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Application of Molecular Topological Descriptors in Chemistry

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Research Article | Open Access

Volume 2021 |Article ID 8792585 | https://doi.org/10.1155/2021/8792585

Hong Yang, Muhammad Naeem, "Topological Descriptors of M-Carbon ", Journal of Chemistry, vol. 2021, Article ID 8792585, 14 pages, 2021. https://doi.org/10.1155/2021/8792585

Topological Descriptors of M-Carbon

Academic Editor: Muhammad Kamran Jamil
Received08 Jun 2021
Accepted02 Oct 2021
Published18 Oct 2021

Abstract

We have studied topological indices of the one the hardest crystal structures in a given chemical system, namely, M-carbon. These structures are based and obtained by the famous algorithm USPEX. The computations and applications of topological indices in the study of chemical structures is growing exponentially. Our aim in this article is to compare and compute some well-known topological indices based on degree and sum of degrees, namely, general Randić indices, Zagreb indices, atom bond connectivity index, geometric arithmetic index, new Zagreb indices, fourth atom bond connectivity index, fifth geometric arithmetic index, and Sanskruti index of the M-carbon . Moreover, we have also computed closed formulas for these indices.

1. Introduction

One of the hardest structures of carbon is diamond. In 2011, Andriy and Artem searches 9500 structures of different system sizes, and they produced a large number of superhard allotropes; these allotropes being as hard as diamonds [1]. One of the superhard carbon allotropes that they studied was M-carbon (Figure 1). Scientists also believe that the synthesis and practical applications of some of these structures may be possible. Some studies also exist giving indications that these types of carbon allotropes such as M-carbon have been obtained by applying cold compression on graphite [2, 3].

In this study, we intend to study and compute the degree-based topological indices of M-carbon structures. One of the first and very old topological indices is the Wiener index [4], and this index is also known as the path number. After that, the scientists of various field started exploring this new technique to study chemical and physical properties of chemical structure, compounds, and molecules. A list of topological indices that we shall discuss in this study is given in Table 1, which includes Randić index, general Randić indices, Zagreb indices, atom bond connectivity index, geometric arithmetic indices, new Zagreb indices, fourth atom bond connectivity index, fifth geometric arithmetic index, and Sanskruti index. For some literature study and results related to these indices, see [513].


S. no.Topological indexNotationFormulaAuthors

1Randić indexRandić [14] in 1975
2General Randić indexBollobás and Erdös [15] and Amic et al. [16] in 1988
3Atom bond connectivity indexEstrada et al. [17]
4Geometric arithmetic indexVukićevič and Furtula. [18]
5First ZagrebGutman and Trinajstić in 1972[19]
6Second ZagrebGutman and Das in 2004 [20]
7New degree-based Zagreb indexShirdel et al. [21] in 2013
8First Zagreb coindexIn 2008, Došlić [22]
9Second Zagreb coindexSame as above
10Fourth atom bond connectivity indexGhorbhani and Hosseinzadeh [23]
11Fifth version of geometric arithmetic indexGraovoc et al. [24]
12Sanskruti indexIn 2016, Hosamani [25]

In Table 1, the number represents the sum and the number represents the sum , where is the degree of vertex and represents the degree of vertex . In 2016, Gutman et al. [26] proved the following theorems for some of the indices in Table 1.

Theorem 1 (See [26]). Let be a graph with vertices and edges. Then,

Theorem 2 (See [26]). Let be a graph with vertices and edges. Then,

2. Construction of for Topological Study

In this section, we shall present our main results about the M-carbon structure denoted as . First, we need to give a brief explanation of the variables in the notation . To find and compute the topological indices of the -carbon structure, we have introduced a way of constructing its structure by the means of these three variables, where represents the unit as shown in Figures 2(a) and 2(b) represents a chain containing three units, where the connection (bond) is shown in blue color. The variable represents the number of connected chains with each having numbers of units (Figure 3). The variable represents the number of connected layers. There are two types of layers odd layer for and even layers for , both are generated by different unit cells. The one depicted in Figure 3 is the odd layer (that is for ) which was generated by the unit of Figure ). The unit cell of an even layer is shown in Figure 4(a), the chain in even layer is shown in Figure 4(b), and Figure 4(c) depicts an even layer. Then, finally, the M-carbon structure is shown in Figure 5, which also depicts how two layers, an even and odd, are connecting. In Figure 5, these connections (bonds) between two layers are shown in red colour. So, in this way, we get structure of M-carbon (Figure 1). By our construction, the graph of M-carbon consists of number vertices and number of edges.

3. Main Results

The graph has , and vertices of degrees , and 4, respectively. The degree-based edge partition of is given in Table 2.


Frequency

(1, 2)2t + 2
(1, 3)2st−2
(2, 2)2
(2, 3)8r + 2s + 2t−8
(2, 4)4st−4s−6t + 6
(3, 3)6rt + 2sr−4r−2s−3t + 5
(3, 4)4rs + 4rt−2s−4t + 2
(4, 4)16rst−8rs−14rt−11st−4r + 7s + 10t−7

In all the theorems in the following, we used Maple for the computations of mathematical expression and graphical comparisons.

Theorem 3. Let be the chemical structural graph of M-carbon , and the general Randić index of is given by

Proof. Let be the chemical structural graph of M-carbon ; then, by using serial number 2 of Table 1 and edge partition given in Table 2, the general Randić index of is computed as follows:The result follows after some simple computations from this equation.
It is very simple and clear to see that from the above Theorem 3, the following Corollary is true.

Corollary 1.

Corollary 2. The second Zagreb index is the same as for , so can be obtained from Corollary 1.

Theorem 4. The first Zagreb index of M-carbon structure is given by

Proof. Let be the chemical structural graph of M-carbon ; then, by using serial number 5 of Table 1 and edge partition given in Table 2, the first Zagreb index of is computed as follows:Thus, the result follows by simple calculations.
The next theorem gives us the new degree-based Zagreb index as defined in [21].

Theorem 5. The new degree-based Zagreb index of M-carbon structure is given by

Proof. Let be the chemical structural graph of M-carbon ; then, by using serial number 7 of Table 1 and edge partition given in Table 2, the new degree-based Zagreb index of is computed as follows:Thus, the result follows by simple calculations.
In the next two theorems, we shall compute the newly defined Zagreb coindex indices which are defined in the form of nonedges of a chemical graph.

Theorem 6. The first Zagreb coindex index of M-carbon structure is given by

Proof. The first Zagreb coindex index of chemical structural graph of M-carbon can be computed by using both serial number 8 of Table 1 and Theorem 1. It is explained and calculated as follows:Thus, the result follows by simple calculations.

Theorem 7. The second Zagreb coindex index of M-carbon structure is given by

Proof. The second Zagreb coindex index of chemical structural graph of M-carbon can be computed by using both serial number 9 of Table 1 and Theorem 2. It is explained and calculated as follows:Thus, the result follows by simple calculations.
In the coming two theorems, we shall find closed formulas for the ABC and GA indices of M-carbon structure.

Theorem 8. Consider the graph of M-carbon with ; then, its index is equal to

Proof. Let be the chemical structural graph of M-carbon ; then, by using serial number 3 of Table 1 and edge partition given in Table 2, the index of is computed as follows:The results now follow after some simple computations of above expression.

Theorem 9. Consider the graph of M-carbon with ; then, its index is equal to

Proof. Let be the chemical structural graph of M-carbon ; then, by using serial number 4 of Table 1 and edge partition given in Table 2, the index of is computed as follows:The results now follow after some simple computations of the above expression.
Table 3 provides the edge partition of the M-carbon structure on the bases of sum of the degrees in the open neighbourhood of the end vertices of each an edge for each edge of . The first of such type of indices were introduced by Ghorbhani and Hosseinzadeh [23], and then, another one was introduced by Hosamani [25]. These are defined in Table 1.
In the following three theorems, we gave closed formula for the three indices, namely, fourth atom bond connectivity index , fifth version of geometric arithmetic index , and Sanskruti index for the graph of .


FrequencyFrequency

(3, 7)4(3, 8)2
(4, 11)2t(4, 12)2st−2s−4t + 6
(4, 13)2st−2s−4t + 4(5, 5)2
(5, 7)2(5, 8)2
(6, 7)2(6, 8)4s−10
(6, 9)8r−8(7, 8)2t−2
(7, 11)2(7, 12)2t−2
(7, 15)2(8, 8)2t−2
(8, 10)2t−2(8, 11)2t−2
(8, 12)4st−10t + 6(8, 14)2
(8, 15)2t−4(9, 10)8r−8
(9, 12)2(9, 14)2r−2
(9, 15)4r−6(10, 10)4rs + 6rt−16r−4s−8t + 18
(10, 11)2(10, 14)4r−6
(10, 15)4rs + 4rt−12r−4s−4t + 12(11, 12)2t−2
(11, 15)2t−2(11, 16)2
(12, 13)2st−2s−4t + 4(12, 15)2t
(12, 16)2st + 2s−4t−6(13, 14)2t−2
(13, 15)4st−4s−10t + 10(14, 15)4r−4
(14, 16)4r−6(15, 15)6rt−2r−7t + 1
(15, 16)10rs−28r−10s + 4rt−4t + 30(16, 16)16rst−24rt−20rs−21st + 38t + 23s + 24r−40

Theorem 10. The index of the graph of M-carbon with is given by

Proof. Let be the chemical structural graph of M-carbon ; then, by using serial number 10 of Table 1 and edge partition given in Table 3, the index of is computed as follows:The results now follow after some simple computation of the above expression.

Theorem 11. Consider the graph of M-carbon with ; then, its index is given by