On Reverse Valency Based Topological Characterization of a Chemical Compound

Rosary, Maria Singaraj; Fufa, Samuel Asefa

doi:https://doi.org/10.1155/2022/5424430

Journal of Mathematics

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Computational and Theoretical Characteristics of Chemical Graph Theory and Applications

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Research Article | Open Access

Volume 2022 | Article ID 5424430 | https://doi.org/10.1155/2022/5424430

On Reverse Valency Based Topological Characterization of a Chemical Compound

Maria Singaraj Rosary¹and Samuel Asefa Fufa²

Academic Editor: Mehar Ali Malik

Received14 Mar 2022

Revised27 Jun 2022

Accepted11 Jul 2022

Published11 Aug 2022

Abstract

Metal-organic frameworks explicit the consequence of these frameworks with adjustable implementations, namely, energy storage gadgets of magnificent electrode materials, gas store, heterogeneous catalysis, environmental hazard, estimation of chemicals, recognizing of definite gases, controlling solids, and supercapacitors. In this paper, we give explicit expression of the reverse general Randic index, the reverse atom bond connectivity index, the reverse geometric arithmetic index, the reverse forgotten index, the reverse Balaban index, the reverse augmented index, and different types of reverse Zagreb indices of the metal-organic framework M1TPyP-M2 (TPyP = 5, 10, 15, 20-tetrakis (4-pyridyl) porphyrin and M1, M2 = Fe and Co). A graphical comparison of the calculated different types of the reverse degree based topological indices with the aid of the numerical values is also included.

1. Introduction

Metal-organic frameworks (MOFs) are valuable in modern chemistry. They are described by their three-dimensional frameworks constructed of metal ions and organic molecules. Chu et al. studied the topological indices of MOFs [1]. Certain topological descriptors of certain metal-organic frameworks are well studied in 2020 [2]. In 1959, the initial metal-organic framework was described by Kinoshita et al. [3]. Metal-organic frameworks acquire a great observation required to the work of reticular chemistry for their synthesis [4]. Later, millenarian of MOFs came to be synthesized and widen the range of their prospective applications. Metal-organic frameworks have exhibited applications in the domain of gas catalysis [5], drug delivery [6], and storage [7].

Let be an ordered pair of graphs, where is a nonempty vertex set and is an edge set. A molecular graph is a set of points denoting the atoms in the molecule and collection of lines denoting the covalent bonds. The topological descriptors are helpful in the forecast of physical and chemical properties and the bioactivity of chemical compounds [8, 9]. They are playing a very important role in the field of chemistry.

Topological indices are important tools for investigating many physicochemical properties of molecules without performing any testing. They are also used to study Quantitative Structure Activity Relationship (QSAR) of pharmaceuticals to determine their molecular characteristics by numerical computation [10]. Various types of topological indices of graphs are classified into distance-based topological indices, degree-based topological indices, and spectrum-based topological indices. Among these, degree-based topological indices play a vital role in theoretical chemistry and pharmacology. Some important degree-based indices are Randic index, Zagreb indices, Harmonic index, and sum connectivity index. For example, Randic index is one of the excellent molecular descriptors in (QSAR) studies and is desirable for measuring the extent of branching of the carbon-atom skeleton of saturated hydrocarbons [11]. The atom-bond connectivity (ABC) index yields a good representation for the stability of linear and branched alkanes as well as the strain energy of cycloalkanes. For particular physical chemical properties, the diagnostic power of GA index is better than the diagnostic power of the Randić connectivity index [12]. Zagreb-type indices relieved to differentiate some alkane isomers boiling points and have aided in the invention, along with other indices, of a few thousand topological graph indices accepted in the chemical databases [13]. There are a number of research studies on the Zagreb indices and their chemical applications [14].

In chemical graph theory, a topological index predicts the biological activity of the molecular graph of chemical compounds [15]. At first, Wiener gave the idea of topological indices during his work on the boiling point of paraffin and mentioned it as “Wiener index,” an initial topological index. In the literature, the most widely used topological indices are Zagreb and Randic [16, 17]. Recently, the topological indices on product graphs and the inverse sum index of hyaluronic acid-paclitaxel molecules applied in anticancer drugs are extensively discussed [18, 19]. Second Zagreb indices of transformation graphs and total transformation graphs and topological indices on hex-derived networks are presented in [20, 21].

The degree of vertex in a graph is the total number of edges incident with vertex , and it is denoted by . A graph can be explained by numerical value, matrix, and polynomial. The concept of reverse vertex degree was given by Kulli [22]. In a graph, the maximum degree of the vertex is given by . Degree-, distance-, eccentric-, and spectrum-based indices are special varieties of topological indices. The computation of topological indices has brought up a wide diversity of dormant uses for the evaluation of (QSAR) antiparasitic products. Reverse degree based molecular descriptors of remdesivir used in the treatment of coronavirus are discussed in [23].

Recently, the reverse degree based indices of polycyclic metal-organic network were discussed by Zhao et al. [24]. Jung et al. explained the reverse degree based indices of some nanotubes [25]. Reverse degree based molecular descriptors of graphene are discussed in [26]. Reversed degree based topological indices of benzenoid systems are extensively studied in [27]. Recently, Rosary and Liu presented the reverse degree based topological analysis on the line graph and paraline graph of remdesivir used for the treatment of coronavirus [28, 29]. In this article, we examine the physical chemical properties and the biological activity of the molecular structure of the metal-organic framework M1TPyP-M2. Also, we discuss the reverse degree based topological indices of the metal-organic framework M1TPyP-M2 and compare the results graphically.”

Milan Randic introduced the first degree based index [17]. Wei et al. defined the reverse Randic index as [23]

Estrada et al. presented the atom bond connectivity index [30]. Wei et al. defined the reverse atom bond connectivity index as [23]

Vukicevic et al. proposed the geometric arithmetic index [31]. Wei et al. defined the reverse geometric arithmetic index as [23]

Gutman discussed the first and second Zagreb indices [16, 32]. Wei et al. defined the reverse first and reverse second Zagreb indices as [23]

Doslic and Gutman et al. [33, 34] presented the first and second Zagreb co-indices. Wei et al. defined the reverse first and reverse second Zagreb co-indices as [23]

Shirdel et al. [35] discussed hyper Zagreb index. Wei et al. defined the reverse hyper Zagreb index as [23]

Wei et al. defined the reverse first multiple and the reverse second multiple Zagreb indices as [23]

Furtula and Gutman [36] introduced forgotten index. Wei et al. defined the reverse forgotten index as [23]

Wei et al. defined the reverse Balban index for a graph of order and size as [23],

For a graph, the redefined first, second, and third Zagreb indices are studied in [37]. Wei et al. defined the reverse redefined first, second, and third Zagreb indices as [23]

Furtula et al. [10] presented augmented Zagreb index. Wei et al. defined the reverse augmented Zagreb index as [23]

2. Main Results

In order to analyse our results, we use the key technique of combinatorial computing, vertices degree counting, edge dividing scheme, logical techniques, and sum of degrees of neighbors’ technique. In addition, we use MATLAB program to do mathematical calculations and plot the results graphically.

The 2-dimensional structure of M1TPyP − M2 MOFs is given in Figure 1. Let G (p, q) be the graph of M1TPyP − M2 MOFs, where p and q are the number of unit cells in each row and column. The 2-dimensional structure of the molecular graph of G (2, 2) is depicted in Figure 1. The cardinality of vertices and edges in M1TPyP-M2 MOFs is and .

Based on the degree of end vertices, the edges of M1TPyP-M2 MOFs (M1, M2 = Fe and Co) can be partitioned into four partitions. The first edge partition includes of edges , where and . The second edge partition includes of edges , where and . The third edge partition includes of edges , where and . The fourth edge partition includes edges , where and . Table 1 shows the edge partition of M1TPyP-M2 MOFs.

The maximum degree of the vertex of M1TPyP-M2 MOFs is 4. By using the definition of reverse vertex degree , the reverse degree based edge partition of M1TPyP-M2 MOFs is given in Table 2.

Now, we calculate the reverse degree based topological indices as follows.

Theorem 1. We consider the graph , then the reverse general Randic index is equal to

Proof. We consider and formula to compute the resultsFor ,For ,For ,For ,

Theorem 2. We consider the graph G(p, q), then the reverse Atom Bond Connectivity index is equal to .