Degree-based topological properties of borophene sheets

1 Introduction

Two-dimensional boron atom-based structures known as borophene include boron cluster sheets. They resemble graphene, a two-dimensional sheet made of carbon atoms that are organised in a hexagonal lattice. Due to its distinctive electrical, mechanical, and thermal characteristics, borophene is a desirable material for a wide range of applications, including catalysis, energy storage, and electronics. Molecular beam epitaxy and chemical vapour deposition are the two methods that can be used to synthesise borophene. Different boron atom layouts, including triangular, honeycomb, and rectangular patterns, can be achieved to produce borophene sheets [1,2,3].

Although there is still much to learn about the characteristics and prospective uses of borophene, several encouraging findings have already been made. For instance, it has been demonstrated that borophene possesses excellent electrical conductivity and may be utilised to make transparent and stretchable conductive films. It might potentially be used as a catalyst for numerous chemical processes and in energy storage systems [4,5,6,7]. Additional information on the boron cluster sheet, known as borophene, may be obtained to gain a more comprehensive understanding of its physical and chemical characteristics, as well as its many uses [8,9,10,11,12,13,14,15].

A prime instance of a molecular descriptor that offers details on the connectivity of atoms in a molecule is degree-based topological numbers. These descriptions are based on the idea of a node’s degree in a molecular network, which denotes how many nearby atoms are directly related to that node [16,17]. There are several types of degree-based topological numbers, including the following:

• Vertex degree: The number of edges (bonds) that are connected to a given atom in a molecule.

• Average degree: The average number of edges that are connected to all atoms in a molecule.

• Eccentricity: The maximum number of edges that must be traversed to reach a given atom from any other atom in the molecule.

• Wiener number: The sum of all pairwise distances between all pairs of atoms in a molecule.

• Zagreb number: The sum of the squares of the vertex degrees of all atoms in a molecule [18,19].

Applications for degree-based topological numbers include drug design, modelling of quantitative structure–activity relationships, and molecular similarity analysis. They can offer a rapid and efficient technique to compare the connectivity of atoms in various compounds and are comparatively easy to compute [20,21]. Chemical species and chemical processes may be represented as nodes and edges in a graph, which can then be used to depict a chemical network. Chemical reactions link the reactants and products of the reaction together at their respective nodes. This makes it possible to describe the chemical network as a directed graph, where each node stands for a particular chemical species and each edge for a particular chemical process [22,23].

The transformation of a molecular system into a graph encompasses many distinct stages: It briefly indicates the substance entities that are participating inside the framework. Such a category encompasses several molecular objects, such as ions, molecules, and radicals in the body, as well as similar constituents. It ensures to indicate the molecular processes that establish connections between the various entities within the framework. Such a process might entail the examination and interpretation of empirical information, mathematical formulas, or various kinds of knowledge. Such compound entities should be allocated to individual vertices within the structure of the graph, while each of the chemical responses should be represented by an edge that directly connects the vertices that correspond to the reactants and products involved in the process [24]. Further data are included in visualization pertaining to relevant parameters for the analysis of the molecular system, such as the rate of reaction, concentrations of substances, or thermodynamic characteristics [25]. After the conversion of the molecular system into the form of a graph, a range of networking-theoretic methods and techniques can potentially be used to examine the structure as well as the motion of the system as a whole [26]. One potential implementation of centralization metrics is the identification of significant vertices within a system. Similarly, neighbourhood discovery techniques may be employed to locate clusters of vertices that exhibit high levels of interconnectedness. The use of this methodology could offer valuable observations on the dynamics of intricate molecular infrastructure, hence facilitating the strategic development of novel molecular interactions and procedures [27].

The study by Ajmal et al. [28] provides a comprehensive analysis of toroidal polyhex networks and generalized prisms within the context of our selected subject matter of geometrical numbers. Maji and Ghorai [29] examined the M-polynomials of the improvement networks, and in the study of Balasubramanian [30], Topochemie-2020 is a computer programming application designed for the calculation of structural numbers. In addition to M-polynomials, the introduction and extensive analysis of omega-sine waves have been undertaken for antiepileptic medicines [31]. The significance of uses in this unique notion is shown by the extensive examination of numbering polynomial functions in the field of chemical reactions [32,33], and topological numbers associated with zero-divisor graphs are noted in in the study of Ahmad and Lopez [34]. The comprehensive elucidation of oxide and silicate networks, together with their respective topological numbers, is provided in the study of Javaid and Jung [35]. The study by Ahmad et al. [36] explores the use of optical transposition interconnect systems to analyse switched systems, introducing the innovative notion of structural numbers. In the study of Munir et al. [37], the titania nanotubes; in the study of Siddiqui et al. [38], nanostar dendrimers; and in the study of Gao et al. [39], the notion of reverse degree-based topological numbers is discussed in relation to dendrimers. For further study on the application part of this work, one can refer to previous studies [40,41,42]. Particularly, the work on the Boron cluster, borophene sheet, and other chemical structures that are closely related to this work is given in the study of Zhang et al. [43].

In addition to numerical values, there are many more subjects that are intricately connected to topological numbers and their corresponding literary work, which may be discovered in previous studies [44,45,46,47,48,49,50,51,52,53,54,55,56].

2 Degree-based results of γ -sheet of boron clusters

Boron cluster sheets are two-dimensional boron atom-based formations called borophene. They are similar to the two-dimensional sheet known as graphene, which is composed of carbon atoms arranged in a hexagonal lattice. The unique electrical, mechanical, and thermal properties of borophene make it a sought-after substance for a variety of uses, such as catalysis, energy storage, and electronics. There are two ways to manufacture borophene: chemical vapour deposition and molecular beam epitaxy. To create borophene sheets, many arrangements of boron atoms, such as triangular, honeycomb, and rectangular patterns, may be created. Figure 1 shows the graph of γ -sheet of boron clusters for p = 6 and q = 3 , and Figure 5 shows a graph of α -sheet of boron clusters for p = 2 and q = 3 .

Figure 1

γ -sheet of boron clusters for p = 6 and q = 3 .

This means that the set V i contains the vertices of degree i . The set of vertices with respect to their degrees are as follows:

as | V 3 | = 2 p + 4 q , | V 4 | = 2 p ( q − 1 ) , | V 5 | = 2 ( p − 1 ) q , and | V 6 | = pq .

Let us divide the edges of G ( p , q ) into partition sets according to the degree of its end vertices. Let

Note that E ( G ( p , q ) ) = Ξ 3,3 ⊔ Ξ 3,4 ⊔ Ξ 3,5 ⊔ Ξ 3,6 ⊔ Ξ 4,4 ⊔ Ξ 4,5 ⊔ Ξ 4,6 ⊔ Ξ 5,5 ⊔ Ξ 5,6 . The number of edges incident to the two vertices of degree 3 is 2 q + 4 , so | Ξ 3,3 | = 2 q + 4 . The number of edges incident to one vertex of degree 3 and other vertices of degrees 4, 5, and 6 are 4 q − 4 , 4 p − 4 , and 2 p + 4 q , respectively. Therefore, | Ξ 3,4 | = 4 q − 4 , | Ξ 3,5 | = 4 p − 4 , and | Ξ 3,6 | = 2 p + q . The number of edges incident to the two vertices of degree 4 is p ( q − 1 ) , so | Ξ 4,4 | = p ( q − 1 ) . The number of edges incident to one vertex of degree 4 and other vertices of degrees 5 and 6 are 4 ( p − 1 ) ( q − 1 ) , and 2 p ( q − 1 ) , respectively. Therefore, | Ξ 4,5 | = 4 ( p − 1 ) ( q − 1 ) and | Ξ 4,6 | = 2 p ( q − 1 ) . The number of edges incident to one vertex of degree 5 and other vertices of degrees 5 and 6 are ( p − 1 ) q and 4 ( p − 1 ) q , respectively. Therefore, | Ξ 5,5 | = ( p − 1 ) q and | Ξ 5,6 | = 4 ( p − 1 ) q . Hence,

In the following theorems, we determined the degree-based topological indices of γ -sheet of boron clusters.

The Randić index is

the second Zagreb index is

the second modified Zagreb index is

the general sum-connectivity index χ α of G ( p , q ) is

the sum-connectivity index is

the first Zagreb index is

and the hyper-Zagreb index is

If α = − 1 2 , then we the get Randić index

If α = 1 , then we obtain the second Zagreb index

If α = − 1 , then we obtain the second modified Zagreb index

For the general sum-connectivity index χ α of G ( p , q ) , we obtain ψ ˆ ( d τ , d σ ) = ( d τ + d σ ) α . Therefore, ψ ˆ ( 3,3 ) = ( 6 ) α , ψ ˆ ( 3,4 ) = ( 7 ) α , ψ ˆ ( 3,5 ) = ( 8 ) α , ψ ˆ ( 3,6 ) = ( 9 ) α , ψ ˆ ( 4,4 ) = ( 8 ) α , ψ ˆ ( 4,5 ) = ( 9 ) α , ψ ˆ ( 4,6 ) = ( 10 ) α , ψ ˆ ( 5,5 ) = ( 10 ) α , and ψ ˆ ( 5,6 ) = ( 11 ) α . Thus, by Lemma 2.1,

If α = − 1 2 , then we obtain the sum-connectivity index

If α = 1 , then we obtain the first Zagreb index χ 1 ( G ( p , q ) )

If α = 2 , then we obtain the hyper-Zagreb index χ 2 ( G ( p , q ) )

the atom-bond connectivity index ABC of G ( p , q ) is

the augmented Zagreb index AZI of G ( p , q ) is

the harmonic index H of G ( p , q ) is

and the symmetric division degree index SDD of G ( p , q ) is

For the atom-bond connectivity index ABC of G ( p , q ) , we obtain ψ ˆ ( d τ , d σ ) = d τ + d σ − 2 d τ d σ . So ψ ˆ ( 3,3 ) = 4 9 , ψ ˆ ( 3,4 ) = 5 12 , ψ ˆ ( 3,5 ) = 6 15 , ψ ˆ ( 3,6 ) = 7 18 , ψ ˆ ( 4,4 ) = 6 16 , ψ ˆ ( 4,5 ) = 7 20 , ψ ˆ ( 4,6 ) = 8 24 , ψ ˆ ( 5,5 ) = 8 25 , and ψ ˆ ( 5,6 ) = 9 30 . Thus, by Lemma 2.1,

For the augmented Zagreb index AZI of G ( p , q ) , we obtain ψ ˆ ( d τ , d σ ) = d τ d σ d τ + d σ − 2 3 . Therefore, ψ ˆ ( 3,3 ) = 729 64 , ψ ˆ ( 3,4 ) = 1,728 125 , ψ ˆ ( 3,5 ) = 3,375 216 , ψ ˆ ( 3,6 ) = 5,832 343 , ψ ˆ ( 4,4 ) = 4,096 216 , ψ ˆ ( 4,5 ) = 8,000 343 , ψ ˆ ( 4,6 ) = 13,824 512 , ψ ˆ ( 5,5 ) = 15,625 512 , and ψ ˆ ( 5,6 ) = 27,000 729 . Thus, by Lemma 2.1,

AZI ( G ( p , q ) ) = 545,166,215 1,580,544 pq − 1,291,939 18,522 p − 74,604,408,241 592,704,000 q + 14,447,819 686,000 = ( 344.92 ) pq − ( 69.752 ) p − ( 125.87 ) q + 21.061 .

For the harmonic index H of G ( p , q ) , we obtain ψ ˆ ( d τ , d σ ) = 2 d τ + d σ . Therefore, ψ ˆ ( 3,3 ) = 2 6 , ψ ˆ ( 3,4 ) = 2 7 , ψ ˆ ( 3,5 ) = 2 8 , ψ ˆ ( 3,6 ) = 2 9 , ψ ˆ ( 4,4 ) = 2 8 , ψ ˆ ( 4,5 ) = 2 9 , ψ ˆ ( 4,6 ) = 2 10 , ψ ˆ ( 5,5 ) = 2 10 , and ψ ˆ ( 5,6 ) = 2 11 . Thus, by Lemma 2.1,

For the symmetric division degree index SDD of G ( p , q ) , we obtain ψ ˆ ( d τ , d σ ) = d τ 2 + d σ 2 d τ d σ . Therefore, ψ ˆ ( 3,3 ) = 18 9 , ψ ˆ ( 3,4 ) = 25 12 , ψ ˆ ( 3,5 ) = 34 15 , ψ ˆ ( 3,6 ) = 45 18 , ψ ˆ ( 4,4 ) = 32 16 , ψ ˆ ( 4,5 ) = 41 20 , ψ ˆ ( 4,6 ) = 52 24 , ψ ˆ ( 5,5 ) = 50 25 , and ψ ˆ ( 5,6 ) = 61 30 . Thus, by Lemma 2.1,

the first redefined Zagreb index Re Z G 1 of G ( p , q ) is

the second redefined Zagreb index Re Z G 2 of G ( p , q ) is

the third redefined Zagreb index ReZ G 3 of G ( p , q ) is

and the variation of the Randić index R ′ of G ( p , q ) is

For the first redefined Zagreb index ReZ G 1 of G ( p , q ) , we obtain ψ ˆ ( d τ , d σ ) = d τ + d σ d τ d σ . Therefore, ψ ˆ ( 3,3 ) = 6 9 , ψ ˆ ( 3,4 ) = 7 12 , ψ ˆ ( 3,5 ) = 8 15 , ψ ˆ ( 3,6 ) = 9 18 , ψ ˆ ( 4,4 ) = 8 16 , ψ ˆ ( 4,5 ) = 9 20 , ψ ˆ ( 4,6 ) = 10 24 , ψ ˆ ( 5,5 ) = 10 25 , and ψ ˆ ( 5,6 ) = 11 30 . Thus, by Lemma 2.1,

For the second redefined Zagreb index ReZ G 2 of G ( p , q ) , we obtain ψ ˆ ( d τ , d σ ) = d τ d σ d τ + d σ . Therefore, ψ ˆ ( 3,3 ) = 9 6 , ψ ˆ ( 3,4 ) = 12 7 , ψ ˆ ( 3,5 ) = 15 8 , ψ ˆ ( 3,6 ) = 18 9 , ψ ˆ ( 4,4 ) = 16 8 , ψ ˆ ( 4,5 ) = 20 9 , ψ ˆ ( 4,6 ) = 24 10 , ψ ˆ ( 5,5 ) = 25 10 , and ψ ˆ ( 5,6 ) = 30 11 . Thus, by Lemma 2.1,

For the third redefined Zagreb index ReZ G 3 of G ( p , q ) , we obtain ψ ˆ ( d τ , d σ ) = d τ d σ ( d τ + d σ ) . Therefore, ψ ˆ ( 3,3 ) = 54 , ψ ˆ ( 3,4 ) = 84 , ψ ˆ ( 3,5 ) = 120 , ψ ˆ ( 3,6 ) = 162 , ψ ˆ ( 4,4 ) = 128 , ψ ˆ ( 4,5 ) = 180 , ψ ˆ ( 4,6 ) = 240 , ψ ˆ ( 5,5 ) = 250 , and ψ ˆ ( 5,6 ) = 330 . Thus, by Lemma 2.1,

For the variation of the Randić index R ′ of G ( p , q ) , we obtain ψ ˆ ( d τ , d σ ) = 1 max { d τ , d σ } . Therefore, ψ ˆ ( 3,3 ) = 1 3 , ψ ˆ ( 3,4 ) = 1 4 , ψ ˆ ( 3,5 ) = 1 5 , ψ ˆ ( 3,6 ) = 1 6 , ψ ˆ ( 4,4 ) = 1 4 , ψ ˆ ( 4,5 ) = 1 5 , ψ ˆ ( 4,6 ) = 1 6 , ψ ˆ ( 5,5 ) = 1 5 , and ψ ˆ ( 5,6 ) = 1 6 . Thus, by Lemma 2.1,