Computational Analysis of topological indices of two Boron Nanotubes

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 M-polynomial of nanotubes produce closed forms of many degree-based topological indices which are numerical parameters of the structure and determine physico-chemical 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 M-polynomial. More importantly we make a graph-theoretic comparison of indices of two types of boron nanotubes namely triangular boron and ∝-boron nanotubes.

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 M-polynomial 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 V-Phenylenic 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 M-polynomial for ∝boron nanotube. Then we derive closed forms of many degree-based topological indices for these tubes. We also draw some conclusions about both types of boron tubes.  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: For general detains about R −1/2 (G) and its generalized please see [23][24][25][26][27][28]

and the inverse Randić index is defined as
. 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: . Thesecond modified Zagreb index is defined as: We refer [32][33][34][35][36] to the readers for comprehensive details of these indices. Other famous indices are Symmetric division index: and augmented Zagreb index: 37,38 .
Tables presented in 7-11 relates some of these well-known degree-based topological indices with M-polynomial with following reserved notations , x y x x y y 0 0

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    The edge set of BNT α [m, n] has following three partitions, And

Conclusions and Discussion
In the present article, we computed closed form of M-polynomial for α-boron nanotubes and then we derived many degree-based 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 α-boron 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 α-boron nanotube. We give a detailed comparative analysis of degree-based 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 alpha-Boron nanotube is less than triangular Boron tubes for m ≤ 9 and for m ≥ 10, total π -electron energy of alpha-Boron 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 alpha-Boron 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 physio-chemical 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 structure-property relations. Figures 12 and 13 shows the graphs of A(G) for both tubes. We use red color for graphs of α-boron 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 α-boron nanotubes. One who wishes to select a boron tube with high A(G) should naturally choose α-boron 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. We believe that all above computed indices show more or less similar results, so in the end we conclude that α-boron nanotubes are far better in obtaining higher values of degree-based 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 α-Boron nanotubes have high correlation coefficient regarding Randic index [41][42][43][44] .