Abstract
There has been a recent debate that boron nanotubes can outperform carbon nanotubes on many grounds. The most stable boron nanotubes are made of a hexagonal lattice with an extra atom added to some of the hexagons called ∝boron nanotubes. Closed forms of Mpolynomial of nanotubes produce closed forms of many degreebased topological indices which are numerical parameters of the structure and determine physicochemical properties of the concerned nanotubes. In this article, we compute and analyze many topological indices of ∝boron nanotubes correlating with the size of structure of these tubes through the use of Mpolynomial. More importantly we make a graphtheoretic comparison of indices of two types of boron nanotubes namely triangular boron and ∝boron nanotubes.
Introduction
Mathematical chemistry provides tools such as polynomials and functions to capture information hidden in the symmetry of molecular graphs and thus predict properties of compounds without using quantum mechanics. A topological index isa numerical parameter of a graph and depicts its topology. It describes the structure of molecules numerically and are used in the development of qualitative structure activity relationships (QSARs). Most commonly known invariants of such kinds are degreebased topological indices. These are actually the numerical values that correlate the structure with various physical properties, chemical reactivity and biological activities^{1,2,3,4,5}. It is an established fact that many properties such as heat of formation, boiling point, strain energy, rigidity and fracture toughness of a molecule are strongly connected to its graphical structure.
Hosoya polynomial, (Wiener polynomial)^{6}, plays a pivotal role in distancebased topological indices. A long list of distancebased indices can be easily evaluated from Hosoya polynomial. A similar breakthrough was obtained recently by Klavzar et. al.^{7}, in the context of degreebased indices. Authors in^{7} introduced Mpolynomial in, 2015, to play a role, parallel to Hosoya polynomial to determine closed form of many degreebased topological indices^{8,9,10,11,12}. The real power of Mpolynomial is its comprehensive nature containing healthy information about degreebased graph invariants. These invariants are calculated on the basis of symmetries present in the 2dmolecular lattices and collectively determine some properties of the material under observation.
Because of increasing interests and developments of new nanomaterials, computations have minimized the burden of experimental labor to some extent. Amongst the nanomaterials, nanocrystals, nanowires and nanotubes, constitute three major categories, the last two being onedimensional. Boron nanotubes are becoming increasingly interesting because of their remarkable properties like structural stability, work function, transport properties, and electronic structure^{13}. Triangular Boron is derived from a triangular sheet as shown in Fig. 1. The first boron nanotubes were created, in 2004, from a buckled triangular latticework^{13,14,15}.
These tubes are discussed recently in^{15,16}. Other wellknown type, αboron, is derived from αsheet. Irrespective of their structures and chiralities, both types are more conductive than carbon nanotubes^{14,17,18,19}. Figure 2 describes basic structure of ∝ boron nanotube. Following Fig. 3 also presents different views of ∝ Boron nanotube, (a) is the planar view whereas (b) is the tabular view.
As for as structure of both tubes are concerned, ∝ Boron nanotube is more complicated than Triangular boron nanotubes with addition of an extra atom to the center of some of the hexagons^{15}. In^{15}, authors proved that this is the most stable known theoretical structure for a boron nanotube. They also showed that, with this pattern, boron nanotubes should have variable electrical properties: wider ones would be metallic conductors, but narrower ones should be semiconductors. So, these tubes boron tubes will be used in Nano devices similar to the diodes and transistors that have already been made from carbon nanotube. In^{16} authors computed some computational facts which are similar in both types of boron nanotubes and carbon nanotubes. Munir et al. computed Mpolynomial and related indices of triangular boron nanotubes in^{11}, polyhex nanotubes in^{12}, nanostar dendrimers in^{8}, titania nanotubes in^{9} and M Polynomials and topological indices of VPhenylenic Nanotubes and Nanotori in^{10}. In all above mentioned articles we presented an analysis of these indices against the parameters of structure involved. In this article, we compute general form of Mpolynomial for ∝ boron nanotube. Then we derive closed forms of many degreebased topological indices for these tubes. We also draw some conclusions about both types of boron tubes.
Basic Definitions and Literature Review
Throughout this article, we assume G to be a connected graph, V (G) and E (G) are the vertex set and the edge set respectively and d_{v} denotes the degree of a vertex v.
Definition 1
. The Mpolynomial of G is defined as: \(M(G,x,y)=\sum _{\delta \le i\le j\le \Delta }{m}_{ij}(G){x}^{i}{y}^{j}\) where δ = Min{d_{v}v ∈ V (G)}, Δ = Max{d_{v}v ∈ V (G)}, and m_{ij}(G) is the edge vu ∈ E(G) such that where ≤j^{7}.
Wiener index and its various applications are discussed in^{20,21,22}. Randić index, R_{−1/2}(G), is introduced by Milan Randić in 1975 defined as: \({R}_{1/2}(G)=\sum _{uv\in E(G)}\frac{1}{\sqrt{{d}_{u}{d}_{v}}}.\) For general detains about R_{−1/2}(G) and its generalized Randić index, \({R}_{\alpha }(G)=\sum _{uv\in E(G)}\frac{1}{{({d}_{u}{d}_{v})}^{\alpha }},\) please see^{23,24,25,26,27,28} and the inverse Randić index is defined as \(R{R}_{\alpha }(G)=\sum _{uv\in E(G)}{({d}_{u}{d}_{v})}^{\alpha }.\) Clearly R_{−1/2}(G) is a special case of R_{α}(G) when \(\alpha =\,\frac{1}{2}\). This index has many applications in diverse areas. Many papers and books such as^{29,30,31} are written on this topological index as well. Gutman and Trinajstić introduced two indices defined as: \({M}_{1}(G)=\sum _{uv\in E(G)}({d}_{u}+{d}_{v})\) and \({M}_{2}(G)=\sum _{uv\in E(G)}({d}_{u}\times {d}_{v})\). Thesecond modified Zagreb index is defined as: \({}^{m}M_{2}(G)=\sum _{uv\in E(G)}\frac{1}{d(u)d(v)}.\) We refer^{32,33,34,35,36} to the readers for comprehensive details of these indices. Other famous indices are Symmetric division index: \({\rm{SDD}}({\rm{G}})=\sum _{uv\in E(G)}\{\frac{{\rm{\min }}({d}_{u},{d}_{v})}{{\rm{\max }}({d}_{u},{d}_{v})}+\frac{{\rm{\max }}({d}_{u},{d}_{v})}{{\rm{\min }}({d}_{u},{d}_{v})}\}\) harmonic index defined \(H(G)=\sum _{vu\in E(G)}\frac{2}{{d}_{u}+{d}_{v}}.\) Inverse sum index i: \(I(G)=\sum _{vu\in E(G)}\frac{{d}_{u}{d}_{v}}{{d}_{u}+{d}_{v}}\) and augmented Zagreb index: \(A(G)=\sum _{vu\in E(G)}{\{\frac{{d}_{u}{d}_{v}}{{d}_{u}+{d}_{v}2}\}}^{3},\)^{37,38}.
Tables presented in^{7,8,9,10,11} relates some of these wellknown degreebased topological indices with Mpolynomial with following reserved notations
Computational Results
In this section, we give our computational results. In terms of chemical graph theory and mathematical chemistry, we associate a graph with the molecular structure where vertices correspond to atoms and edges to bonds. Following the same lines, we represent a ∝ boron nanotube, by a planar graph, BNT_{t}[m, n] of order n × m, as the in Fig. 3 demonstrates. Clearly, from Fig. 4, there are \(\frac{3nm}{2}\) vertices and \(\frac{3}{2}mn+3{m}^{2}+\frac{9}{2}m+n3\) edges in 2D graph model of ∝boron nanotubes.
Theorem 1.
Let BNT_{α}[m, n] is ∝boron Nanotubes. Then
Proof.
Let BNT_{α}[m, n] be ∝boron nanotubes, where m is the number of columns and n is the number of rows. From Fig. 4 it is easy to observe that the graph of BNT_{α}[m, n] has \(\frac{4nm}{3}\) number of vertices and \(\frac{3}{2}mn+3{m}^{2}+\frac{9}{2}m+n3\) edges where n is multiple of 3.
The edge set of BNT_{α}[m, n] has following three partitions,
And
and the vertex set V(BNT_{α}[m, n]) of BNT_{α}[m, n] has six partitions:
Now
and
Thus the Mpolynomial of BNTα[m, n] is:
Above Fig. 5 presents the Maple 13 plot of the Mpolynomial of ∝ Boron Nanotubes. Clearly, values drastically increases as Χ → ±1, Y → ±2 For the most part of [−1, 1] × [−2, 2], values remain stable whereas triangular boron nanotubes show opposite trends^{11}, shown in Fig. 6 below
Now we derive formulas for many degreebased topological indices using Mpolynomial.
Proposition 1
Let BNT_{α}[m, n] is the boron ∝ nanotube, then

1.
\({M}_{1}(G)=94m+5m(3n11)+33{m}^{2}+12n36.\)

2.
\({M}_{2}(G)=220m+\frac{25}{2}m(3n11)+90{m}^{2}+36n108.\)

3.
\({}^{m}M_{2}(G)=\frac{1}{10}{m}^{2}+(\frac{943}{3600}+\frac{3}{50}n)m+\frac{1}{36}n\frac{1}{12}.\)

4.
\({R}_{\alpha }(G)=3\times {16}^{\alpha }m+2m{20}^{\alpha }+4m{24}^{\alpha }+\frac{1}{2}{25}^{\alpha }m(3n11)+3{m}^{2}{30}^{\alpha }+{36}^{\alpha }(m+n3).\)

5.
\({R}_{\alpha }(G)=\frac{3}{{16}^{\alpha }}m+\frac{2}{{20}^{\alpha }}m+\frac{4}{{24}^{\alpha }}m+\frac{1}{2\times {25}^{\alpha }}m(3n11)+\frac{3}{{30}^{\alpha }}{m}^{2}+\frac{1}{{36}^{\alpha }}(m+n3).\)

6.
\(SSD(G)=\frac{623}{30}m+m(3n11)+\frac{61}{10}{m}^{2}+2n6.\)

7.
\(H(G)=\frac{6}{11}{m}^{2}+(\frac{191}{180}+\frac{3}{10}n)m+\frac{1}{6}n\frac{1}{2}.\)

8.
\(I(G)=\frac{90}{11}{m}^{2}+(\frac{1673}{180}+\frac{15}{4}n)m+3n9.\)

9.
\(A(G)=\frac{1000}{9}{m}^{2}+(\frac{35698735591}{395136000}+\frac{46875}{1024}n)m+\frac{5832}{125}n\frac{17496}{125}.\)
Proof.
Let
Then

1.
First Zagreb Index
$${M}_{1}(G)={({D}_{x}+{D}_{y})f(x,y)}_{x=y=1}=94m+5m(3n11)+33{m}^{2}+12n36.$$(31) 
2.
Second Zagreb Index
$${M}_{2}(G)={{D}_{y}{D}_{x}(f(x,y))}_{x=y=1}=220m+\frac{25}{2}m(3n11)+90{m}^{2}+36n108.$$(32) 
3.
Modified Second Zagreb Index
$${}^{m}M_{2}(G)={{S}_{x}{S}_{y}(f(x,y))}_{x=y=1}=\frac{1}{10}{m}^{2}+(\frac{943}{3600}+\frac{3}{50}n)m+\frac{1}{36}n\frac{1}{12}.$$(33) 
4.
Generalized Randic’ Index
$$\begin{array}{rcl}{R}_{\alpha }(G) & = & {{D}_{x}^{\alpha }{D}_{y}^{\alpha }(f(x,y))}_{x=y=1}=3\times {16}^{\alpha }m+2m{20}^{\alpha }+4m{24}^{\alpha }\\ & & +\frac{1}{2}{25}^{\alpha }m(3n11)+3{m}^{2}{30}^{\alpha }+{36}^{\alpha }(m+n3).\end{array}$$(34) 
5.
Inverse Randic’ Index
$$\begin{array}{rcl}R{R}_{\alpha }(G) & = & {{S}_{x}^{\alpha }{S}_{y}^{\alpha }(f(x,y))}_{x=y=1}=\frac{3}{{16}^{\alpha }}m+\frac{2}{{20}^{\alpha }}m\\ & & +\frac{4}{{24}^{\alpha }}m+\frac{1}{2\times {25}^{\alpha }}m(3n11)+\frac{3}{{30}^{\alpha }}{m}^{2}+\frac{1}{{36}^{\alpha }}(m+n3).\end{array}$$(35) 
6.
Symmetric Division Index
$$SSD(G)={({S}_{y}{D}_{x}+{S}_{x}{D}_{y})(f(x,y))}_{x=y=1}=\frac{623}{30}m+m(3n11)+\frac{61}{10}{m}^{2}+2n6.$$(36) 
7.
Harmonic Index
$$H(G)=2{S}_{x}J(f(x,y)){}_{x=1}=\frac{6}{11}{m}^{2}+(\frac{191}{180}+\frac{3}{10}n)m+\frac{1}{6}n\frac{1}{2}.$$(37) 
8.
Inverse Sum Index
$$I(G)={S}_{x}J{D}_{x}{D}_{y}{(f(x,y))}_{x=1}=\frac{90}{11}{m}^{2}+(\frac{1673}{180}+\frac{15}{4}n)m+3n9.$$(38) 
9.
Augmented Zagreb Index
Conclusions and Discussion
In the present article, we computed closed form of Mpolynomial for αboron nanotubes and then we derived many degreebased topological indices as well. Some other degree based topological indices of boron nanotubes are given in^{39}. Topological indices thus calculated for these nanotubes can help us to understand the physical features, chemical reactivity, and biological activities. In this point of view, a topological index can be regarded as a score function which maps each molecular structure to a real number and is used as descriptors of the molecule under testing. These results can also play a vital part in the determination of the significance of ∝ boron nanotubes in electronics and industry. We also want to remark that a thorough comparison of αboron nanotubes can be made with triangular boron Nanotubes^{11}. For the rest of this article we reserve symbol T for Triangular boron tube and P for αboron nanotube.
We give a detailed comparative analysis of degreebased topological indices of both boron tubes. It has been experimentally verified that the first Zagreb index is directly related with total π electron energy of the structure^{33,40} and references therein. So structure having high values of First Zagreb Index have higher total π electron energy. From the following Fig. 6 it is evident that total π electron energy of alphaBoron nanotube is less than triangular Boron tubes for m ≤ 9 and for m ≥ 10, total π electron energy of alphaBoron nanotube rises sharply as compared to triangular Boron tubes with increase in m.
Similarly the given Figs 7 and 8 elaborates that total π electron energy of alphaBoron nanotube is larger than triangular Boron tubes for n ≤ 11 and for n ≥ 11, total π electron energy of triangular Boron tubes rises sharply as compared to alpha Boron tubes with increase in n.
Similarly Randic index is useful for determining physiochemical properties of alkanes as noticed by chemist Melan Randic in 1975. He noticed the correlation between the Randic index R and several physico–chemical properties of alkanes like, enthalpies of formation, boiling points, chromatographic retention times, vapor pressure and surface areas. Following Fig. 9 is adapted from^{21} relating to boiling point of some Alkanes and its correlation with Randic index.
Now we give some comparative remarks showing some correlations. The next Fig. 10 clearly depicts that green color rises sharply as compared with red indicating that alpha tube have significant correlation coefficients of above said properties over the triangular boron tubes with the rise in n and m.
It is noticeable from above Fig. 11 that boiling point and other above properties are correlated with Randic index. Subsequent years of research showed that Randic index has a variety of applications specially in medicinal and pharmacological issues. For the results about Triangular boron, we refer to^{11}. We give comparative analysis of both tubes for Augmented Zagreb index, Inverse sum index and Harmonic index denoted by A, I and H respectively. Remaining can be traced out in similarly. We start with A(G), the Augmented Zagreb index. Recently it has been proved that this index has relatively high correlation coefficient so it can be used for designing quantitative structureproperty relations. Figures 12 and 13 shows the graphs of A(G) for both tubes. We use red color for graphs of αboron nanotubes whereas blue is used for triangular boron tubes. Figure 12 shows that A(G) decrease linearly with the rise in m for Triangular boron whereas it rises sharply for αboron nanotubes. One who wishes to select a boron tube with high A(G) should naturally choose αboron nanotubes. Figure 13 shows that if we fix m instead of n, A(G) rises linearly with both types of tube although for n < 4, A(T) < A(P) and A(T) > A(P) for n > 4. Critical fact is the value n = 4 where A(T) = A(P). These figures also suggest the length and width of tubes for the desired values of A(G).
Now we discuss the Inverse sum index.
As the Fig. 14 suggests that I(P) rises sharply with the rise in m but I(T) slopes downward very slowly with the same rise if we fix n = 2. Whereas both I(P) and I(T) rise linearly with rise in n although rise in I(T) seems to be negligible see Fig. 15.
For H(G), we can see that again H(P) ≥ H(T) for all positive values of m with n = 2, see Fig. 16. Fixing m = 2, we can see that for n > 3, H(P) ≥ H(T) and H(P) < H(T) for n > 3 but for n = 3, we get H(P) = H(T), see Fig. 17. We believe that all above computed indices show more or less similar results, so in the end we conclude that αboron nanotubes are far better in obtaining higher values of degreebased topological indices than triangular boron nanotubes.
Above graphs of topological indices show the correlation of M_{1}(G), R_{α}(G), A(G), I(G) and H(G) with m and n. It is clear that all topological indices have linear relation with n whereas graphs are parabolas in relation to m. These facts give an insight idea to control topological indices with length and width of these tubes. Moreover we can find extreme values of topological indices for some definite values of m and n. We give structural analysis for only three indices as all other indices discussed above show similar trends. We also conclude that αBoron nanotubes have high correlation coefficient regarding Randic index^{41,42,43,44}.
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Y.C. Kwun designed the problem, M. Munir, and W. Nazeer proved the results and S. Rafique and S.M. Kang verified the results and wrote his paper.
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Kwun, Y.C., Munir, M., Nazeer, W. et al. Computational Analysis of topological indices of two Boron Nanotubes. Sci Rep 8, 14843 (2018). https://doi.org/10.1038/s4159801833081y
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DOI: https://doi.org/10.1038/s4159801833081y
Keywords
 Boron Nanotubes
 Topological Index
 Triangular Boron
 Hosoya Polynomial
 Qualitative Structure Activity Relationships (QSARs)
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