Some Computational Aspects of Triangular Boron Nanotubes

1 Division of Science and Technology, University of Education, Lahore 54000, Pakistan; mmunir@ue.edu.pk (M.M.); nazeer.waqas@ue.edu.pk (W.N.); arnizami@ue.edu.pk (A.R.N.) 2 Center for Excellence in Molecular Biology, Punjab University Lahore, Lahore 53700, Pakistan, shahziarafique@gmail.com 3 Department of Mathematics and RINS, Gyeongsang National University, Jinju 52828, Korea 4 Center for General Education, China Medical University, Taichung 40402, Taiwan * Correspondence: smkang@gnu.ac.kr; Tel.: +82-55-772-1420


Introduction
Nanoscience has become the flavor of the modern era because of its increasing applications and uses.Amongst the nanomaterials, nanocrystals, nanowires and nanotubes, constitute three major categories, the last two being one-dimensional.Since the discovery of the carbon nanotubes in 1991, interest in one-dimensional nanomaterials has grown remarkable and a phenomenal number of research articles have been published on nanotubes as well as on nanowires.
The numerical tendencies of a certain property depend on the molecular structure which is, in fact, a graph where vertices represent atoms of nanomaterials and edges correspond to chemical bonds.Chemical graph theory is contributing a lion's share in predicting chemical properties of a nanomaterial without going into wet labs.Cheminformatics is another emerging field in which quantitative structure-activity (QSAR) and Structure-property (QSPR) relationships predict the biological activities and properties of nanomaterial see [2][3][4][5].In these studies, some physio-chemical properties and topological indices are used to predict bioactivity of the chemical compounds see [13,14,15].Boron nanotubes are coming increasingly interesting because of their remarkable properties, like: structural stability, work function, transport properties, and electronic structure [30].Two structural classes of boron nanotubes are extremely important.First one is a nanotube derived from triangular sheet and the second one is derived from α − sheet.Regardless of their structure and chiralities, both boron nanotubes are more conductive than carbon nanotubes.We refer the readers [27,28,29] for further study about nanomaterials.
Several algebraic polynomials have useful applications in chemistry such as Hosoya Polynomial (also called Wiener polynomial) [25]  In this article, we calculated different degree-based topological indices of Boron nanotubes by using M-polynomial.Before this, we need to recall a few concepts from chemical graph theory.
M-polynomial is recently introduced in [26].Throughout this paper we fixed following notations: This polynomial has been one of the key areas of interest in computational aspects of materials.
From this M-polynomial, we can calculate many topological indices.The topological index of a molecule structure can be considered as a non-empirical numerical quantity which quantifies the molecular structure and its branching pattern in many ways.In this point of view, the topological index can be regarded as a score function which maps each molecular structure to a real number and is used as a descriptor of the molecule under testing [6,7,9,10,11].
, for more details see [16].Zagreb indices have been introduced by I. Gutman and N. Trinajstic [21].
d d , [22,23].Second modified Zagreb index is defined by ) , where α is an arbitrary real number see [24].Symmetric division index is defined by ( ) ( ) min( , ) max( , ) .max( , ) min( , ) The harmonic index H(G) is another variant of Randic index defined as The inverse sum topological index ISI is defined as and it is useful for computing heat of formation of alkanes [33,34].These indices can help to characterize the chemical and physical properties of molecules see [6-10, 18-22,24-26].Most recently M. Munir et al. computed M-polynomials and related topological indices for Nanostar dendrimers [1] and Titania Nanotubes in [19].Some degree-based topological indices are derived from M-polynomial l [5].The following table-1 relates these derivations.Augmented Zagreb Index In the present article, we compute the closed forms of M-polynomials triangular boron nanotubes and represent them graphically using Maple.As consequences, we derive as many as nine different topological degree-based indices.We start by defining M-polynomial of a general graph, see [25].Recently a lot of research is in progress to find closed forms of certain topological indices.M-polynomial, introduced in 2015 [25], is an approach to obtain closed forms of many degree-based topological indices.

RESULTS AND DISCUSSION
In this section, we use symmetric structures of Boron triangular Nanotubes to determine closed parametric form of M-polynomials many topological indices for these tubes.

Figure 4 .
Figure 4. Plot of M-polynomial of [4]he concept of degree is some what closely related to the concept of valence in chemistry.For details on basics of graph theory, any standard text such as[4]can be of great help.
that plays a vital role in determining distance-based topological indices.Among Other algebraic polynomials, M-polynomial introduced recently in [26], plays the same role in determing many degree-based topological d

Table 1 .
Derivation of some degree-based topological indices from M-polynomial

Table 2 .
Edge partition of edge set of