}{3}) \\ + (24 cd - 12 c - 12 d + 12) (\frac{4}{9}) + (12 cd - 10 c - 10 d + 10) (\frac{3}{\sqrt{14}}) \end{matrix}$ $\begin{matrix} = (4 c + 4 d - 4) (\frac{1}{\sqrt{10}} + \frac{2 \sqrt{5}}{2} + \sqrt{\frac{11}{40}} + \frac{\sqrt{13}}{3 \sqrt{6}} + \frac{\sqrt{7}}{3 \sqrt{2}}) \\ + (12 cd - 10 c - 10 d + 10) (\frac{3}{\sqrt{14}} + \frac{1}{\sqrt{5}} + \frac{\sqrt{3}}{2 \sqrt{2}}) \\ + (24 cd - 16 c - 16 d + 16) (\sqrt{\frac{31}{56}} + \frac{\sqrt{5}}{2 \sqrt{6}}) \\ + (24 cd - 12 c - 12 d + 12) (\frac{2 \sqrt{2}}{\sqrt{21}} + \frac{4}{9}) + (36 cd - 18 c - 18 d + 18) (\frac{\sqrt{2}}{3}) . \end{matrix}$

Theorem 8

The ${GA}_{5}$ index of graph $G_{1} (c, d)$ is $\begin{matrix} {GA}_{5} (G_{1} (a, b)) = & (4 c + 4 d - 4) (\frac{2 \sqrt{30}}{11} + 5 + \frac{4 \sqrt{10}}{13} + \frac{6 \sqrt{6}}{15} + \frac{2 \sqrt{2}}{3}) \\ + & (12 cd - 10 c - 10 d + 10) (\frac{2 \sqrt{70}}{17} + \frac{4 \sqrt{5}}{9} + \frac{4 \sqrt{6}}{7}) \\ + & (24 cd - 16 c - 16 d + 16) (\frac{4 \sqrt{14}}{15} + \frac{12 \sqrt{2}}{17}) \\ + & (24 cd - 12 c - 12 d + 12) (\frac{\sqrt{21}}{5} + 1) \\ + & (36 cd - 18 c - 18 d + 18) (\frac{3 \sqrt{7}}{8}) . \end{matrix}$

Proof

From what we have given in Table 3 and the definition of ${GA}_{5}$ index as in Table 1, the value of this index can be workout as follows: $\begin{matrix} {GA}_{5} (G_{1} (c, d)) = (4 c + 4 d - 4) (\frac{2 \sqrt{30}}{11}) + (4 c + 4 d - 4) (5) + (4 c + 4 d - 4) (\frac{4 \sqrt{10}}{13}) \\ + (4 c + 4 d - 4) (\frac{6 \sqrt{6}}{15}) + (4 c + 4 d - 4) (\frac{2 \sqrt{2}}{3}) \\ + (12 cd - 10 c - 10 d + 10) (\frac{4 \sqrt{5}}{9}) + (12 cd - 10 c - 10 d + 10) (\frac{4 \sqrt{6}}{7}) \\ + (24 cd - 16 c - 16 d + 16) (\frac{4 \sqrt{14}}{15}) + (24 cd - 16 c - 16 d + 16) (\frac{12 \sqrt{2}}{17}) \\ + (24 cd - 12 c - 12 d + 12) (\frac{\sqrt{21}}{5}) + (36 cd - 18 c - 18 d + 18) (\frac{3 \sqrt{7}}{8}) \\ + (24 cd - 12 c - 12 d + 12) (1) + (12 cd - 10 c - 10 d + 10) (\frac{2 \sqrt{70}}{17}) \end{matrix}$ $\begin{matrix} = (4 c + 4 d - 4) (\frac{2 \sqrt{30}}{11} + 5 + \frac{4 \sqrt{10}}{13} + \frac{6 \sqrt{6}}{15} + \frac{2 \sqrt{2}}{3}) \\ + (12 cd - 10 c - 10 d + 10) (\frac{2 \sqrt{70}}{17} + \frac{4 \sqrt{5}}{9} + \frac{4 \sqrt{6}}{7}) \\ + (24 cd - 16 c - 16 d + 16) (\frac{4 \sqrt{14}}{15} + \frac{12 \sqrt{2}}{17}) \\ + (24 cd - 12 c - 12 d + 12) (\frac{\sqrt{21}}{5} + 1) \\ + (36 cd - 18 c - 18 d + 18) (\frac{3 \sqrt{7}}{8}) . \end{matrix}$

Table 3 The edge partition of graph

G_{1} (c, d)

based on degree sum of neighbor vertices of end vertices of each edge.

$(S_{x}, S_{y})$ where $xy \in E (G_{1} (c, d))$	Number of edges
$(3, 6)$	$4 c + 4 d - 4$
$(3, 7)$	$24 cd - 12 c - 12 d + 12$
$(3, 8)$	$12 cd - 10 c - 10 d + 10$
$(5, 5)$	$4 c + 4 d - 4$
$(5, 6)$	$4 c + 4 d - 4$
$(5, 8)$	$4 c + 4 d - 4$
$(6, 9)$	$4 c + 4 d - 4$
$(7, 8)$	$24 cd - 16 c - 16 d + 16$
$(7, 9)$	$36 cd - 18 c - 18 d + 18$
$(7, 10)$	$12 cd - 10 c - 10 d + 10$
$(8, 9)$	$24 cd - 16 c - 16 d + 16$
$(8, 10)$	$12 cd - 10 c - 10 d + 10$
$(9, 9)$	$24 cd - 12 c - 12 d + 12$

3 Topological aspects of 2D metal-organic superlattice

There is an appreciable progress in the synthesis of various types of metal–organic frameworks and in the research of their applications in several fields (Kambe et al., 2013; Sheberla et al., 2014). Inspired by all their theoretical and experimental studies, C. Chakravarty et al. (Chakravarty et al., 2016) have forecasted a novel metal–organic frameworks with the support of first-principles computations. The suggested $2 D$ sheet contains an -NH substituted coronene moiety covalently connected with distinct transition metals. In the $2 D$ formation, the transition metals are encircled by four -NH groups, producing a square planar geometry encircling them. The unit cell has a coronene molecule functionalized by the transition metal and eight -NH moieties. Fig. 2 shows the introduced the $2 D$ sheet consisting of NH substituted coronene moiety covalently connected to various transition metals. The unit cell of this 2D lattice is shown in the rectangular box. The spheres with white, brown, blue and cyan color are hydrogen, transition metal, carbon and nitrogen respectively. We denote the molecular graph of 2D metal–organic superlattice with b unit cells in each row and a unit cell in each column by $G_{2} (a, b)$ . Fig. 2 depicts the molecular graph $G_{2} (2, 2)$ .

Theorem 9

The Randic connectivity index of the graph $G_{2} (a, b)$ is $R_{α} (G_{2} (a, b)) = \{\begin{matrix} 474 ab - 32 a - 32 b, & if α = 1, \\ (28 \sqrt{3} + 114) ab + (2 \sqrt{2} - 6 \sqrt{3} + 2 \sqrt{6} - 6) a \\ + (2 \sqrt{2} - 6 \sqrt{3} + 2 \sqrt{6} - 6) b, & if α = \frac{1}{2} \\ \frac{160 ab + 5 a + 5 b}{18}, & if α = - 1, \\ \frac{\sqrt{3} (48 ab + 38 \sqrt{3} ab + 3 \sqrt{6} a + 3 \sqrt{2} a - 9 a - 2 \sqrt{3} a)}{9} \\ + \frac{\sqrt{3} (3 \sqrt{6} b + 3 \sqrt{2} b - 9 b - 2 \sqrt{3} b)}{9} & if α = \frac{- 1}{2} . \end{matrix}$

Proof

By using the values of Table 4 and the definition of Randic connectivity index presented in Table 1, this index can be determined in the following way: For $α = 1$ $\begin{matrix} R_{1} (G_{2} (a, b)) = & \sum_{xy \in E_{(1, 2)} (G_{2} (a, b))} η_{x} η_{y} + \sum_{xy \in E_{(1, 3)} (G_{2} (a, b))} η_{x} η_{y} + \sum_{xy \in E_{(2, 3)} (G_{2} (a, b))} η_{x} η_{y} \\ + & \sum_{xy \in E_{(3, 3)} (G_{2} (a, b))} η_{x} η_{y} + \sum_{xy \in E_{(3, 4)} (G_{2} (a, b))} η_{x} η_{y} \\ = & (2 a + 2 b) (2) + (12 ab - 2 a - 2 b) (3) + (38 ab - 2 a - 2 b) (9) \\ + & (2 a + 2 b) (6) + (8 ab - 2 a - 2 b) (12) \\ = & 474 ab - 32 a - 32 b . \end{matrix}$ For $α = \frac{1}{2}$ $\begin{matrix} R_{\frac{1}{2}} (G_{2} (a, b)) = & \sum_{xy \in E_{(1, 2)} (G_{2} (a, b))} \sqrt{η_{x} η_{y}} + \sum_{xy \in E_{(1, 3)} (G_{2} (a, b))} \sqrt{η_{x} η_{y}} + \sum_{xy \in E_{(2, 3)} (G_{2} (a, b))} \sqrt{η_{x} η_{y}} \\ + & \sum_{xy \in E_{(3, 3)} (G_{2} (a, b))} \sqrt{η_{x} η_{y}} + \sum_{xy \in E_{(3, 4)} (G_{2} (a, b))} \sqrt{η_{x} η_{y}} \\ = & (2 a + 2 b) (\sqrt{2}) + (12 ab - 2 a - 2 b) (\sqrt{3}) + (2 a + 2 b) (\sqrt{6}) \\ + & (38 ab - 2 a - 2 b) (3) + (8 ab - 2 a - 2 b) (2 \sqrt{3}) \\ = & (28 \sqrt{3} + 114) ab + (2 \sqrt{2} - 6 \sqrt{3} + 2 \sqrt{6} - 6) a + (2 \sqrt{2} - 6 \sqrt{3} + 2 \sqrt{6} - 6) b . \end{matrix}$ For $α = - 1$ $\begin{matrix} R_{- 1} (G_{2} (a, b)) = & \sum_{xy \in E_{(1, 2)} (G_{2} (a, b))} \frac{1}{η_{x} η_{y}} + \sum_{xy \in E_{(1, 3)} (G_{2} (a, b))} \frac{1}{η_{x} η_{y}} + \sum_{xy \in E_{(2, 3)} (G_{2} (a, b))} \frac{1}{η_{x} η_{y}} \\ + & \sum_{xy \in E_{(3, 3)} (G_{2} (a, b))} \frac{1}{η_{x} η_{y}} + \sum_{xy \in E_{(3, 4)} (G_{2} (a, b))} \frac{1}{η_{x} η_{y}} \end{matrix}$ $\begin{matrix} R_{- 1} (G_{2} (a, b)) = & (2 a + 2 b) (\frac{1}{2}) + (12 ab - 2 a - 2 b) (\frac{1}{3}) + (2 a + 2 b) (\frac{1}{6}) \\ = & (2 a + 2 b) (\frac{1}{2}) + (12 ab - 2 a - 2 b) (\frac{1}{3}) + (2 a + 2 b) (\frac{1}{6}) \\ + & (38 ab - 2 a - 2 b) (\frac{1}{9}) + (8 ab - 2 a - 2 b) (\frac{1}{12}) \\ = & \frac{160 ab + 5 a + 5 b}{18} . \end{matrix}$ For $α = \frac{- 1}{2}$ $\begin{matrix} R_{\frac{- 1}{2}} (G_{2} (a, b)) = \sum_{xy \in E_{(1, 2)} (G_{2} (a, b))} \frac{1}{\sqrt{η_{x} η_{y}}} + \sum_{xy \in E_{(1, 3)} (G_{2} (a, b))} \frac{1}{\sqrt{η_{x} η_{y}}} + \sum_{xy \in E_{(2, 3)} (G_{2} (a, b))} \frac{1}{\sqrt{η_{x} η_{y}}} \\ + \sum_{xy \in E_{(3, 3)} (G_{2} (a, b))} \frac{1}{\sqrt{η_{x} η_{y}}} + \sum_{xy \in E_{(3, 4)} (G_{2} (a, b))} \frac{1}{\sqrt{η_{x} η_{y}}} \\ = (2 a + 2 b) (\frac{1}{\sqrt{2}}) + (12 ab - 2 a - 2 b) (\frac{1}{\sqrt{3}}) + (2 a + 2 b) (\frac{1}{\sqrt{6}}) \end{matrix}$ $\begin{matrix} + (38 ab - 2 a - 2 b) (\frac{1}{3}) + (8 ab - 2 a - 2 b) (\frac{1}{2 \sqrt{3}}) \\ = \frac{\sqrt{3} (48 ab + 38 \sqrt{3} ab + 3 \sqrt{6} a + 3 \sqrt{2} a - 9 a - 2 \sqrt{3} a)}{9} \\ + \frac{\sqrt{3} (3 \sqrt{6} b + 3 \sqrt{2} b - 9 b - 2 \sqrt{3} b)}{9} . \end{matrix}$

Theorem 10

The sum conductivity index of graph $G_{2} (a, b)$ is $\begin{matrix} S (G_{2} (a, b)) = & \frac{\sqrt{15} (2 (\sqrt{5} + \sqrt{3}) a + 2 (\sqrt{5} + \sqrt{3}) b)}{15} + 6 ab - a - b \\ + & \frac{\sqrt{6} (19 ab - a - b)}{3} + \frac{\sqrt{7} (8 ab - 2 a - 2 b)}{7} . \end{matrix}$

Proof

In the light of Table 4 and the definition of sum conductivity index given in Table 1, this index can be computed as follows: $\begin{matrix} S (G_{2} (a, b)) = & \sum_{xy \in E_{(1, 2)} (G_{2} (a, b))} \frac{1}{\sqrt{η_{x} + η_{y}}} + \sum_{xy \in E_{(1, 3)} (G_{2} (a, b))} \frac{1}{\sqrt{η_{x} + η_{y}}} \\ + & \sum_{xy \in E_{(2, 3)} (G_{2} (a, b))} \frac{1}{\sqrt{η_{x} + η_{y}}} + \sum_{xy \in E_{(3, 3)} (G_{2} (a, b))} \frac{1}{\sqrt{η_{x} + η_{y}}} \\ + & \sum_{xy \in E_{(3, 4)} (G_{2} (a, b))} \frac{1}{\sqrt{η_{x} + η_{y}}} \\ = & (2 a + 2 b) (\frac{1}{\sqrt{3}}) + (12 ab - 2 a - 2 b) (\frac{1}{2}) + (2 a + 2 b) (\frac{1}{\sqrt{5}}) \\ + & (38 ab - 2 a - 2 b) (\frac{1}{\sqrt{6}}) + (8 ab - 2 a - 2 b) (\frac{1}{\sqrt{7}}) \\ = & \frac{\sqrt{15} (2 (\sqrt{5} + \sqrt{3}) a + 2 (\sqrt{5} + \sqrt{3}) b)}{15} + 6 ab - a - b \\ + & \frac{\sqrt{6} (19 ab - a - b)}{3} + \frac{\sqrt{7} (8 ab - 2 a - 2 b)}{7} . \end{matrix}$

Theorem 11

The ABC index of graph $G_{2} (a, b)$ is $\begin{matrix} ABC (G_{2} (a, b)) = & 4 \sqrt{6} ab + (2 \sqrt{2} - 2 \sqrt{\frac{2}{3}}) (a + b) + \frac{76 ab - 4 a - 4 b}{3} \\ + & \frac{\sqrt{15} (4 ab - a - b)}{3} . \end{matrix}$

Proof

By using the values of Table 4 and the definition of ABC index manifested in Table 1, this index can be determined as follows: $\begin{matrix} ABC (G_{2} (a, b)) = & (2 a + 2 b) (\frac{1}{\sqrt{2}}) + (12 ab - 2 a - 2 b) (\sqrt{\frac{2}{3}}) + (2 a + 2 b) (\frac{1}{\sqrt{2}}) \\ + & (38 ab - 2 a - 2 b) (\frac{2}{3}) + (8 ab - 2 a - 2 b) (\sqrt{\frac{5}{12}}) \\ = & 4 \sqrt{6} ab + (2 \sqrt{2} - 2 \sqrt{\frac{2}{3}}) (a + b) + \frac{76 ab - 4 a - 4 b}{3} \\ + & \frac{\sqrt{15} (4 ab - a - b)}{3} . \end{matrix}$

Theorem 12

The GA index of graph $G_{2} (a, b)$ is $\begin{matrix} GA (G_{2} (a, b)) = & 6 \sqrt{3} ab - \sqrt{3} a - \sqrt{3} b + \frac{2 (10 \sqrt{2} + 6 \sqrt{6}) a + 2 (10 \sqrt{2} + 6 \sqrt{6}) b}{15} \\ + & 38 ab - 2 a - 2 b + \frac{32 \sqrt{3} ab - 8 \sqrt{3} a - 8 \sqrt{3} b}{7} . \end{matrix}$

Proof

From what we have computed in Table 4 and the definition of GA index given in Table 1, this index can be computed as follows: $\begin{matrix} GA (G_{2} (a, b)) = & \sum_{xy \in E_{(1, 2)} (G_{2} (a, b))} \frac{2 \sqrt{η_{x} η_{y}}}{η_{x} + η_{y}} + \sum_{xy \in E_{(1, 3)} (G_{2} (a, b))} \frac{2 \sqrt{η_{x} η_{y}}}{η_{x} + η_{y}} + \sum_{xy \in E_{(2, 3)} (G_{2} (a, b))} \frac{2 \sqrt{η_{x} η_{y}}}{η_{x} + η_{y}} \\ + & \sum_{xy \in E_{(3, 3)} (G_{2} (a, b))} \frac{2 \sqrt{η_{x} η_{y}}}{η_{x} + η_{y}} + \sum_{xy \in E_{(3, 4)} (G_{2} (a, b))} \frac{2 \sqrt{η_{x} η_{y}}}{η_{x} + η_{y}} \end{matrix}$ $\begin{matrix} = & (2 a + 2 b) (\frac{2 \sqrt{2}}{3}) + (12 ab - 2 a - 2 b) (\frac{\sqrt{3}}{2}) + (2 a + 2 b) (\frac{2 \sqrt{6}}{5}) \\ + & (38 ab - 2 a - 2 b) (1) + (8 ab - 2 a - 2 b) (\frac{4 \sqrt{3}}{7}) \\ = & 6 \sqrt{3} ab - \sqrt{3} a - \sqrt{3} b + \frac{2 (10 \sqrt{2} + 6 \sqrt{6}) a + 2 (10 \sqrt{2} + 6 \sqrt{6}) b}{15} \\ + & 38 ab - 2 a - 2 b + \frac{32 \sqrt{3} ab - 8 \sqrt{3} a - 8 \sqrt{3} b}{7} . \end{matrix}$

Molecular Graph of 2D metal–organic superlattice.

Table 4 The edge partition of graph

G_{2} (a, b)

based on degrees of end vertices of each edge.

$(η_{x}, η_{y})$ where $xy \in E (G_{2} (a, b))$	Number of edges
$(1, 2)$	$2 a + 2 b$
$(1, 3)$	$12 ab - 2 a - 2 b$
$(2, 3)$	$2 a + 2 b$
$(3, 3)$	$38 ab - 2 a - 2 b$
$(3, 4)$	$8 ab - 2 a - 2 b$

In the next results, we find the exact values of Zagreb indices.

Theorem 13

The first Zagreb index of graph $G_{2} (a, b)$ is $M_{1} (G_{2} (a, b)) = 332 ab - 18 a - 18 b .$

Proof

The edge partitioning given in Table 4 and the definition of first Zagreb index depicted in Table 1, this index can be computed as follows: $\begin{matrix} M_{1} (G_{2} (a, b)) = & \sum_{xy \in E_{(1, 2)} (G_{2} (a, b))} (η_{x} + η_{y}) + \sum_{xy \in E_{(1, 3)} (G_{2} (a, b))} (η_{x} + η_{y}) + \sum_{xy \in E_{(2, 3)} (G_{2} (a, b))} (η_{x} + η_{y}) \\ + & \sum_{xy \in E_{(3, 3)} (G_{2} (a, b))} (η_{x} + η_{y}) + \sum_{xy \in E_{(3, 4)} (G_{2} (a, b))} (η_{x} + η_{y}) . \end{matrix}$ $\begin{matrix} = & (2 a + 2 b) (3) + (12 ab - 2 a - 2 b) (4) + (2 a + 2 b) (5) \\ + & (38 ab - 2 a - 2 b) (6) + (8 ab - 2 a - 2 b) (7) \\ = & 332 ab - 18 a - 18 b . \end{matrix}$

Theorem 14

The second Zagreb index of graph $G_{2} (a, b)$ is $M_{2} (G_{2} (a, b)) = 1284 ab - 682 a + 682 - 682 b .$

Proof

From what we have provided in Table 4 and the definition of second Zagreb index illustrated in Table 1, this index can be computed as follows: $\begin{matrix} M_{2} (G_{2} (a, b)) = & \sum_{xy \in E_{(1, 2)} (G_{2} (a, b))} η_{x} η_{y} + \sum_{xy \in E_{(1, 3)} (G_{2} (a, b))} η_{x} η_{y} + \sum_{xy \in E_{(2, 3)} (G_{2} (a, b))} η_{x} η_{y} \\ + & \sum_{xy \in E_{(3, 3)} (G_{2} (a, b))} η_{x} η_{y} + \sum_{xy \in E_{(3, 4)} (G_{2} (a, b))} η_{x} η_{y} . \end{matrix}$ $\begin{matrix} = & (2 a + 2 b) (2) + (12 ab - 2 a - 2 b) (3) + (2 a + 2 b) (6) \\ + & (38 ab - 2 a - 2 b) (9) + (8 ab - 2 a - 2 b) (12) \\ = & 474 ab - 32 a - 32 b . \end{matrix}$

Theorem 15

The ${ABC}_{4}$ index of graph $G_{2} (a, b)$ is $\begin{matrix} {ABC}_{4} (G_{2} (a, b)) = & \sqrt{\frac{5}{2}} (2 a + 2 b) + \frac{\sqrt{14} (a + b) + \sqrt{10} (a + b)}{3} + \frac{\sqrt{5} (a + b)}{2} \\ + & \frac{8 \sqrt{42} ab}{21} + \sqrt{6} (2 ab - a - b) \frac{\sqrt{182} b}{14} + \frac{\sqrt{78} (b + a)}{9} \\ + & \frac{\sqrt{30} (4 nm - b - a)}{6} + 2 \sqrt{3} nm - \sqrt{3} b - \sqrt{3} a \\ + & \frac{8 \sqrt{2} nm - 2 \sqrt{2} b - 2 \sqrt{2} a}{3} + \frac{88 nm - 8 b}{9} . \end{matrix}$

Proof

The edge partition of graph $G_{2} (a, b)$ based on the degree sum of neighbor vertices of end vertices of each edge is provided in Table 5. By using Table 5 and the definition of ${ABC}_{4}$ index which is formulated in Table 1, we compute ${ABC}_{4} (G_{2} (a, b))$ in the following way: $\begin{matrix} {ABC}_{4} (G_{2} (a, b)) = & (2 a + 2 b) (\sqrt{\frac{5}{2}}) + (2 a + 2 b) (\sqrt{\frac{7}{18}}) + (2 a + 2 b) (\sqrt{\frac{10}{32}}) \\ + & (4 ab) (\sqrt{\frac{8}{21}}) + (8 ab - 4 a - 4 b) (\frac{3}{\sqrt{24}}) + (2 a + 2 b) (\frac{\sqrt{10}}{6}) \\ + & (2 b) (\sqrt{\frac{13}{56}}) + (8 ab - 2 a - 2 b) (\sqrt{\frac{14}{63}}) + (8 ab - 2 a - 2 b) (\frac{\sqrt{15}}{6 \sqrt{2}}) \\ + & (2 a + 2 b) (\frac{\sqrt{13}}{3 \sqrt{6}}) + (8 ab - 4 a - 4 b) (\frac{\sqrt{3}}{4}) + (22 ab - 2 b) (\frac{4}{9}) . \end{matrix}$ After some simplifications, we have the required result.

Theorem 16

The ${GA}_{5}$ index of graph $G_{2} (a, b)$ is $\begin{matrix} {GA}_{5} (G_{2} (a, b)) = & \frac{(3360 \sqrt{6} + 924 \sqrt{21}) ab + (1320 \sqrt{3} + 3080 \sqrt{2} + 2310 - 756 \sqrt{6}) b}{1155} \\ + & \frac{616 \sqrt{14} b + (1320 \sqrt{3} + 3080 \sqrt{2} + 2310 - 756 \sqrt{6}) a}{1155} \\ + & \frac{3 \sqrt{7} (4 ab - a - b)}{4} + \frac{96 \sqrt{2} ab - 24 \sqrt{2} a - 24 \sqrt{2} b}{17} \\ + & \frac{16 \sqrt{6} ab - 8 \sqrt{6} a - 8 \sqrt{6} b}{5} + 22 ab - 2 b . \end{matrix}$

Proof

By using the values of Table 5 and the definition of ${GA}_{5}$ index as in Table 1, this index can be calculated as follows: $\begin{matrix} {GA}_{5} (G_{2} (a, b)) = (2 a + 2 b) (\frac{4}{7} \sqrt{3}) + (2 a + 2 b) (\frac{2}{3} \sqrt{2}) + (8 ab - 4 a - 4 b) (\frac{4}{11} \sqrt{6}) \\ + (4 ab) (\frac{\sqrt{21}}{5}) + (2 a + 2 b) (\frac{2}{3} \sqrt{2}) + (2 a + 2 b) (1) + (2 a + 2 b) (\frac{2 \sqrt{6}}{5}) \\ + (2 b) (\frac{4}{15} \sqrt{14}) + (8 ab - 2 a - 2 b) (\frac{3}{8} \sqrt{7}) + (8 ab - 2 a - 2 b) (\frac{12}{17} \sqrt{2}) \\ + (8 ab - 4 a - 4 b) (\frac{2}{5} \sqrt{6}) + (22 ab - 2 b) (1) \end{matrix}$ $\begin{matrix} = \frac{(3360 \sqrt{6} + 924 \sqrt{21}) ab + (1320 \sqrt{3} + 3080 \sqrt{2} + 2310 - 756 \sqrt{6}) b}{1155} \\ + \frac{616 \sqrt{14} b + (1320 \sqrt{3} + 3080 \sqrt{2} + 2310 - 756 \sqrt{6}) a}{1155} \\ + \frac{3 \sqrt{7} (4 ab - a - b)}{4} + \frac{96 \sqrt{2} ab - 24 \sqrt{2} a - 24 \sqrt{2} b}{17} \\ + \frac{16 \sqrt{6} ab - 8 \sqrt{6} a - 8 \sqrt{6} b}{5} + 22 ab - 2 b . \end{matrix}$

Table 5 The edge partition based on the degree sum of neighbor vertices of each edge of graph

G_{2} (a, b)

$(S_{x}, S_{y})$ where $xy \in E (G_{2} (c, d))$	Number of edges
$(3, 4)$	$2 a + 2 b$
$(3, 6)$	$2 a + 2 b$
$(3, 7)$	$4 ab$
$(3, 8)$	$8 ab - 4 a - 4 b$
$(4, 8)$	$2 a + 2 b$
$(6, 6)$	$2 a + 2 b$
$(6, 9)$	$2 a + 2 b$
$(7, 8)$	$2 b$
$(7, 9)$	$8 ab - 2 a - 2 b$
$(8, 9)$	$8 ab - 2 a - 2 b$
$(8, 12)$	$8 ab - 4 a - 4 b$
$(9, 9)$	$22 ab - 2 b$

4 Conclusion

A topological descriptor can be assumed to be a function that provides the information in numerical form about any underline molecular structure. Topological descriptors capture the symmetry of chemical compounds and present the facts in the numerical form including the presence of heteroatoms, molecular size, multiple bonds, shape, and branching. In this paper, important chemical structures are considered, and with the help of edge partitioning technique and molecular graph structure analysis, the precise formulas of some valuable degree-based indices are computed.

Declaration of Competing Interest

None.

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Introduction

Topological aspects of 2 D trans- Pd - ( NH 2 ) S lattice

Topological aspects of 2D metal-organic superlattice

Conclusion

Declaration of Competing Interest

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[30] Yang H., Imran M., Akhter S., Iqbal Z., Siddiqui M.K., . On distance-based topological descriptors of subdivision vertex-edge join of three graphs. IEEE Access. 2019;7:143381-143391.
[Google Scholar]

[31] Zhao D., Iqbal Z., Irfan R., Chaudhry M.A., Ishaq M., Jamil M.K., Fahad A., . Comparison of irregularity indices of several dendrimers structures. Processes.. 2019;7:662.
[Google Scholar]

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[Google Scholar]

[33] Zhou B., Trinajstić N., . On general sum-connectivity index. J. Math. Chem.. 2010;47:210-218.
[Google Scholar]

[34] Zhou Q., Wang J., Chwee T.S., Wu G., Wang X., Ye Q., Xu J., Yang S.W., . Topological insulators based on 2D shape-persistent organic ligand complexes. Nanoscale.. 2015;7(2):727-735.
[Google Scholar]

Topological aspects of 2D structures of trans- Pd ( NH 2 ) S lattice and a metal-organic superlattice

Abstract

Abstract

Keywords

Trans-pd-(NH2) lattice

Metal-organic superlattice

Degree-based topological descriptors

1 Introduction

2 Topological aspects of 2 D trans- Pd - ( NH 2 ) S lattice

3 Topological aspects of 2D metal-organic superlattice

4 Conclusion

Declaration of Competing Interest

References

Suggested read for related articles:

Topological aspects of 2D structures of trans- $Pd ({NH}_{2}) S$ lattice and a metal-organic superlattice

2 Topological aspects of $2 D$ trans- $Pd - ({NH}_{2}) S$ lattice