Volume 2022, Issue 1 3340057

Research Article

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[Retracted] Eccentricity-Based Topological Descriptors of First Type of Hex-Derived Network

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Usman Babar,

Usman Babar

Department of Mathematics, Government College University, Lahore, Pakistan gcu.edu.pk

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Asim Naseem,

Asim Naseem

Department of Mathematics, Government College University, Lahore, Pakistan gcu.edu.pk

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Hani Shaker,

Hani Shaker

orcid.org/0000-0003-2500-2179

Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore, Pakistan comsats.edu.pk

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Mian Muhammad Zobair,

Corresponding Author

Mian Muhammad Zobair

[email protected]

orcid.org/0000-0002-5749-0608

Department of Mathematics, National University of Sciences and Technology, Islamabad, Pakistan nust.edu.pk

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Haidar Ali,

Haidar Ali

orcid.org/0000-0002-9815-7158

Department of Mathematics, Riphah International University, Faisalabad Campus, Pakistan riphah.edu.pk

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Usman Babar,

Usman Babar

Department of Mathematics, Government College University, Lahore, Pakistan gcu.edu.pk

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Asim Naseem,

Asim Naseem

Department of Mathematics, Government College University, Lahore, Pakistan gcu.edu.pk

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Hani Shaker,

Hani Shaker

orcid.org/0000-0003-2500-2179

Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore, Pakistan comsats.edu.pk

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Mian Muhammad Zobair,

Corresponding Author

Mian Muhammad Zobair

[email protected]

orcid.org/0000-0002-5749-0608

Department of Mathematics, National University of Sciences and Technology, Islamabad, Pakistan nust.edu.pk

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Haidar Ali,

Haidar Ali

orcid.org/0000-0002-9815-7158

Department of Mathematics, Riphah International University, Faisalabad Campus, Pakistan riphah.edu.pk

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First published: 09 May 2022

https://doi.org/10.1155/2022/3340057

Citations: 1

Academic Editor: Andrea Penoni

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Abstract

Graph theory has made significant progress in mathematical chemistry and has gained a lot of traction among scientists due to its numerous applications in mathematical chemistry. The numerical invariants of a molecular structure are known as molecular topological descriptors, and they are highly effective for predicting their bioactivity. A number of such indices are examined and applied in pharmaceutical research, chemistry, medication development, and other fields. The eccentricity-based Zagreb indices, total eccentricity, geometric arithmetic (GA₄), atom-bond connectivity (ABC₅), and average eccentricity indices of a hex-derived network of first type are computed in this article. We also provide analytically closed formulas for these descriptors, which may be used to investigate the underlying topologies.

1. Introduction

Graph theory is a vital organ of scientific tools to study the mechanism of different molecular structures and networks without going into the labor of laboratory work. It provides valid reasons to adopt the techniques of topological indices to cater for the difficult molecular structures which might not be studied by any other technique. It has several applications in social sciences, computer science, neural network, genomics, molecular biology, and chemistry. Chemical graph theory is a hot topic among researchers and allows us to estimate some physicochemical features of chemical substances just by looking at their visual representations [1, 2]. A molecular frame can be described by covalent bonds which connects the atoms. In case of graph theory, atoms are vertices and these covalent bonds are edges between atoms. Information sciences, mathematics, and chemistry are combined in cheminformatics. This is a new research field that attracts the attention of researchers. Chemical graph theory is a branch of chemistry that uses graph theory to represent chemical substances mathematically. There has been a lot of study on this topic during the last few decades. The importance of graph theory in chemistry is demonstrated by few references [3, 4].

Consider a graph G = (V, E), where V and E are nonempty sets of vertices and edges. Chemical graph theory uses graph theory to model molecular phenomena mathematically, which is useful for studying molecular structures. Pharmaceutical researchers study and apply a wide range of TI’s in theoretical chemistry. There are several TI’s for graph in chemical graph theory that are useful in the study of chemical compounds. This approach has had a significant impact on chemical science advancement.

d(i, j) is distance between i and j and can be defined as the length of shortest path in G if i, j ∈ V(G). Eccentricity is distance between vertex i and the farthest vertex in the graph.

In a numerical format,

(1)

Farooq and Malik [5] proposed the total eccentricity index as

(2)

where is the eccentricity of the vertex s.

Average eccentricity of a graph

(3)

Many researchers like Hinz, Ilic, Dankelmann, and Tang have worked on average eccentricity index, demonstrated by few references [6–9].

Ghorbani and Khaki [10] proposed the geometric-arithmetic index based on eccentricity, as

(4)

Additional details on the average eccentricity and eccentricity-based geometric-arithmetic indices can be seen in [11]. Farahani [12] proposed the eccentricity version of the ABC index, expressed as

(5)

Imran et al. computed ABC₅ and GA₄ indices of copper oxide in [13], and Gao et al. [14] shows the results of linear polycene parallelogram benzenoid.

Some modified eccentric versions of Zagreb indices mentioned in [15, 16] are

(6)

Second and third eccentric versions of Zagreb indices are as follows, respectively:

(7)

(8)

2. Applications and Motivation

Because degree-based TI’s are beneficial for analyzing the chemical properties of various molecule structures. As a result of this inspiration, we concentrate on eccentricity-based TI’s. Topological index based on eccentricity is a useful method to predict physicochemical and toxicological properties of a substance [17–19]. Recent activity on hex-derived networks can be studied in [20–22].

Zobair et al. studied TI’s of carbon nanocones as well as TI’s of different families of David-derived networks in [23, 24]. One index, known as the ABC index, has extremely good results for determining the stability and strain energy of hydrocarbons. [25–28]. The GA index has much stronger prediction potential than the Randic connectivity index [29–31]. Within specified approximate formulas, the Zagreb indices used for computation of total Π-electron energy of the molecules [32].

3. Hex-Derived Network

Hexagonal structures had observed a lot of applications in different fields including chemical engineering, transportation, mechanical engineering, architecture, nanofabrication, and recently bioinformatics. The greatest challenge in this research direction is to comprehend the unique characteristics of hex-derived structures, relying on their structures, scales, and the materials used to allow the minimization of the materials to get minimum weight and maximum strength. Hex-derived networks are under consideration in the present research article which are further extracted from hexagonal structures. Figure 1(a)shows a hex-derived network for two dimension, Figure 1(c) demonstrates thatHX(2)has faces, and Figure 2 shows an example of HDN₁(3) [33].

Details are in the caption following the image — Open in figure viewer PowerPoint

We adopt the combinatorial computing approach, the vertex and edge partitioning technique, tools for graphs, analytic methods, degree counting, and the sum of degrees of neighbours techniques to compute our results [34, 35]. Furthermore, for numerical calculations and verifications, we use MATLAB software. We also plot mathematical conclusions using the Maple software.

(i)
For major findings, based on eccentricity, we divide the vertices of the HDN₁(n) into two sets. The eccentricity of the set V₁ is , and the number of vertices in the set is 1, β = 1, and n ≥ 2. The set V₂ contains the vertices with the eccentricity , and the number of vertices in the set V₂ is 6(3β − 4), 2 ≤ β ≤ n, n ≥ 2. In addition, the variable β denotes the distance between two vertices, which assists in the creation of this vertex division. Additionally, β denotes the range of the total number of vertices having that eccentricity. Table 1 shows the vertex division of a HDN₁ depending on the eccentricity of each vertex.

(ii)
Now, we divide the edges of the HDN₁(n) based on the eccentricity into two sets of the end vertices. The set E₁ contains edges with eccentricities , 2 ≤ β ≤ n, n ≥ 2, and E₁ set is (30m − 18), 1 ≤ m ≤ n − 1. The set E₂ contains the edges with eccentricities , 2 ≤ β ≤ n, n ≥ 2, and E₂ set is (24m − 6), 1 ≤ m ≤ n − 1. In addition, β denotes the total number of pairing with that eccentricities. Table 2 shows the edge division of a HDN₁(n) Â depending on the eccentricity of end vertices.

1. Vertex partition of HDN1(n) based on range and n.

Sets		Number of vertices	Range of β and n
V₁	n − 1	1	β = 1, n ≥ 2
V₂	n + β − 2	6(3β − 4)	2 ≤ β ≤ n, n ≥ 2

2. Eccentricity-based edge partition of HDN₁(n).

Sets		Edge frequency	β value range	Range of m and n
E₁	(n + β − 3, n + β − 2	30m − 18	2 ≤ β ≤ n	1 ≤ m ≤ n − 1
E₂	n + β − 2, n + β − 2	24m − 6	2 ≤ β ≤ n	1 ≤ m ≤ n − 1

3.1. Main Results for HDN₁(n)

The close approximations of some TI’s for this network were calculated in this section. We have founnd here the closed results for total eccentricity, average eccentricity, eccentricity version of Zagreb, eccentric ABC, and GA indices for hex-derived networks of first type.

Theorem 3.1.1. Consider HDN₁(n), ∀n ∈ N, n ≥ 2, be the graph of first type of hex-derived network, then ζ index of HDN₁(n) is

(9)

Proof. Consider HDN₁(n), ∀n ∈ N, n ≥ 2, be the first type Hex-derived network.

We calculated the total eccentricity index as follows, using the vertices partitioned from Table 1 and Equation (2):

(10)

After calculations, we have

(11)

Theorem 3.1.2. Consider HDN₁(n), ∀n ∈ N, n ≥ 2, be the graph of first type of hex-derived network, then the avec index of HDN₁(n) is

(12)

Proof. Consider HDN₁(n), ∀n ∈ N and n ≥ 2 containing 9n² − 15n + 7 vertices and 27n² − 51n + 24 edges.

We calculated the average eccentricity index as follows, using the vertices partitioned from Table 1 and Equation (3):

(13)

After calculations, we have

(14)

Theorem 3.1.3. Consider HDN₁(n), ∀n ∈ N, n ≥ 2, be the graph of first type of hex-derived network, then of HDN₁(n) is

(15)

Proof. Consider HDN₁(n), ∀n ∈ N and n ≥ 2. We calculated as follows, using Table 2 edge partitioning and Equation (6):

(16)

After calculations, we have

(17)

Theorem 3.1.4. Consider HDN₁(n), ∀n ∈ N, n ≥ 2, be the graph of first type of hex-derived network, then index of HDN₁(n) is

(18)

Proof. Consider HDN₁(n), ∀n ∈ N, n ≥ 2, be the hex-derived network of first type. We calculated the second Zagreb eccentricity index as follows, using the vertices partitioned from Table 1 and Equation (7); the proof is analogue of Theorem 3.1.3.

Theorem 3.1.5. Consider HDN₁(n), ∀n ∈ N, n ≥ 2, be the graph of first type of hex-derived network, then the index of HDN₁(n) is

(19)

Proof. Consider HDN₁(n), ∀n ∈ N, n ≥ 2, be the hex-derived network of first type.

We calculated the third Zagreb eccentricity index as follows, using Table 2 edge partitioning and Equation (8):

(20)

After calculations, we have

(21)

Theorem 3.1.6. Consider HDN₁(n), ∀n ∈ N, n ≥ 2, be the graph of first type of hex-derived network, then the GA₄ index of HDN₁(n) is

(22)

Proof. Consider HDN₁(n), ∀n ∈ N, n ≥ 2, be the hex-derived network of first type.

We calculated the eccentric GA₄ index as follows, using Table 2 edge partitioning and Equation (4):

(23)

After calculations, we have

(24)

Theorem 3.1.7. Consider HDN₁(n), ∀n ∈ N, n ≥ 2, be the graph of first type of hex-derived network, then the ABC₅ index of HDN₁(n) is

(25)

Proof. Consider HDN₁(n), ∀n ∈ N, n ≥ 2, be the hex-derived network of first type.

We calculated the eccentric ABC₅ index as follows, using Table 2 edge partitioning and Equation (5); the proof is analogue of Theorem 3.1.6.

4. Comparison and Discussion

The eccentricity-based TI’s of the HDN₁(n) first type of hex-derived network for specific values of m and β are compared in Figures 3–5. It is obvious from the graphs that the values of indices are increasing for different m and β. So that it provides an indication that the results provided for different indices are true.

5. Conclusion

In this article, eccentricity-based TI’s, the average eccentricity, eccentric version of Zagreb indices, total eccentricity index, eccentricity version of geometric-artithmatic index, and atom-bond connectivity index based on eccentricity for first type of hex-derived network are contemplated and investigate the basic topologies of these networks. Furthermore, we made a comparison in Figures 3–5 by the use of graph comparison. This analysis will facilitate researchers engaged in network science in recognizing the topology of the abovementioned networks. Furthermore, the results presented in this paper might be helpful in the QSPR/QSAR analysis to predict further structural and physicochemical properties of molecular graphs under discussion.

Conflicts of Interest

There is no conflict of interest among the authors.

Open Research

Data Availability

The data used to support the findings of this study are included within the article.

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[Retracted] Eccentricity-Based Topological Descriptors of First Type of Hex-Derived Network

Retraction(s) for this article

Retracted: Eccentricity-Based Topological Descriptors of First Type of Hex-Derived Network

Abstract

1. Introduction

2. Applications and Motivation

3. Hex-Derived Network

3.1. Main Results for HDN₁(n)

4. Comparison and Discussion

5. Conclusion

Conflicts of Interest

Open Research

Data Availability

References

Citing Literature

Figures

References

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[Retracted] Eccentricity-Based Topological Descriptors of First Type of Hex-Derived Network

Retraction(s) for this article

Retracted: Eccentricity-Based Topological Descriptors of First Type of Hex-Derived Network

Abstract

1. Introduction

2. Applications and Motivation

3. Hex-Derived Network

3.1. Main Results for HDN1(n)

4. Comparison and Discussion

5. Conclusion

Conflicts of Interest

Open Research

Data Availability

References

Citing Literature

Figures

References

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3.1. Main Results for HDN₁(n)