M-Polynomials and Degree-Based Topological Indices of the Molecule Copper(I) Oxide

Chaudhry, Faryal; Shoukat, Iqra; Afzal, Deeba; Park, Choonkil; Cancan, Murat; Farahani, Mohammad Reza

doi:https://doi.org/10.1155/2021/6679819

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Computational Invariant of Chemical Structures and their Applications

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Volume 2021 | Article ID 6679819 | https://doi.org/10.1155/2021/6679819

M-Polynomials and Degree-Based Topological Indices of the Molecule Copper(I) Oxide

Faryal Chaudhry,¹Iqra Shoukat,¹Deeba Afzal,¹Choonkil Park,²Murat Cancan,³and Mohammad Reza Farahani⁴

Academic Editor: Teodorico C. Ramalho

Received04 Dec 2020

Revised20 Dec 2020

Accepted28 Jan 2021

Published25 Feb 2021

Abstract

Topological indices are numerical parameters used to study the physical and chemical residences of compounds. Degree-based topological indices have been studied extensively and can be correlated with many properties of the understudy compounds. In the factors of degree-based topological indices, M-polynomial played an important role. In this paper, we derived closed formulas for some well-known degree-based topological indices like first and second Zagreb indices, the modified Zagreb index, the symmetric division index, the harmonic index, the Randić index and inverse Randić index, and the augmented Zagreb index using calculus.

1. Introduction

1.1. Application Background

A graph that represents the construction of a molecule and also their connectivity is known as a molecular graph, and such a representation is generally known as topological representations of molecule. Molecular graphs are normally characterized by means of exclusive topological basis for parallel of chemicals shape of a molecule with organic, chemical, or bodily homes. Study of graph has some programs of various topological indices in quantitative structure-activity relationship (QSAR) and quantitative structure-property relationship (QSPR), digital screenings, and computational drug designing citations as shown in [1, 2]. Thus far, several exclusive topological indices have been established, and maximum of them are most effective graph descriptors in [3, 4]; apart, some indices have proven their parallel with organic, chemical, or physical residences of secure molecules in [5–17].

In the field of mathematics, any graph has vertices and edges that are represented by the atoms and chemical bonds. Graph that represents the construction of molecules and their connectivity is known as a molecular graph, and such representation is usually referred as topological representation of molecules. There are some significant topological indices like distance-based topological indices, degree-based topological indices, and primarily based topological indices. Among these works, distance primarily based topological indices unit works out a crucial task in a chemical graph started, specifically in chemistry [18,19]. Many fields have many features that can be solved with the help of graphs. In the physiochemical compounds or network systems, we have a tendency to abstractly outline exclusive ideas in modeling of mathematics. We have a tendency to refer to as the distinctive names, such as Randić index and national capital index.

A topological index is a numerical parameter of a graph and describes its topology. It describes the molecular shape numerically and is applied within the advancement of qualitative structure-activity relationships (QSARs). The following are the 3 types of topological indices:(1)Degree-based.(2)Distance-based.(3)Spectral-based.

Degree-based topological indices were studied extensively and may be correlated with many residences of the understudy molecular compounds. There is a strong relationship among distance-based and degree-based topological indices in [20]. Most commonly known invariants of such kinds are degree-based topological indices. These are actually the numerical values that correlate the structure with various physical properties, chemical reactivities, and biological activities. Topological indices are sincerely the numerical values that relate the shape to one of a kind of physical residences, artificial reactivity, and natural biological activities [21,22].

Loads of research has been executed inside the course of M-polynomial, as in the case of Munir et al., processed M-polynomial and related lists of triangular boron nanotubes in [6], polyhex nanotubes in [23], nanostar dendrimers in [4], and titania nanotubes in [5]. M-Polynomials and topological lists of V-phenylenic nanotubes and nanotori. In this paper, the objective is to process the M-polynomial of the crystallographic realistic structure of the atom copper(I) oxide (Cu₂O) [8,24].

1.2. Crystallographic Structure of Cu₂O(m; n)

Copper oxide is a p-type semiconductor and inorganic compound. Copper oxide is a chemical element with formula Cu₂O(m; n). Cu₂O(m; n) is a certainly happening reddish coral that is particularly used in chemical sensors and solar orientated cells in [8, 24]. It has many advantages such as photochemical effects, stability, pigment, a fungicide, nontoxicity, and low cost. It has potential applications in new energy, sensing, sterilization, and other fields. It has narrow band gap and is easily excited by visible light.

Cu₂O(m; n) is additionally responsible for the pink shading in Benedict’s test and is the essential cause to select Cu₂O (see Figures 1 and 2). The promising projects of Cu₂O(m; n) are mainly on chemical sensors, sunlight-based cells, photocatalysis, lithium particle batteries, and catalysis. Here, we have taken into consideration a monolayer of Cu₂O(m; n) for satisfaction. To ultimate the basis for Cu₂O(m; n), we pick out the setting of this graph as Cu₂O(m; n) be the chemical graph of copper(I) oxide with (m; n) unit cells within the aircraft.

(a)

(b)

2. Definitions and Literature Review

2.1. M-Polynomial

M-Polynomial is defined by S. Klavžar or E. Deutsch in 2015 [3, 8]. Within the factors of degree-based topological indices, we compete necessary role of M-polynomial. Readers can refer to [9–17, 27–35]. It is the foremost general progressive polynomial and an additionally closed formula alongside 10 distance-based topological indices is given by M-polynomial. It is explained asand we have and where m_ij(G) is the edge E(G), where i ≤ j.

2.2. Degree-Based Topological Indices

Any purpose on a graph which does not build upon numbering of its vertices is molecular descriptor. This is also called as topological index. Topological indices are most useful in the field of isomeric discrimination, chemical validation, QSAR, QSPR, and a pharmaceutical drug form. Topological indices are accessed from the system of molecule.

There are some important degree-based topological indices defined, and the first Zagreb index was introduced by Gutman and Trinajstić as follows:

Gutman and Trinajstić proposed the second Zagreb index in 1972, which is stated as

The second modified Zagreb index is defined as

General 1^st and 2^nd multiplicative Zagreb indices are introduced by Kulli, Stone, Wang, and Wei and are stated as

The general 1^st and 2^nd Zagreb indices proposed by Kulli, Stone, Wang, and Wei are stated as

In 1987, Fajtlowicz in [36] proposed the harmonic index and stated

The inverse sum index is defined:

Symmetric division index is described as

SU and XU recognized general Randić index or general multiplicative Randić index stated as follows (Table 1):

Theorem 1. Crystallographic structure of the graph of copper(I) oxide G ≈ Cu₂0[m; n], where n; m ≥ 1. We have

Proof. suppose G be the crystallographic structure of Cu₂O[l; m; n]. The edge set of Cu₂O[l; m; n] has the following three partitions by Figures 1 and 2:such thatThus, the M-polynomial of Cu₂O[l; m; n] is

Theorem 2. Crystallographic structure of the graph of copper(I) oxide G ≈ Cu₂0[m; n], where n; m ≥ 1. We have .

Proof. supposeWe have to findMultiply a on both sides:Similarly,Now, the first Zagreb index isAfter solving, the result isThe 3D plot of first Zagreb index is given in Figure 3 (f or u = 1 left, = 1 middle, and = 1 right), and we see the dependent variables of the first Zagreb index on the involved parameters.

(a)

(b)

Theorem 3. Crystallographic structure of the graph of copper(I) oxide G ≈ Cu₂0[m; n], where n; m ≥ 1. We have .

Proof. supposeWe have to find D_bD_a; first, we take D_a:Now, take D_b:The second Zagreb index isAfter solving, the result isThe 3D plot of second Zagreb index is given in Figure 4 (f or u = 1 left, = 1 middle, and = 1 right), and we see the dependent variables of the second Zagreb index on the involved parameters.

(a)

(b)

Theorem 4. Crystallographic structure of the graph of copper(I) oxide G ≈ Cu₂0[m; n], where n; m ≥ 1, and we have

Proof. supposeNow, we have to find S_aS_b; first, we find S_a:Taking integration on both sides,Now, take S_b and thenNow, the second modified Zagreb index isAfter solving, the result isThe 3D plot of modified second Zagreb index is given in Figure 5 (f or u = 1 left, = 1 middle, and = 1 right), and we see the dependent variables of the modified second Zagreb index on the involved parameters.

(a)

(b)

Theorem 5. Crystallographic structure of the graph of copper(I) oxide G ≈ Cu₂0[m; n], where n; m ≥ 1, and we have

Proof. supposeWe have to find D_aD_b first, and we find D_a:Now, take D_b:Take α on the above equation:Now, the general Randić index isThe result isThe 3D plot of Randić index is given in Figure 6 (f or u = 1 left, = 1 middle, and = 1 right), and we see the dependent variables of the Randić index on the involved parameters.

(a)

(b)

Theorem 6. Crystallographic structure of the graph of copper(I) oxide G ≈ Cu₂0[m; n], where n; m ≥ 1, and we have

Proof. supposeNow, we have to find S_aS_b, and first, we find S_a:Similarly, take S_b:Take α on the above equation:The inverse Randić isThe 3D plot of inverse Randić index is represented in Figure 7 (f or u = 1 left, = 1 middle, and = 1 right), and we see the dependent variables of the inverse Randić index on the involved parameters.

(a)

(b)

Theorem 7. Crystallographic structure of the graph of copper(I) oxide G ≈ Cu₂0[m; n], where n; m ≥ 1, and we have .

Proof. supposeFirst, we have to find S_b:Now, take D_a:Similarly,Take D_b:Now, the symmetric division index isPut the valuesAfter the calculation, the result isThe 3D plot of symmetric division index is given in Figure 8 (f or u = 1 left, = 1 middle, and = 1 right), and we see the dependent variables of the symmetric division index on the involved parameters.

(a)

(b)

Theorem 8. Crystallographic structure of the graph of copper(I) oxide G ≈ Cu₂0[m; n], where n; m ≥ 1, and we have

Proof. supposeFirst, we have to find J_f(a;b):Take S_a:The harmonic index isNow, the result isThe 3D plot of harmonic index is given in Figure 9 (f or u = 1 left, = 1 middle, and = 1 right), and we see the dependent variables of the harmonic index on the involved parameters.

(a)

(b)

Theorem 9. Crystallographic structure of the graph of copper(I) oxide G ≈ Cu20[m; n], where n; m ≥ 1, and we have

Proof. supposeFirst, we have to find D_b:Take D_a:Take J_f(a; b):Take S(a):The inverse sum index isAfter the calculation, the result isThe 3D plot of inverse sum index is given in Figure 10 (f or u = 1 left, = 1 middle, and = 1 right), and we see the dependent variables of the inverse sum index on the involved parameters.

(a)

(b)

Data Availability

No data were used in this study.

Disclosure

All authors have not any fund, grant, and sponsor for supporting publication charges.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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Copyright © 2021 Faryal Chaudhry et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Journal of Chemistry

Computational Invariant of Chemical Structures and their Applications

M-Polynomials and Degree-Based Topological Indices of the Molecule Copper(I) Oxide

Abstract

1. Introduction

1.1. Application Background

1.2. Crystallographic Structure of Cu2O(m; n)

2. Definitions and Literature Review

2.1. M-Polynomial

2.2. Degree-Based Topological Indices

Data Availability

Disclosure

Conflicts of Interest

References

Copyright

1.2. Crystallographic Structure of Cu₂O(m; n)