Advances in Materials Science and Engineering

Volume 2018, Article ID 5729291, 11 pages

https://doi.org/10.1155/2018/5729291

## Topological Aspects of Boron Nanotubes

^{1}School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China^{2}Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore, Pakistan

Correspondence should be addressed to Hani Shaker; moc.liamg@teu.inah

Received 6 April 2018; Accepted 16 May 2018; Published 4 July 2018

Academic Editor: Jamal Berakdar

Copyright © 2018 Jia-Bao Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The degree-based topological indices are used to correlate the physical and chemical properties of a molecule with its chemical structure. Boron nanotubular structures are high-interest materials due to the presence of multicenter bonds and have novel electronic properties. These materials have some important issues in nanodevice applications like mechanical and thermal stability. Therefore, they require theoretical studies on the other properties. In this paper, we present certain degree-based topological indices such as , the fourth , , and the fifth indices for boron triangular and boron-*α* nanotubes.

#### 1. Introduction

Mathematical chemistry is a branch of theoretical chemistry in which we get information about the molecular structure by using mathematical techniques without assigning that structure to quantum mechanics [1, 2]. Chemical graph theory is a branch of mathematical chemistry which implements graph theory to study mathematical modeling of chemical aspects [3]. This theory shows a prominent effect on the extension of the chemical sciences [4].

The chemical structure of a molecule is strongly related to its chemical properties such as strain energy, boiling point, and heat of formation. Molecular graphs can be used to model the chemical structures of molecules and molecular compounds by considering atoms as vertices and the chemical bonds between the atoms as edges. Consider molecular graph *G* having vertex set and edge set . Let be the set of edges of *G* that are incident with a vertex , then the degree of *p* is defined as the cardinality of the set and , where set consists of all neighbor vertices of *p*, that is, .

A topological index is the graph invariant which is used to correlate the physical and chemical properties of a chemical compound with its molecular graph. In this sense, topological indices are based on several topological aspects of the corresponding molecular graph. The use of topological indices is particularly important when using experimental methods leads to waste of time and financial expenditures in large amounts and theoretical methods have not been successful. Topological indices are used to correlate physical properties of chemical structures in QSPR/QSAR studies and provide a measure of structural similarity/stability/diversity of chemical databases. The relative stability of the fullerenes has been correlated with topological indices in [5]. In [6], topological indices are also used to predict the stable isomers of a given fullerene, and for detailed study, we refer [7].

Generally, topological indices can be categorized in three classes: degree-based, distance-based, and spectrum-based indices. In this paper, certain degree-based topological indices are going to be discussed because of their great applications in chemical graph theory. For recent study of distance-based indices, we refer [8, 9], and for spectrum-based indices, we refer [10, 11].

The first degree-based topological index is the Randić connectivity index which was presented by Randić [12] and is defined as

This index has been shown to reflect molecular branching and is deeply examined by chemists and mathematicians [13, 14]. Many physical and chemical properties depend on such factors which are different rather than branching. With this motivation, Estrada et al. [15] presented the atom-bond connectivity index, which is defined as

It is reported in [15–17] that this index can be applied in modeling thermodynamic features of organic chemical compounds. In addition, Estrada [16] interpreted a new quantum theory which presented that this index implements a model for consideration of 1,2-, 1,3-, and 1,4-interactions in the carbon-atom skeleton of saturated hydrocarbons, and in this sense, it can be applied for rationalizing steric effects in such compounds.

Recently, Ghorbani and Hosseinzadeh [18] presented the fourth atom-bond connectivity index which is defined as

The sense for presenting a new index is to increase prediction of some property of molecules. One of the successors of the Randić connectivity index called the geometric-arithmetic connectivity index which is presented by Vukičević and Furtula [19] is defined as

It is reported in [19, 20] that this index provides better prediction rate than the Randić index in various physicochemical properties such as entropy, enthalpy of formation, and standard enthalpy of vaporization. Furthermore, the enhancement in the prediction accuracy rate of the *GA* index comparing to the Randić index is more than 9% for the standard enthalpy of vaporization. In [21], it also presents that there forms excellent correlation between heat of formation of benzenoid hydrocarbons and the index (the correlation coefficient is 0.972).

Recently, the fifth index is presented by Graovac et al. [22] and is defined as

In this paper, we present , , , and topological indices for new classes of nanotubes fabricated from carbon hexagonal nanotubes called the boron triangular and boron-*α* nanotubes.

#### 2. Boron Nanotubes

In last 20 years, various types of boron-containing nanomaterials such as boron nanoclusters, boron nanowires, boron nanotubes, boron nanobelts, boron nanoribbons, boron nanosheets, and boron fullerenes have been experimentally synthesized and identified. Boron nanomaterials have been considered as excellent materials for enhancing the characteristics of optoelectronic nanodevices because of their broad elastic modulus, high-melting point, excessive conductivity, great emission uniformity, and low turn-on field. These materials can carry excessive emission current, which recommends that they may have great prospective applications in the field emission area [23]. Furthermore, boron nanomaterials also have some better properties compared to carbon nanomaterials such as excessive resistance to oxidation at high temperatures and great chemical stability and are stable broad band-gap semiconductor [24, 25]. Due to these properties, boron nanomaterials may have great applications at high temperatures or in corrosive environments functioning as supercapacitors, solid lubricants, fuel cells, and batteries [26]. Moreover, the extensive range of boron nanomaterials themselves could be the building blocks for combining with other existing nanomaterials to design and create materials with new properties. For this reason, the boron-nitrogen clusters have been developed in [27, 28], and boron-nitrogen cages have been computed in [29]. The boron fullerenes are the cage molecules [30] which can serve as the building blocks for fabrication of new hybrid nanostructures with novel properties.

The boron triangular nanotube was created in 2004 [23] and obtained from a carbon hexagonal nanotube by adding an extra atom to the center of each hexagon. Also, a special boron nanotube was fabricated from a carbon hexagonal nanotube in 2008, by adding an extra atom to the center of certain hexagons [31, 32]. This nanotube is designed by generating a mixture of hexagons and triangles called the boron-*α* nanotube. It is reported in [33, 34] that these nanotubes are important materials for optical, electronic, chemical, and biosensing applications. The comparison study about some computational aspects of boron triangular and boron-*α* nanotubes has been investigated in [35]. The perceptions of boron triangular and boron-*α* nanotube are presented in Figure 1.