Computational Analysis of Imbalance-Based Irregularity Indices of Boron Nanotubes

Abstract

Molecular topology provides a basis for the correlation of physical as well as chemical properties of a certain molecule. Irregularity indices are used as functions in the statistical analysis of the topological properties of certain molecular graphs and complex networks, and hence help us to correlate properties like enthalpy, heats of vaporization, and boiling points etc. with the molecular structure. In this article we are interested in formulating closed forms of imbalance-based irregularity measures of boron nanotubes. These tubes are known as α-boron nanotube, triangular boron nanotubes, and tri-hexagonal boron nanotubes. We also compare our results graphically and come up with the conclusion that alpha boron tubes are the most irregular with respect to most of the irregularity indices.

1. Introduction

Mathematical chemistry is evidently proving useful with efficient and time saving tools to assist in determining the properties of chemical compounds without going into laborious laboratory details. Topological indices are the structure preserving maps which capture some key combinatorial and topological aspects of the structure and determine properties of chemical compounds without the help of quantum mechanics as the final output [1,2]. Among such invariants, most are known as degree-based indices, and are widely used nowadays, [3,4,5,6,7]. Some relations of topological indices with properties of compounds can be seen in [1,2,3,4,5,6]. In short, these indices actually forecast some mathematical formulation of the properties of the substances under discussion. Wiener used these indices to guess the boiling points of parafin [4] and Randic determined the degree of molecular branching [3]. Estrada determined energies and atom bond connectives of branched alkanes [5] and determined the enthalpy of formation of alkanes using atom-bond connectivity index [6]. Molecular connectivity and its relationship with structure analysis have been given by Kier et al. [7,8].

Nanomaterials at present are being used in as electronics and cosmetics. Many other sectors, for example, polymeric composite materials, are carrying out a lot of scientific and technological works, and there are also plans for a wide range of projects involving nanomaterials. It has caught tremendous attention and interest for the promotion of nanostructured compounds. All this is because of the unique properties that are at hand, offering possibilities of multi-functionality, reduction of thickness, and a great spectrum of applications relating to technology. Indeed, recent works on nanoparticles showcase potential risks of nano-object releases and highlight the interest in the management of nano-risk in workplaces by making possible the removal of some lab tests in a research process [9]. For example, many studies highlight nano-object emissions due to nanomaterial use [10,11,12]. Cases of nano-object exposure in the field of occupational hygiene at workplaces have been reported [13]. These aspects are addressed according to metrological challenges related to particle characterization [14].

In QSAR (Quantitative structure-activity relationship), scientists use topological indices to study relationships of different quantities with chemical properties of nanomaterials. A broad class of the indices is the irregularity indices of the molecular graph of a certain molecule. Irregularity indices have been selected to study the topological pattern of the molecular structure recently [15,16,17,18,19,20,21,22,23,24]. Term network irregularity was put forth by Collatz and Sinogowitz in 1957 [18]. Actually, these indices are used in complex networks which represent disparate systems. Key topological features possessed by these systems are small-worldness, network motifs, scale-freeness, and self-similarity, [19,20,21]. In a nutshell, the power-law degree distribution of complex networks contrasts sharply with the regularity observed in random models like the one proposed by Erdös and Rényi [23]. Reti et al. used these indices to test the physiochemical properties like entropy, entropy of vaporization, boiling point, standard enthalpy of vaporization, and the acentric factor of octane isomers [15]. In fact, they provided that these properties of four octane isomers can be predicted with good accuracy by using irregularity indices. Estrada et al. analyzed the irregularity of 10 protein-protein interaction networks in different organisms ranging from 50 to 3000 nodes [16]. They, in fact, proved that these networks attain higher irregularity than a regular random network. A graph is said to be regular if all its vertices are of same degree, otherwise it is irregular. However, one is deeply interested to know the degree of irregularity in the graph [23,24,25,26,27]. These structures are used as long infinite chains macromolecules in chemistry and related areas. Total irregularity index has been introduced recently and some well stabilized graphs have been studied. Due to the ever-increasing interest in the complexity of molecular structures, one is interested to know the degree of complexity in the underlined structure. One way is to compute the irregularity indices of the structure [24,25,26,27]. Bell discussed the most irregular graphs according to some irregularity measures [24]. Gutman gave some fundamental results for irregularity indices of some graphs in [26].

In this article, we are interested in characteristic study of irregularity determinants of three types nanotube structures. The first nanotube is the tri-hexagonal boron nanotube, and the other two are boron nanotubes which can be formulated easily on to the structure of carbon nanotubes. Boron nanotubes are considered as an effective replacement of carbon nanotubes because of their high conductivity and thermal properties [28,29,30,31,32,33,34]. Figure 1 presents a tabular view of these tubes. Figure 1a is carbon nanotubes, whereas Figure 1b is triangular boron nanotube structure, and Figure 1c is α-boron nanotubes.

These tubes and their computational aspects have been treated in [29] by Manuel et al. Because of increasing interest and the development 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 one-dimensional [29,30]. Boron nanotubes are becoming increasingly interesting because of their remarkable properties like structural stability, work function, transport properties, and electronic structure. Triangular Boron is derived from a triangular sheet as shown in Figure 1. The first boron nanotubes were created in 2004 from a buckled triangular latticework [30,31,32,33,34]. Another well-known type, α-boron, is derived from α-sheet. Irrespective of their structures and chiralities, both types are more conductive than carbon nanotubes. Figure 2 presents molecular graphs of these three tubes in the same order as given in order of Figure 1.

As far as the structure of both tubes is concerned, the α-Boron nanotube is more complicated than the triangular boron nanotubes with the addition of an extra atom to the center of some of the hexagons. The 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 [30,31,32,33,34].

Other degree-based indices are also recently studied on a large scale. Several authors preferred to use the approach of a general polynomial, and to then compute these indices using variety of differential and integral operators. The authors computed M-polynomials and different famous degree based indices for different graphs, [35,36,37,38,39]. Other different popular indices and their relations have been showcased in [36,37,38].

2. Preliminaries and Notations

Let G be a simple connected graph with vertex V, edge set E, du and dv the degree of vertices u and v. A topological invariant is an isomorphism of the graph that preserves the topology of the graph. A graph is said to be regular if every vertex of the graph has the same degree. A topological invariant is called an irregularity index if this index vanishes for a regular graph, and is non-zero for a non-regular graph. Regular graphs have been extensively investigated, particularly in mathematics. Their applications in chemical graph theory came to be known after the discovery of nanotubes and fullerenes. Paul Erdos emphasized this in the study of irregular graphs for the first time in history in [40]. In the Second Krakow Conference on Graph Theory (1994), Erdos officially posed it as an open problem, “The determination of extreme size of highly irregular graphs of given order” [41]. Since then, irregular graphs and the degree of irregularity have become one of the core open problems of graph theory. A graph in which each vertex has a different degree then the other vertices is known as a perfect graph. The authors of [42] demonstrated that no graph is perfect. The graphs lying in between are called quasi-perfect graphs, in which all except two vertices have different degrees [41]. Simplified ways of expressing the irregularities are irregularity indices. These irregularity indices have been studied recently in a novel way [43,44]. The first such irregularity index was introduced in [45]. Most of these indices used the concept of the imbalance of an edge defined as $imbal{l}_{uv}=\left|du-dv\right|$, [46,47]. The Albertson index, AL(G), was defined by Alberston in [47] as $AL\left(G\right)={\displaystyle \sum }_{UV\in E}\left|d_u-d_v\right|$. In this index, the imbalance of edges is computed. The irregularity index IRL(G) and IRLU(G) is introduced by Vukicevic and Gasparov, [47] as $IRL\left(G\right)={\displaystyle \sum }_{UV\in E}\left|ln{d}_u-ln{d}_v\right|$, and $IRLU\left(G\right)={\displaystyle \sum }_{UV\in E}\frac{\left|d_u-d_v\right|}{min\left(d_{u,}{d}_v\right)}$. Recently, Abdoo et al. introduced the new term “total irregularity measure of a graph G”, which is defined as, [46,48,49], $IR{R}_t\left(G\right)=\frac{1}{2}{\displaystyle \sum }_{UV\in E}\left|d_u-d_v\right|$. Recently, Gutman et al. introduced the IRF(G) irregularity index of the graph G, which is described as $IRF\left(G\right)={\displaystyle \sum }_{UV\in E}{\left(d_u-d_v\right)}^2$ in [50]. The Randic index itself is directly related to an irregularity measure, which is described as $IRA\left(G\right)={\displaystyle \sum }_{UV\in E}{\left(d_u^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-d_v^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}^2$ in [50]. Further irregularity indices of similar nature can be traced in [50] in detail. These indices are given as $IRDIF\left(G\right)={\displaystyle \sum }_{UV\in E}\left|\frac{d_u}{d_v}-\frac{d_v}{d_v}\right|$, $IRLF\left(G\right)={\displaystyle \sum }_{UV\in E}\frac{\left|d_u-d_v\right|}{\sqrt{\left(d_u{d}_v\right)}}$, $LA\left(G\right)=2{\displaystyle \sum }_{UV\in E}\frac{\left|d_u-d_v\right|}{\left(d_u+d_v\right)}$, $IRD1={\displaystyle \sum }_{UV\in E}ln\left\{1+\left|d_v-d_v\right|\right\}$, $IRGA\left(G\right)={\displaystyle \sum }_{UV\in E}ln\frac{d_u+d_v}{2\sqrt{\left(d_u{d}_v\right)}}$, and $IRB\left(G\right)={\displaystyle \sum }_{UV\in E}{\left(d_u^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-d_v^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}^2$.

Recently, Zahid et al. computed the irregularity indices of a nanotube [51]. Gao et al. recently computed irregularity measures of some dendrimer structures in [52] and molecular structures in [53]. These structures are used as long infinite chain macromolecules in chemistry and related areas.

Hussain et al. computed some irregularity indices of benzenoid systems [54] and dendrimers in [55]. Liu et al. computed Zagreb indices and multiplicative Zagreb indices of Eulerian graphs in [56].

In the current article, we are interested in finding the degree of imbalance-based irregularity of the three boron nanotubes discussed in Figure 3, Figure 4 and Figure 5. The main motivation comes from the fact that graphs of the irregularity indices show close accurate results about properties like entropy, standard enthalpy, vaporization, and acentric factors of octane isomers [52]. The molecular pattern and topology of these three boron nanotubes are shown in these figures.

3. Main Results

In this section we put together our main computational results. We start with the tri-hexagonal boron nanotube structure $CN{T}_H\left[p,q\right]$.

Theorem 1.

For tri-hexagonal boron nanotube$CN{T}_H\left[p,q\right]$, for positive values of p and q, we have

1. 

$IRDIF\left(CN{T}_H\left[p,q\right]\right)=\frac{37}{10}q+\frac{37}{10}pq$

2. 

$AL\left(CN{T}_H\left[p,q\right]\right)=6q+12pq$

3. 

$IRL\left(CN{T}_H\left[p,q\right]\right)=1.7260887q+2.677722616pq$

4. 

$IRLU\left(CN{T}_H\left[p,q\right]\right)=\frac{5}{2}q+3pq$

5. 

$IRLU\left(CN{T}_H\left[p,q\right]\right)=\frac{2\sqrt{15}-3\sqrt{5}}{5}q+\frac{6\sqrt{5}}{5}pq$

6. 

$IRF\left(CN{T}_H\left[p,q\right]\right)=18q+12pq$

7. 

$IRLA\left(CN{T}_H\left[p,q\right]\right)=\frac{5}{3}q+\frac{8}{3}pq$

8. 

$IRD1\left(CN{T}_H\left[p,q\right]\right)=2.432790649q+8.317766167pq$

9. 

$IRA\left(CN{T}_H\left[p,q\right]\right)=0.8489489603q+\frac{27-12\sqrt{5}}{2}pq$

10. 

$IRGA\left(CN{T}_H\left[p,q\right]\right)=0.1563480034q+0.0745351199pq$

11. 

$IRB\left(CN{T}_H\left[p,q\right]\right)=1.189831306q+0.66873708pq$

Proof. 

In order to prove the above theorem we have to consider Figure 3. Following

Table 1. Edge partition of tri-hexagonal boron nanotube structure $CN{T}_H\left[p,q\right]$.

$\mathbf{Number}\mathbf{of}\mathbf{Edges}({\mathit{d}}_{\mathit{u}},{\mathit{d}}_{\mathit{v}})$	Number of Indices
(3,5)	6q
(4,4)	2pq − q
(4,5)	6q (2p − 1)
(5,5)	4pq

contains information for the distribution of edges into different classes.

Now using above Table 1 and definitions we have,

• $IRDIF\left(G\right)={\displaystyle \sum }_{UV\in E}\left|\frac{d_u}{d_v}-\frac{d_v}{d_v}\right|$

  $$\begin{array}{cc}\hfill IRDIF\left(CN{T}_H\left[p,q\right]\right)& =6q\left|\frac{5}{3}-\frac{3}{5}\right|+2pq-q\left|\frac{4}{4}-\frac{4}{4}\right|+6q\left(2p-1\right)\left|\frac{5}{4}-\frac{4}{5}\right|+4pq\left|\frac{5}{5}-\frac{5}{5}\right|\hfill \\ & =6q\left|\frac{5}{3}-\frac{3}{5}\right|+6q\left(2p-1\right)\left|\frac{5}{4}-\frac{4}{5}\right|\hfill \end{array}$$
• $AL\left(G\right)={\displaystyle \sum }_{UV\in E}\left|d_u-d_v\right|$

  $$\begin{array}{cc}\hfill AL\left(CN{T}_H\left[p,q\right]\right)& =6q\left|5-3\right|+2pq-q\left|4-4\right|+6q\left(2p-1\right)\left|5-4\right|+4pq\left|5-5\right|\hfill \\ & =12p+6q\left(2p-1\right)\hfill \end{array}$$
• $IRL\left(G\right)={\displaystyle \sum }_{UV\in E}\left|ln{d}_u-ln{d}_v\right|$

  $$\begin{array}{cc}\hfill IRL\left(CN{T}_H\left[p,q\right]\right)& =6q\left|ln5-ln3\right|+2pq-q\left|ln4-ln4\right|+6q\left(2p-1\right)\left|ln5-ln4\right|+4pq\left|ln5-ln5\right|\hfill \\ & =6qln\frac{5}{3}+6q\left(2p-1\right)ln\frac{5}{4}\hfill \end{array}$$
• $IRLU\left(G\right)={\displaystyle \sum }_{UV\in E}\frac{\left|d_u-d_v\right|}{min\left(d_u{d}_v\right)}$

  $$\begin{array}{cc}\hfill IRLU\left(CN{T}_H\left[p,q\right]\right)& =6q\frac{\left|5-3\right|}{3}+2pq-q\frac{\left|4-4\right|}{4}+6q\left(2p-1\right)\frac{\left|5-4\right|}{4}+4pq\frac{\left|5-5\right|}{5}\hfill \\ & =\frac{12q}{3}+\frac{6q\left(2p-1\right)}{4}\hfill \end{array}$$
• $IRLU\left(G\right)={\displaystyle \sum }_{UV\in E}\frac{\left|d_u-d_v\right|}{\sqrt{\left(d_u{d}_v\right)}}$

  $$\begin{array}{cc}\hfill IRLU\left(CN{T}_H\left[p,q\right]\right)& =6q\frac{\left|5-3\right|}{\sqrt{15}}+2pq-q\frac{\left|4-4\right|}{\sqrt{16}}+6q\left(2p-1\right)\frac{\left|5-4\right|}{\sqrt{20}}+4pq\frac{\left|5-5\right|}{\sqrt{25}}\hfill \\ & =\frac{6q}{\sqrt{15}}+\frac{6q\left(2p-1\right)}{\sqrt{20}}\hfill \end{array}$$
• $IRF\left(G\right)={\displaystyle \sum }_{UV\in E}{\left(d_u-d_v\right)}^2$

  $$\begin{array}{cc}\hfill IRF\left(CN{T}_H\left[p,q\right]\right)& =6q{\left(5-3\right)}^2+2pq-q{\left(4-4\right)}^2+6q\left(2p-1\right){\left(5-4\right)}^2+4pq{\left(5-5\right)}^2\hfill \\ & =24q+6q\left(2p-1\right)\hfill \end{array}$$
• $IRLA\left(G\right)=2{\displaystyle \sum }_{UV\in E}\frac{\left|d_u-d_v\right|}{\left(d_u+d_v\right)}$

  $$\begin{array}{cc}\hfill IRLA\left(CN{T}_H\left[p,q\right]\right)& =2\left[6q\frac{\left|5-3\right|}{8}+2pq-q\frac{\left|4-4\right|}{8}+6q\left(2p-1\right)\frac{\left|5-4\right|}{9}+4pq\frac{\left|5-5\right|}{10}\right]\hfill \\ & =\frac{24q}{8}+\frac{12q\left(2p-1\right)}{9}\hfill \end{array}$$
• $IRD1={\displaystyle \sum }_{UV\in E}ln\left\{1+\left|d_v-d_v\right|\right\}$

  $$\begin{array}{cc}\hfill IRD1\left(CN{T}_H\left[p,q\right]\right)& =6qln\left\{1+\left|5-3\right|\right\}+2pq-qln\left\{1+\left|4-4\right|\right\}+6q\left(2p-1\right)ln\left\{1+\left|5-4\right|\right\}\hfill \\ & +4pqln\left\{1+\left|5-5\right|\right\}=6qln3+6q\left(2p-1\right)ln2\hfill \end{array}$$
• $IRA\left(G\right)={\displaystyle \sum }_{UV\in E}{\left(d_u^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-d_v^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}^2$

  $$\begin{array}{cc}\hfill IRA\left(CN{T}_H\left[p,q\right]\right)& =6q{\left(\frac{1}{\sqrt{5}}-\frac{1}{\sqrt{3}}\right)}^2+2pq-q{\left(\frac{1}{\sqrt{4}}-\frac{1}{\sqrt{4}}\right)}^2+6q\left(2p-1\right){\left(\frac{1}{\sqrt{5}}-\frac{1}{\sqrt{4}}\right)}^2\hfill \\ & +4pq{\left(\frac{1}{\sqrt{5}}-\frac{1}{\sqrt{5}}\right)}^2=6q{\left(\frac{1}{\sqrt{5}}-\frac{1}{\sqrt{3}}\right)}^2+6q\left(2p-1\right){\left(\frac{1}{\sqrt{5}}-\frac{1}{\sqrt{4}}\right)}^2\hfill \end{array}$$
• $IRGA\left(G\right)={\displaystyle \sum }_{UV\in E}ln\frac{d_u+d_v}{2\sqrt{\left(d_u{d}_v\right)}}$

  $$\begin{array}{cc}\hfill IRGA\left(CN{T}_H\left[p,q\right]\right)& =6qln\frac{5+3}{2\sqrt{15}}+2pq-qln\frac{4+4}{2\sqrt{16}}+6q\left(2p-1\right)ln\frac{5+4}{2\sqrt{20}}+4pqln\frac{5+5}{2\sqrt{25}}\hfill \\ & =6nln\frac{5+3}{2\sqrt{15}}+6n\left(2m-1\right)ln\frac{5+4}{2\sqrt{20}}\hfill \end{array}$$
• $IRB\left(G\right)={\displaystyle \sum }_{UV\in E}{\left(d_u^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-d_v^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}^2$

  $$\begin{array}{cc}\hfill IRB\left(CN{T}_H\left[p,q\right]\right)& =6q{\left(\sqrt{5}-\sqrt{3}\right)}^2+2pq-q{\left(\sqrt{4}-\sqrt{4}\right)}^2+6q\left(2p-1\right){\left(\sqrt{5}-\sqrt{4}\right)}^2\hfill \\ & +4pq{\left(\sqrt{5}-\sqrt{5}\right)}^2=6q{\left(\sqrt{5}-\sqrt{3}\right)}^2+6q\left(2p-1\right){\left(\sqrt{5}-\sqrt{4}\right)}^2\hfill \end{array}$$

Following

Table 2. Irregularity indices for tri-hexagonal boron nanotube structure $CN{T}_H\left[p,q\right]$.

Irregularity Indices	p = 1, q = 1	p = 2, q = 2	p =3, q = 3	p = 4, q = 4	p = 5, q = 5
$IRDIF\left(G\right)={\displaystyle {\displaystyle \sum }_{UV\in E}}\left|\frac{d_u}{d_v}-\frac{d_v}{d_v}\right|$	7.40	22.20	44.40	74	111
$AL\left(G\right)={\displaystyle {\displaystyle \sum }_{UV\in E}}\left|d_u-d_v\right|$	18	60	126	216	330
$IRL\left(G\right)={\displaystyle {\displaystyle \sum }_{UV\in E}}\left|ln{d}_u-ln{d}_v\right|$	4.41	14.17	29.28	49.75	75.58
$IRLU\left(G\right)={\displaystyle {\displaystyle \sum }_{UV\in E}}\frac{\left|d_u-d_v\right|}{min\left(d_{u,}{d}_v\right)}$	5.50	17	34.50	58	87.50
$IRLU\left(G\right)={\displaystyle {\displaystyle \sum }_{UV\in E}}\frac{\left|d_u-d_v\right|}{\sqrt{\left(d_u{d}_v\right)}}$	2.89	11.15	24.78	43.77	68.12
$IRF\left(G\right)={\displaystyle {\displaystyle \sum }_{UV\in E}}{\left(d_u-d_v\right)}^2$	30	84	162	264	390
$IRLA\left(G\right)=2{\displaystyle {\displaystyle \sum }_{UV\in E}}\frac{\left|d_u-d_v\right|}{\left(d_u+d_v\right)}$	4.34	14	29	49.34	75
$IRD1={\displaystyle {\displaystyle \sum }_{UV\in E}}ln\left\{1+\left|d_v-d_v\right|\right\}$	10.76	38.14	82.16	142.82	220.109
$IRA\left(G\right)={\displaystyle {\displaystyle \sum }_{UV\in E}}{\left(d_u^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-d_v^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}^2$	0.94	2.033	3.299	4.74	6.34
$IRGA\left(G\right)={\displaystyle {\displaystyle \sum }_{UV\in E}}ln\frac{d_u+d_v}{2\sqrt{\left(d_u{d}_v\right)}}$	0.24	0.62	1.14	1.82	2.65
$IRB\left(G\right)={\displaystyle {\displaystyle \sum }_{UV\in E}}{\left(d_u^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-d_v^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}^2$	1.86	5.055	9.59	15.46	22.67
	7.40	22.20	44.40	74	111

contains some calculated results of these indices for $CN{T}_H\left[p,q\right]$ for different values of p and q where p > 0. □

Now, we move towards triangular boron nanotubes. We represent an arbitrary structure of this nanotube with $BN{T}_{\alpha }\left[p, q\right]$ with p and q be the numbers of boron atoms in a row and a column respectively.

Theorem 2.

Let$BN{T}_{\alpha }\left[p, q\right]$with p and q both positive, then

1. 

$IRDIF\left(BN{T}_{\alpha }\left[p, q\right]\right)=\frac{127}{30}p+\frac{33}{30}{p}^2$

2. 

$AL\left(BN{T}_{\alpha }\left[p, q\right]\right)=10p+3p^2$

3. 

$IRL\left(BN{T}_{\alpha }\left[p, q\right]\right)=2.0866p+0.5469646p^2$

4. 

$IRLU\left(BN{T}_{\alpha }\left[p, q\right]\right)=\frac{5}{2}p+\frac{3}{5}{p}^2$

5. 

$IRLU\left(BN{T}_{\alpha }\left[p, q\right]\right)=\frac{10\sqrt{6}+3\sqrt{5}}{15}p+\frac{\sqrt{30}}{10}{p}^2$

6. 

$IRF\left(BN{T}_{\alpha }\left[p, q\right]\right)=18p+3p^2$

7. 

$IRLA\left(BN{T}_{\alpha }\left[p, q\right]\right)=\frac{92}{45}p+\frac{6}{11}{p}^2$

8. 

$IRD1\left(BN{T}_{\alpha }\left[p, q\right]\right)=5.7807435p+2.07944p^2$

9. 

$IRA\left(BN{T}_{\alpha }\left[p, q\right]\right)=0.0392463138p+\frac{11-2\sqrt{30}}{10}{p}^2$

10. 

$IRGA\left(BN{T}_{\alpha }\left[p, q\right]\right)=0.0940665090p+0.01244820422p^2$

11. 

$IRB\left(BN{T}_{\alpha }\left[p, q\right]\right)=0.9196202955p+33-6\sqrt{30}{p}^2$

Proof. 

We prove these results on the same lines as then in the case of Theorem 1. In order to prove the above theorem we have to consider the following edge partition of the $BN{T}_{\alpha }\left[p, q\right]$ in the

Table 3. Edge partition of alpha (α)-Boron Nano tube.

$\mathbf{Number}\mathbf{of}\mathbf{Edges}({\mathit{d}}_{\mathit{u}},{\mathit{d}}_{\mathit{v}})$	Number of Indices
(4,4)	$3p$
(4,5)	$2p$
(4,6)	$4p$
(5,5)	$\frac{p}{2}\left(3q-11\right)$
(5,6)	$3p^2$
(6,6)	$p+q-3$

as depicted in the Figure 4.

Now, using above Table 3 and definitions we have,

• $IRDIF\left(G\right)={\displaystyle \sum }_{UV\in E}\left|\frac{d_u}{d_v}-\frac{d_v}{d_v}\right|$

  $$\begin{array}{cc}\hfill IRDIF\left(BN{T}_{\alpha }\left[p,q\right]\right)& =3p\left|\frac{4}{4}-\frac{4}{4}\right|+2p\left|\frac{5}{4}-\frac{4}{5}\right|+4p\left|\frac{6}{4}-\frac{4}{6}\right|+\frac{p}{2}\left(3q-11\right)\left|\frac{5}{5}-\frac{5}{5}\right|+3p^2\left|\frac{6}{5}-\frac{5}{6}\right|\hfill \\ & +\left(p+q-3\right)\left|\frac{6}{6}-\frac{6}{6}\right|=\frac{127}{30}p+\frac{33}{30}{p}^2\hfill \end{array}$$
• $AL\left(G\right)={\displaystyle \sum }_{UV\in E}\left|d_u-d_v\right|$

  $$\begin{array}{cc}\hfill AL\left(BN{T}_{\alpha }\left[p,q\right]\right)& =3p\left|4-4\right|+2p\left|5-4\right|+4p\left|6-4\right|+\frac{p}{2}\left(3q-11\right)\left|5-5\right|+3p^2\left|6-5\right|\hfill \\ & +\left(p+q-3\right)\left|6-6\right|=3p^2+10p\hfill \end{array}$$
• $IRL\left(G\right)={\displaystyle \sum }_{UV\in E}\left|ln{d}_u-ln{d}_v\right|$

  $$\begin{array}{cc}\hfill IRL\left(BN{T}_{\alpha }\left[p,q\right]\right)& =3p\left|ln4-ln4\right|+2p\left|ln5-ln4\right|+4p\left|ln6-ln4\right|+\frac{p}{2}\left(3q-11\right)\left|ln5-ln5\right|\hfill \\ & +3p^2\left|ln6-ln5\right|+\left(p+q-3\right)\left|ln6-ln6\right|=2pln\frac{5}{4}+4pln\frac{6}{4}+3p^{2}ln\frac{6}{5}\hfill \end{array}$$
• $IRLU\left(G\right)={\displaystyle \sum }_{UV\in E}\frac{\left|d_u-d_v\right|}{min\left(d_u{d}_v\right)}$

  $$\begin{array}{cc}\hfill IRLU\left(BN{T}_{\alpha }\left[p,q\right]\right)& =3p\frac{\left|4-4\right|}{4}+2p\frac{\left|5-4\right|}{4}+4p\frac{\left|6-4\right|}{4}+\frac{p}{2}\left(3q-11\right)\frac{\left|5-5\right|}{5}+3p^2\frac{\left|6-5\right|}{5}\hfill \\ & +\left(p+q-3\right)\frac{\left|6-6\right|}{6}=2p\frac{1}{4}+4p\frac{1}{2}+3p^2\frac{1}{5}\hfill \end{array}$$
• $IRLU\left(G\right)={\displaystyle \sum }_{UV\in E}\frac{\left|d_u-d_v\right|}{\sqrt{\left(d_u{d}_v\right)}}$

  $$\begin{array}{cc}\hfill IRLU\left(BN{T}_{\alpha }\left[p,q\right]\right)& =3p\frac{\left|4-4\right|}{\sqrt{16}}+2p\frac{\left|5-4\right|}{\sqrt{20}}+4p\frac{\left|6-4\right|}{\sqrt{24}}+\frac{p}{2}\left(3q-11\right)\frac{\left|5-5\right|}{\sqrt{25}}+3p^2\frac{\left|6-5\right|}{\sqrt{30}}\hfill \\ & +\left(p+q-3\right)\frac{\left|6-6\right|}{\sqrt{36}}=\frac{2p}{\sqrt{20}}+\frac{8p}{\sqrt{24}}+\frac{3p^2}{\sqrt{30}}\hfill \end{array}$$
• $IRF\left(G\right)={\displaystyle \sum }_{UV\in E}{\left(d_u-d_v\right)}^2$

  $$\begin{array}{cc}\hfill IRF\left(BN{T}_{\alpha }\left[p,q\right]\right)& =3p{\left(4-4\right)}^2+2p{\left(5-4\right)}^2+4p{\left(6-4\right)}^2+\frac{p}{2}\left(3q-11\right){\left(5-5\right)}^2+3p^2{\left(6-5\right)}^2\hfill \\ & +\left(p+q-3\right){\left(6-6\right)}^2=18p+3p^2\hfill \end{array}$$
• $IRLA\left(G\right)=2{\displaystyle \sum }_{UV\in E}\frac{\left|d_u-d_v\right|}{\left(d_u+d_v\right)}$

  $$\begin{array}{cc}\hfill IRLA\left(BN{T}_{\alpha }\left[p,q\right]\right)& =2[3p\frac{\left|4-4\right|}{\left(8\right)}+2p\frac{\left|5-4\right|}{\left(9\right)}+4p\frac{\left|6-4\right|}{\left(10\right)}+\frac{p}{2}\left(3q-11\right)\frac{\left|5-5\right|}{\left(10\right)}+3p^2\frac{\left|6-5\right|}{\left(11\right)}+(p\hfill \\ & +q-3)\frac{\left|6-6\right|}{\left(12\right)}]=\frac{92p}{45}+\frac{6p^2}{11}\hfill \end{array}$$
• $IRD1={\displaystyle \sum }_{UV\in E}ln\left\{1+\left|d_v-d_v\right|\right\}$

  $$\begin{array}{cc}\hfill IRD1& =3pln\left\{1+\left|4-4\right|\right\}+2pln\left\{1+\left|5-4\right|\right\}+4pln\left\{1+\left|6-4\right|\right\}+\frac{p}{2}\left(3q-11\right)ln\left\{1+\left|5-5\right|\right\}\hfill \\ & +3p^{2}ln\left\{1+\left|6-5\right|\right\}+\left(p+q-3\right)ln\left\{1+\left|6-6\right|\right\}\hfill \\ & =2pln\left(2\right)+4pln\left(3\right)+3p^{2}ln\left(2\right)\hfill \end{array}$$
• $IRA\left(G\right)={\displaystyle \sum }_{UV\in E}{\left(d_u^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-d_v^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}^2$

  $$\begin{array}{cc}\hfill IRA\left(BN{T}_{\alpha }\left[p,q\right]\right)& =3p{\left(\frac{1}{\sqrt{4}}-\frac{1}{\sqrt{4}}\right)}^2+2p{\left(\frac{1}{\sqrt{5}}-\frac{1}{\sqrt{4}}\right)}^2+4p{\left(\frac{1}{\sqrt{6}}-\frac{1}{\sqrt{4}}\right)}^2+\frac{p}{2}\left(3q-11\right){\left(\frac{1}{\sqrt{5}}-\frac{1}{\sqrt{5}}\right)}^2\hfill \\ & +3p^2{\left(\frac{1}{\sqrt{6}}-\frac{1}{\sqrt{5}}\right)}^2+\left(p+q-3\right){\left(\frac{1}{\sqrt{6}}-\frac{1}{\sqrt{6}}\right)}^2\hfill \\ & =2p{\left(\frac{1}{\sqrt{5}}-\frac{1}{\sqrt{4}}\right)}^2+4p{\left(\frac{1}{\sqrt{6}}-\frac{1}{\sqrt{4}}\right)}^2+3p^2{\left(\frac{1}{\sqrt{6}}-\frac{1}{\sqrt{5}}\right)}^2\hfill \end{array}$$
• $IRGA\left(G\right)={\displaystyle \sum }_{UV\in E}ln\frac{d_u+d_v}{2\sqrt{\left(d_u{d}_v\right)}}$

  $$\begin{array}{cc}\hfill IRGA\left(BN{T}_{\alpha }\left[p,q\right]\right)& =3pln\frac{4+4}{2\sqrt{\left(16\right)}}+2pln\frac{5+4}{2\sqrt{\left(20\right)}}+4pln\frac{6+4}{2\sqrt{\left(24\right)}}+\frac{p}{2}\left(3q-11\right)ln\frac{5+5}{2\sqrt{\left(25\right)}}\hfill \\ & +3p^{2}ln\frac{6+5}{2\sqrt{\left(30\right)}}+\left(p+q-3\right)ln\frac{6+6}{2\sqrt{\left(36\right)}}\hfill \\ & =2pln\frac{9}{2\sqrt{\left(20\right)}}+4pln\frac{10}{2\sqrt{\left(24\right)}}+3p^{2}ln\frac{11}{2\sqrt{\left(30\right)}}\hfill \end{array}$$
• $IRB\left(G\right)={\displaystyle \sum }_{UV\in E}{\left(d_u^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-d_v^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}^2$

  $$\begin{array}{cc}\hfill IRB\left(BN{T}_{\alpha }\left[p,q\right]\right)& =3p{\left(\sqrt{4}-\sqrt{4}\right)}^2+2p{\left(\sqrt{5}-\sqrt{4}\right)}^2+4p{\left(\sqrt{6}-\sqrt{4}\right)}^2+\frac{p}{2}\left(3q-11\right){\left(\sqrt{5}-\sqrt{5}\right)}^2\hfill \\ & +3p^2{\left(\sqrt{6}-\sqrt{5}\right)}^2+\left(p+q-3\right){\left(\sqrt{6}-\sqrt{6}\right)}^2\hfill \\ & =2p{\left(\sqrt{5}-\sqrt{4}\right)}^2+4p{\left(\sqrt{6}-\sqrt{4}\right)}^2+3p^2{\left(\sqrt{6}-\sqrt{5}\right)}^2\hfill \end{array}$$

Following

Table 4. Irregularity indices for Alpha (α) Boron Nanotube.

Irregularity Indices	p = 1, q = 1	p = 2, q = 2	p = 3, q = 3	p = 4, q = 4	p = 5, q = 5
$IRDIF\left(G\right)={\displaystyle {\displaystyle \sum }_{UV\in E}}\left|\frac{d_u}{d_v}-\frac{d_v}{d_v}\right|$	5.33	12.87	22.60	34.53	48.67
$AL\left(G\right)={\displaystyle {\displaystyle \sum }_{UV\in E}}\left|d_u-d_v\right|$	13	32	57	88	125
$IRL\left(G\right)={\displaystyle {\displaystyle \sum }_{UV\in E}}\left|ln{d}_u-ln{d}_v\right|$	2.6329646	6.3598584	11.1806814	17.0954336	24.1041150
$IRLU\left(G\right)={\displaystyle {\displaystyle \sum }_{UV\in E}}\frac{\left|d_u-d_v\right|}{min\left(d_{u,}{d}_v\right)}$	3.10	7.40	12.90	19.60	27.50
$IRLU\left(G\right)={\displaystyle {\displaystyle \sum }_{UV\in E}}\frac{\left|d_u-d_v\right|}{\sqrt{\left(d_u{d}_v\right)}}$	2.627930	6.351306	11.170128	17.084396	24.094110
$IRF\left(G\right)={\displaystyle {\displaystyle \sum }_{UV\in E}}{\left(d_u-d_v\right)}^2$	21	48	81	120	165
$IRLA\left(G\right)=2{\displaystyle {\displaystyle \sum }_{UV\in E}}\frac{\left|d_u-d_v\right|}{\left(d_u+d_v\right)}$	2.59	6.27	11.04	16.90	23.86
$IRD1={\displaystyle {\displaystyle \sum }_{UV\in E}}ln\left\{1+\left|d_v-d_v\right|\right\}$	7.8601835	19.87924	36.0571905	56.3940140	80.8897175
$IRA\left(G\right)={\displaystyle {\displaystyle \sum }_{UV\in E}}{\left(d_u^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-d_v^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}^2$	0.0438011	0.0967118	0.1587321	0.2298620	0.3101015
$IRGA\left(G\right)={\displaystyle {\displaystyle \sum }_{UV\in E}}ln\frac{d_u+d_v}{2\sqrt{\left(d_u{d}_v\right)}}$	0.106514	0.23792	0.39423	0.57543	0.78153
$IRB\left(G\right)={\displaystyle {\displaystyle \sum }_{UV\in E}}{\left(d_u^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-d_v^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}^2$	1.46734	4.0301	7.6883	12.4420	18.2911
	5.33	12.87	22.60	34.53	48.67

computes some values of the above indices of $BN{T}_{\alpha }\left[p,q\right]$ for p and q both positive. □

The following theorem contains results about triangular boron nanotubes $BN{T}_t\left[p, q\right]$.

Theorem 3.

Let$BN{T}_t\left[p, q\right]$be the triangular boron nanotube with p and q both positive, we have

1. 

$IRDIF\left(BN{T}_t\left[p, q\right]\right)=5p$

2. 

$AL\left(BN{T}_t\left[p, q\right]\right)=12p$

3. 

$IRL\left(BN{T}_t\left[p, q\right]\right)=2.432790p$

4. 

$IRLU\left(BN{T}_t\left[p, q\right]\right)=3p$

5. 

$IRLU\left(BN{T}_t\left[p, q\right]\right)=\sqrt{6}p$

6. 

$IRF\left(BN{T}_t\left[p, q\right]\right)=24p$

7. 

$IRLA\left(BN{T}_t\left[p, q\right]\right)=\frac{24}{10}p$

8. 

$IRD1\left(BN{T}_t\left[p, q\right]\right)=6.591673732p$

9. 

$IRA\left(BN{T}_t\left[p, q\right]\right)=\frac{5-2\sqrt{6}}{2}p$

10. 

$IRGA\left(BN{T}_t\left[p, q\right]\right)=0.122465p$

11. 

$IRB\left(BBN{T}_t\left[p, q\right]\right)=60-24\sqrt{6}p$

Proof. 

In order to prove the above theorem, we have to consider Figure 5 and

Table 5. Edge partition of Triangular Boron Nanotube.

$\mathbf{Number}\mathbf{of}\mathbf{Edges}({\mathit{d}}_{\mathit{u}},{\mathit{d}}_{\mathit{v}})$	Number of Indices
(4,4)	$3p$
(4,6)	$6p$
(6,6)	$\frac{p}{2}\left(9q-24\right)$

which contain facts relating edge distributions of this graph.

Now using above Table 5 and definitions we have,

• $IRDIF\left(G\right)={\displaystyle \sum }_{UV\in E}\left|\frac{d_u}{d_v}-\frac{d_v}{d_v}\right|$

  $$IRDIF\left(BN{T}_t\left[p,q\right]\right)=3p\left|\frac{4}{4}-\frac{4}{4}\right|+6p\left|\frac{6}{4}-\frac{4}{6}\right|+\frac{p}{2}\left(9q-24\right)\left|\frac{6}{6}-\frac{6}{6}\right|=6p\left|\frac{6}{4}-\frac{4}{6}\right|$$
• $AL\left(G\right)={\displaystyle \sum }_{UV\in E}\left|d_u-d_v\right|$

  $$AL\left(BN{T}_t\left[p,q\right]\right)=3p\left|4-4\right|+6p\left|6-4\right|+\frac{p}{2}\left(9q-24\right)\left|6-6\right|=12p$$
• $IRL\left(G\right)={\displaystyle \sum }_{UV\in E}\left|ln{d}_u-ln{d}_v\right|$

  $$IRL\left(BN{T}_t\left[p,q\right]\right)=3p\left|ln4-ln4\right|+6p\left|ln6-ln4\right|+\frac{p}{2}\left(9q-24\right)\left|ln6-ln6\right|=6pln\frac{6}{4}$$
• $IRLU\left(G\right)={\displaystyle \sum }_{UV\in E}\frac{\left|d_u-d_v\right|}{min\left(d_u{d}_v\right)}$

  $$IRLU\left(BN{T}_t\left[p,q\right]\right)=3p\frac{\left|4-4\right|}{4}+6p\frac{\left|6-4\right|}{4}+\frac{p}{2}\left(9q-24\right)\frac{\left|6-6\right|}{6}=3p$$
• $IRLU\left(G\right)={\displaystyle \sum }_{UV\in E}\frac{\left|d_u-d_v\right|}{\sqrt{\left(d_u{d}_v\right)}}$

  $$IRLU\left(BN{T}_t\left[p,q\right]\right)=3p\frac{\left|4-4\right|}{\sqrt{16}}+6p\frac{\left|6-4\right|}{\sqrt{24}}+\frac{p}{2}\left(9q-24\right)\frac{\left|6-6\right|}{\sqrt{36}}=\frac{12p}{\sqrt{24}}$$
• $IRF\left(G\right)={\displaystyle \sum }_{UV\in E}{\left(d_u-d_v\right)}^2$

  $$IRF\left(BN{T}_t\left[p,q\right]\right)=3p{\left(4-4\right)}^2+6p{\left(6-4\right)}^2+\frac{p}{2}\left(9q-24\right){\left(6-6\right)}^2=24p$$
• $IRLA\left(G\right)=2{\displaystyle \sum }_{UV\in E}\frac{\left|d_u-d_v\right|}{\left(d_u+d_v\right)}$

  $$IRLA\left(BN{T}_t\left[p,q\right]\right)=2\left[3p\frac{\left|4-4\right|}{\left(8\right)}+6p\frac{\left|6-4\right|}{\left(10\right)}+\frac{p}{2}\left(9q-24\right)\frac{\left|6-6\right|}{\left(12\right)}\right]=\frac{24p}{10}$$
• $IRD1={\displaystyle \sum }_{UV\in E}ln\left\{1+\left|d_v-d_v\right|\right\}$

  $$IRD1=3pln\left\{1+\left|4-4\right|\right\}+6pln\left\{1+\left|6-4\right|\right\}+\frac{p}{2}\left(9q-24\right)ln\left\{1+\left|6-6\right|\right\}=6pln3$$
• $IRA\left(G\right)={\displaystyle \sum }_{UV\in E}{\left(d_u^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-d_v^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}^2$

  $$IRA\left(BN{T}_t\left[p,q\right]\right)=3p{\left(\frac{1}{\sqrt{4}}-\frac{1}{\sqrt{4}}\right)}^2+6p{\left(\frac{1}{\sqrt{6}}-\frac{1}{\sqrt{4}}\right)}^2+\frac{p}{2}\left(9q-24\right){\left(\frac{1}{\sqrt{6}}-\frac{1}{\sqrt{6}}\right)}^2=6p{\left(\frac{1}{\sqrt{6}}-\frac{1}{\sqrt{4}}\right)}^2$$
• $IRGA\left(G\right)={\displaystyle \sum }_{UV\in E}ln\frac{d_u+d_v}{2\sqrt{\left(d_u{d}_v\right)}}$

  $$IRGA\left(BN{T}_t\left[p,q\right]\right)=3pln\frac{4+4}{2\sqrt{\left(16\right)}}+6pln\frac{6+4}{2\sqrt{\left(24\right)}}+\frac{p}{2}\left(9q-24\right)ln\frac{6+6}{2\sqrt{\left(36\right)}}=6pln\frac{10}{2\sqrt{\left(24\right)}}$$
• $IRB\left(G\right)={\displaystyle \sum }_{UV\in E}{\left(d_u^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-d_v^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}^2$

  $$IRB\left(BN{T}_t\left[p,q\right]\right)=3p{\left(\sqrt{4}-\sqrt{4}\right)}^2+6p{\left(\sqrt{6}-\sqrt{4}\right)}^2+\frac{p}{2}\left(9q-24\right){\left(\sqrt{6}-\sqrt{6}\right)}^2=6p{\left(\sqrt{6}-\sqrt{4}\right)}^2$$

The following

Table 6. Irregularity indices for Triangular Boron Nanotube.

Irregularity Indices	p = 1, q = 1	p = 2, q = 2	p =3, q = 3	p = 4, q = 4	p = 5, q = 5
$IRDIF\left(G\right)={\displaystyle {\displaystyle \sum }_{UV\in E}}\left|\frac{d_u}{d_v}-\frac{d_v}{d_v}\right|$	5	10	15	20	25
$AL\left(G\right)={\displaystyle {\displaystyle \sum }_{UV\in E}}\left|d_u-d_v\right|$	12	24	36	48	60
$IRL\left(G\right)={\displaystyle {\displaystyle \sum }_{UV\in E}}\left|ln{d}_u-ln{d}_v\right|$	2.44	4.87	7.30	9.75	12.17
$IRLU\left(G\right)={\displaystyle {\displaystyle \sum }_{UV\in E}}\frac{\left|d_u-d_v\right|}{min\left(d_{u,}{d}_v\right)}$	3	6	9	12	15
$IRLU\left(G\right)={\displaystyle {\displaystyle \sum }_{UV\in E}}\frac{\left|d_u-d_v\right|}{\sqrt{\left(d_u{d}_v\right)}}$	2.45	4.90	7.35	9.80	12.25
$IRF\left(G\right)={\displaystyle {\displaystyle \sum }_{UV\in E}}{\left(d_u-d_v\right)}^2$	24	48	72	96	120
$IRLA\left(G\right)=2{\displaystyle {\displaystyle \sum }_{UV\in E}}\frac{\left|d_u-d_v\right|}{\left(d_u+d_v\right)}$	2.4	4.8	7.2	9.6	12
$IRD1={\displaystyle {\displaystyle \sum }_{UV\in E}}ln\left\{1+\left|d_v-d_v\right|\right\}$	6.59	13.18	19.78	26.37	32.98
$IRA\left(G\right)={\displaystyle {\displaystyle \sum }_{UV\in E}}{\left(d_u^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-d_v^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}^2$	0.05051	0.10102	0.15153	0.20204	0.2526
$IRGA\left(G\right)={\displaystyle {\displaystyle \sum }_{UV\in E}}ln\frac{d_u+d_v}{2\sqrt{\left(d_u{d}_v\right)}}$	0.123	0.245	0.3674	0.489	0.612
$IRB\left(G\right)={\displaystyle {\displaystyle \sum }_{UV\in E}}{\left(d_u^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-d_v^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}^2$	1.2122	2.4244	3.6368	4.8490	6.0612
	5	10	15	20	25

provides some values of the irregularity indices of $BN{T}_t\left[p,q\right]$ for positive values of p and q. □

4. Conclusions, Graphical Analysis and Discussions

In this part we give a graphical analysis of the irregularity indices of three tubes, and compare the results presented in the above section. We use two types of graphs as tools for our comparisons. One is the 2D graph where the dependence of a certain irregularity index is plotted against one parameter of the structure p or q while other is kept fixed. Second tool is the 3D graph where dependence of a certain irregularity index is plotted against both parameter of the structure and obtained surface represent the trends of irregularity index against both parameters p and q at the same time. We use blue color to show the surface of irregularity indices of alpha (α)-boron nanotube, black color to show the surface of irregularity index of triangular boron nanotube system, and green color to show the surface of irregularity indices of tri-hexagonal boron nanotube structure $CN{T}_H\left[p,q\right]$. All surfaces plotted in the Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11 below show that the alpha boron tube structure is the most irregular whereas triangular boron tubes are somewhat in the middle and tri-hexagonal boron nanotube structure $CN{T}_H\left[p,q\right]$ is the most regular structure with respect to most irregular indices discussed above.

However, the associated surfaces for IRA and IRGA behave differently, as shown in the following Figure 12 and Figure 13.

In the following Figure 14, we demonstrate 2D graph for q = 1 and change p. Here, red color shows the behavior of the irregularity index for the alpha boron nanotube. Figure 15 shows that alpha boron nanotube becomes rapidly irregular with respect to change in p than triangular and carbon boron tubes.

This is just a model of the behavior of almost all irregularity indices described above.

Only one of the irregularity indices, namely IRA, shows a different trend. For some values of p, triangular boron becomes more irregular than alpha boron nanotubes.

Finally, we conclude that alpha boron tubes are the most irregular in pattern, as compared to the other two structures. These facts contribute towards the complexity and the pattern of structures, and can be effectively used in nano-engineering and nano-devices.