Research Article Topological Aspects of Boron Nanotubes Jia-Bao Liu , 1 Hani Shaker , 2 Imran Nadeem , 2 and Muhammad Hussain 2 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; hani.uet@gmail.com Received 6 April 2018; Accepted 16 May 2018; Published 4 July 2018 Academic Editor: Jamal Berakdar Copyright © 2018 Jia-Bao Liu et al. is 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. e 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. ese materials have some important issues in nanodevice applications like mechanical and thermal stability. erefore, they require theoretical studies on the other properties. In this paper, we present certain degree-based topological indices such as ABC, the fourth ABC, GA, and the fifth GA 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 im- plements graph theory to study mathematical modeling of chemical aspects [3]. is theory shows a prominent effect on the extension of the chemical sciences [4]. e 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 V G and edge set E G . Let I G be the set of edges of G that are incident with a vertex p ∈ V p , then the degree of p is defined as the cardinality of the set I G and δ p �  q∈N p d q , where set N p consists of all neighbor vertices of p, that is, N p � q ∈ V G | pq ∈ E G  . 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. e 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. e relative stability of the fullerenes has been correlated with topological indices in [5]. In [6], topological indices are also usedtopredictthestableisomersofagivenfullerene,andfor 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 in- dices are going to be discussed because of their great ap- plications in chemical graph theory. For recent study of distance-based indices, we refer [8, 9], and for spectrum- based indices, we refer [10, 11]. e first degree-based topological index is the Randi´ c connectivity index which was presented by Randi´ c [12] and is defined as χ(G)�  pq∈E G 1 ���� d p d q  . (1) is 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 Hindawi Advances in Materials Science and Engineering Volume 2018, Article ID 5729291, 11 pages https://doi.org/10.1155/2018/5729291