# Computational Aspects of Carbon and Boron Nanotubes

## Abstract

**:**

## 1. Introduction

^{st}century as space, entertainment and communication technology revolutionized the 20

^{th}century. It involves different structures of nanotubes. The most significant nano structures are carbon nanotubes, boron triangular nanotubes and boron α-nanotubes. See Figure 1a,b,c. Nanotubes are three dimensional cylindrical structures formed out of the two dimensional sheets.

## 2. Basic Properties of Nanotubes of Armchair Model

**Theorem**

**2.1:**

**Proof:**

**Lemma**

**2.2:**

**Theorem 2.3:**

- (1)
- A carbon hexagonal nanotube of order n × m has nm vertices and m(3n–2)/2 edges.
- (2)
- A boron triangular nanotube of order n × m has 3nm/2 vertices and 3m(3n–2)/2 edges.
- (3)
- A boron α-nanotube of order n × m has 4 nm/3vertices and m(7n–4)/2 edges when n is a multiple of 3.

## 3. Independent Set of Three Nanotubes

^{0.304n}) exponential algorithm for solving maximum independent set problem for general graphs. Soares and Stefanes [19] have given a polynomial algorithm to find maximum independent set of convex bipartite graphs. Algorithms are proposed to solve maximum independent set problem of planar graphs [20] and apple-free graphs [21]. In this section we show that the maximum independent set of three nanotubes are the same.

**Theorem**

**3.1:**

**Proof:**

## 4. Perfect Matching and Matching Ratio

**Theorem**

**4.1:**

**Proof**:

## 5. Broadcasting Problem of Carbon and Boron Nanotubes

#### 5.1. Broadcasting Algorithm for Carbon Hexagonal Nanotubes

- Step 1:
- Let O be the source node with the message and E denote the eccentric node of O. Unfold the carbon hexagonal nanotube into a rectangular sheet in such a way that the eccentric vertex E lies on the perimeter of the rectangular sheet. See Figure 6.
- Step 2:
- Draw lines TOZ (at angle 30°), ROX (vertical), and SOY (at angle 150°). These lines create six zones, namely, zones ROS, SOT, TOX, XOY, YOZ and ZOR. See Figure 7.
- Step 3:
- Delete all the edges of zone ROS which are perpendicular to OS. Similarly, delete all the edges perpendicular to OT in zone SOT, edges perpendicular to OX in zone TOX, edges perpendicular to OY in zone XOY, edges perpendicular to OZ in zone YOZ, edges perpendicular to OR in zone ZOR. The resulting tree is the broadcasting tree of the carbon hexagonal nanotubes. See Figure 8.
- Step 4:
- Message is disseminated from source O based on farthest-distance-first protocol where a node with the message chooses an uninformed adjacent node which leads to longest path in the tree. If a node has label i, it means that the node receives the message from its neighbor at i
^{th}time unit. See Figure 9.

**Proof of Correctness**:

**Lemma 5.1**:

**Proof**:

**Theorem 5.2**:

**Proof**:

#### 5.2. Broadcasting Problem of Boron Triangular Nanotubes

- Step 1:
- Let O be the source node with the message and E denote the eccentric node of O. Unfold the boron triangular nanotube into a rectangular sheet in such a way that the eccentric vertex E lies on the perimeter of the rectangular sheet. See Figure 10.
- Step 2:
- Draw lines SOY (horizontal), TOZ (at angle 60°), and XOR (at angle 120°). These lines create six zones, namely, zones ROS, SOT, TOX, XOY, YOZ and ZOR. See Figure 11.
- Step 3:
- Delete all the edges of zone ROS except the edges parallel to OS. Similarly retain only the edges parallel to OT in zone SOT, edges parallel to OX in zone TOX, edges parallel to OY in zone XOY, edges parallel to OZ in zone YOZ, and edges parallel to OR in zone ZOR. The resulting tree is the broadcasting tree of the boron triangular nanotube. See Figure 12.
- Step 4:
- The message is disseminated from source O based on farthest-distance-first protocol where a node with the message chooses an uninformed adjacent node which leads to longest path in the tree. If a node has label i, it means that the node receives the message from its neighbor at i
^{th}time unit. See Figure 13.

**Theorem 5.3**:

## Conclusions

## Acknowledgements

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Sample Availability: Contact the authors. |

**Figure 1.**a. Carbon hexagonal sheet (1991); b. Boron triangular sheet (2004); c. Boron α-sheet (2008).

**Figure 3.**This is an armchair carbon hexagonal nanotube of order 11 × 6. There are six columns and each column has 11 rows of vertices. Each vertex is labeled based on its location with respect to row and column. The first column and the last column of the carbon hexagonal sheet are merged to form a carbon nanotube.

**Figure 4.**The maximum independent set is the same for all the three nanotubes. The set of red vertices is a maximum independent set of three nanotubes. The set of blue vertices is another maximum independent set.

**Figure 5.**All three nanotubes have perfect matching. The edges of blue dotted lines form a perfect matching.

**Figure 6.**A rectangular sheet of carbon hexagonal nanotube of order 19 × 8. Node O is the source and node E is the eccentric node of O. The wrapping edges between the first column and the last column are not drawn.

**Figure 7.**The rectangular sheet is divided into 6 zones: zones ROS, SOT, TOX, XOY, YOZ and ZOR. For example, zone ROS is a subgraph induced by the edges lying between the lines OR and OS.

**Figure 8.**Broadcasting tree of the nanotube. Broadcasting is based on farthest-distance-first protocol.

**Figure 9.**A node of label 5 means that the node receives the message from its neighbor at 5

^{th}time unit.

**Figure 10.**Node O is the source and node E is the eccentric node of O. The wrapping edges between the first column and the last column are not drawn.

**Figure 12.**Broadcasting tree of boron nanotube. Broadcasting is based on farthest-distance-first protocol.

**Figure 13.**A node of label 5 means that the node receives the message from its neighbor at 5

^{th}time unit.

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Manuel, P.
Computational Aspects of Carbon and Boron Nanotubes. *Molecules* **2010**, *15*, 8709-8722.
https://doi.org/10.3390/molecules15128709

**AMA Style**

Manuel P.
Computational Aspects of Carbon and Boron Nanotubes. *Molecules*. 2010; 15(12):8709-8722.
https://doi.org/10.3390/molecules15128709

**Chicago/Turabian Style**

Manuel, Paul.
2010. "Computational Aspects of Carbon and Boron Nanotubes" *Molecules* 15, no. 12: 8709-8722.
https://doi.org/10.3390/molecules15128709