Comparative study of vertex-edge based indices for semi-capped carbon nanotubes

1.   Introduction

Carbon nanotubes have been widely used in research since their initial development by the authors of ^[1]. Carbon nanotubes are remarkable compounds as compared to others. Their thermal conductivity is higher than that of other materials, and their electrical conductivity is comparable to copper. According to the manufacturer, they are more robust, can tackle larger currents and have a much better tensile strength than steel ^[2].

Carbon nanotubes that are capped or semi-capped are one of the nanotube types. For distinct armchair carbon nanotube symmetries, semi-capped implies one-end open nanotubes, whereas capped refers to both-end closed nanotubes ^[3]. While carbon nanotubes are well-known for their applications, semi-capped and capped nanotubes are equally valuable in a variety of scientific fields. The overall uses of semi-capped, capped and carbon nanotubes are presented in the following literature: the detection of subsurface defects ^[4], a material in photovoltaics and saturable absorbers ^[5], thin films ^[6] fuel cells ^[7], a material for CVD growth with a palm oil precursor ^[8], applications in regenerative medicine and pharmaceuticals ^[9], as a memory device ^[10], a material in tissue and biomedical engineering ^[11,12], a material for self-cleaning solar cell coating and anti-reflection ^[13], a material to immobilize and improve the function of certain biomolecules ^[14], a material in biochemical sensors for the analysis of pyrophosphate, cholesterol, lactate, glucose, alcohols and other analytes ^[15], carbon nanotubes are used in the applications of DNA and proteins and found in ^[16]. Additionally, for nanostructure details, particularly carbon nanotubes, we refer the reader to ^{[17,18,19,20]}; also, apart from the applications listed above, we recommend ^{[21,22,23,24,25]}.

Limited to the scope of the paper, we recommend ^[24,25] for more information on chiral and zigzag nanotubes. The nanotubes divided into single- and multi-walled nanotubes resulting from the drifting of single- and multi-walled sheets of graphene ^[13].

In the fields of chemical graph theory, molecular topology and mathematical chemistry, a topological index, also known as a connectivity index is a type of molecular descriptor that is calculated based on the molecular graph of a chemical compound. Topological indices are numerical parameters of a graph that characterize its topology, and they are usually graph-invariant. Topological indices are used, for example, in the development of quantitative structure-activity relationships in which the biological activity or other properties of molecules are correlated with their chemical structure. Wiener first used a topological index in science to investigate paraffin breaking points ^[26]; this was followed by other efforts to elucidate physico-chemical features, culminating in the presentation of a variety of topological descriptors. The authors of ^[27] put forward a new idea for degree-based topological indices, which they nambed "$ve$-degree and $ev$-degree". The interesting basics of this new idea are detailed in ^{[28,29,30,31]}.

Many papers have been written and various studies have been performed on topological indices as a result of the wide range of applications of topological indices in biology and chemistry ^[32,33,34]. For a molecular structure, the edge-splitting approach was applied in ^[35]; honeycomb networks are discussed in ^[36,37]; a hexagon star network was developed in ^[38]; a hex-derived network is described in ^[39]; interconnection system topological indices are discussed in ^[40]. In addition, some biological, chemical and other degree-based properties of structures are discussed in ^[41,42]. The articles solely focused on the vertex-edge and edge-vertex topological indices are given here in detail. For some examples, ^[43] discusses some general graphs on this topic, titanium dioxide nanotubes are discussed with this idea of indices in ^[44], ^[45] focuses on some metal trihalides networks and the hexagon star network have been studied ^[46]. Structures devised by the generalizations of two graph operations and their topological descriptors detailed in ^[47], computation of edge- and vertex-degree-based topological indices for tetrahedral sheets of clay minerals found in ^[48]. Few studies on the edge-vertex- and vertex-edge-related topological indices are presented in ^[49,50,51]. Some new and linked topics are dicussed in ^{[52,53,54,55]}.

With the exception of the examples given above, topological indices have other uses, particularly in chemistry ^[56,57]. The main focus of this paper is on expanding the applicability of armchair carbon nanotubes ($ACNT\left({p}, {q}\right)$) and investigating various degree-based topological indices for armchair carbon semi-capped nanotubes ($ACSCNT\left({p}, {q}\right)$) and armchair carbon capped nanotubes ($ACCNT\left({p}, {q}\right)$).

A graphical representation of an armchair carbon capped nanotube is shown in Figure 1. Figure 2 depicts a three-dimensional view of capped and semi-capped armchairs.

2.   Preliminaries

A combination of $V$, which is referred to as a vertex set that is usually non-empty, and $E$, which is the edge set for vertices, is called a graph and denoted by $G = \left(V, E\right).$ Let say that the order and size or the number of members in the vertex and edge sets are denoted by $x$ and $y$, respectively. The non-empty vertex set is transferred from atoms, and the edges are usually denoted by the bonds in a molecular graph. Let a notation $\Psi\left(\theta\right)$ be the count of edges attached to a vertex $\theta$; this context is named as a degree of the $\theta$-vertex. A graph can also be represented by a sequence of numbers or numerical quantities; sometimes, a matrix is also replaced by a graph. A numerical way of representing a graph is a topological index, and this context shows the different characterizations of the graph. There are many different types of topological indices, and they are distinguished by their methodologies. But, here, we will follow the method for degree-based topological indices, and it is very popular in fields ranging from chemistry through chemical graph theory. Given below are some topological indices based on the degree relation of a graph that we implement to evaluate the main theorems in this paper.

The term $N\left(\theta\right)$ is called the open neighborhood degree of a vertex $\theta$, and this concept is defined as the count of vertices attached to a $\theta$-vertex. We take union of a vertex $\theta$ in the $N\left(\theta\right)$ set, which is defined as a closed neighborhood of the $\theta$-vertex and labeled as $N[\theta].$ The $ev$-degree of any edge $e = \theta\vartheta\in E$ is labeled as $\Psi_{ev}\left(e\right)$ and defined as the total count of unions of the closed neighborhoods of vertices $\theta$ and $\vartheta.$ The $ve$-degree written is as $\Psi_{ve}\left(\theta\right)$ and defined for any vertex $\theta \in V$ by the total count of different edges that are attached to any closed neighborhood vertex of $\theta.$ The intense literature on this topic and clearly view of these definitions we refer to see ^[55]. Moreover, few of the topological indices for the above defined methods are formulated as below.

The $ev$-degree Zagreb $\alpha$ index is defined in Eq (2.1); the first $ve$-degree Zagreb $\alpha$ index and first $ve$-degree Zagreb $\beta$ index are defined in Eqs (2.2) and (2.3); the second $ve$-degree Zagreb index is described in the Eq (2.4); the $ve$- and $ev$-degree Randić indices are respectively defined in Eqs (2.5) and (2.6); the $ve$-degree atom-bond connectivity index is described by the Eq (2.7). Equation (2.8) presents the $ve$-degree geometric-arithmetic index, while Eqs (2.9) and (2.10) described the $ve$-degree harmonic index and $ve$-degree sum-connectivity index, respectively.

3.   Results for $ACSCNT\left({p}, {q}\right)$

Table 2 shows the different types of vertices by degree, while Tables 1 and 3 show that for edges; note that these specifications represent the semi-capped nanotubes made of carbon, i.e., $ACSCNT\left({p}, {q}\right).$ In this section, we determine the $ev$-degree Zagreb index, first $ve$-degree Zagreb $\alpha$/$\beta$ index, second $ve$-degree Zagreb $\beta$ index, $ve$-degree atom-bond connectivity ($\mathbb{ABC}_{ve}$) index, $ve$-degree geometric-arithmetic ($\mathbb{GA}_{ve}$) index, $ve$-degree Randic index, $ev$-degree Randic index, $ve$-degree sum-connectivity ($\mathbb{\chi}_{ve}$) index and $ve$-degree harmonic index ($\mathbb{H}_{ve}$) for semi-capped nanotubes made of carbon, i.e., $ACSCNT\left({p}, {q}\right)$; note that, throughout the results we labeled this structural graph with $G_{p, q}.$

3.1.   $ev$-degree Zagreb index

The $ev$-degree Zagreb index can be computed by using Table 1, as follows:

3.2.   First $ve$-degree Zagreb $\alpha$ index

The first $ve$-degree Zagreb $\alpha$-index can be computed by using Table 2, as follows:

3.3.   First $ve$-degree Zagreb $\beta$ index

The first $ve$-degree Zagreb $\beta$ index can be determined by using Table 3, follows:

3.4.   Second $ve$-degree Zagreb index

The second $ve$-degree Zagreb index can be calculated by using Table 3, as follows:

3.5.   $ve$-degree Randic index

The $ve$-degree Randić index can be calculated by using Table 3, as follows:

3.6.   $ve$-degree atom-bond connectivity index

The $ve$-degree atom-bond connectivity index can be determined by using the values in Table 3, as follows:

3.7.   $ve$-degree geometric-arithmetic index

The $ve$-degree geometric-arithmetic index can be calculated by using the values defined in Table 3, as follows:

3.8.   $ve$-degree harmonic index

The $ve$-degree harmonic index can be determined by using the values defined in Table 3, as follows:

3.9.   $ve$-degree sum-connectivity index

The $ve$-degree sum-connectivity index can be determined by using the values in Table 3, as follows:

4.   Conclusions and discussion

The numerical values of all of the above-mentioned parameters, i.e., ${M}_{ev},$ ${M}^1_{\alpha ve},$ ${M}^1_{\beta ve},$ ${M}^2_{ve},$ ${R}_{ve},$ ${ABC}_{ve},$ ${GA}_{ve},$ ${H}_{ve}$ and ${\chi}_{ve},$ are defined in Tables 4 and 5. We present these results graphically in Figures 3 and 4.

Acknowledgements

This work was supported by the National Science Foundation of China (11961021 and 11561019), Guangxi Natural Science Foundation (2020GXNSFAA159084), and Hechi University Research Fund for Advanced Talents (2019GCC005).

Conflict of interest

The authors declare that there is no conflict of interest.