Research Article  Open Access
Ansheng Ye, Aisha Javed, Muhammad Kamran Jamil, Kanza Abdul Sattar, Adnan Aslam, Zahid Iqbal, Asfand Fahad, "On Computation of Face Index of Certain Nanotubes", Discrete Dynamics in Nature and Society, vol. 2020, Article ID 3468426, 6 pages, 2020. https://doi.org/10.1155/2020/3468426
On Computation of Face Index of Certain Nanotubes
Abstract
Topological index is a number that can be used to characterize the graph of a molecule. Topological indices describe the physical, chemical, and biological properties of a chemical structure. In this paper, we derive the analytical closed formulas of face index of some planar molecular structures such as , , , , and armchair .
1. Introduction
In this time of rapid technological development, the pharmacological techniques have evolved rapidly during the recent years. Consequently, a large number of new drugs and chemical compounds have been obtained. A huge amount of work is required to study the biological, chemical, and pharmacological aspects of these new drugs and chemical compounds. This workload is becoming more and more cumbersome as it requires sufficient tools, reagents, human resources, and a lot of time to check the performance of these new chemical compounds. However, the developing countries cannot afford these equipment and reagents to check up these biochemical properties and are resultantly unable to compete with the developed world in the areas of medical science and industry. To some extent, the chemical graph theory solved this problem as it assists to measure the pharmaceutical, chemical, and physical properties of the chemical compounds. Fortunately, previous research has revealed that chemical properties of a molecule such as boiling point, melting point, and toxicity are closely related to their molecular structures (see [1, 2]). This relationship is one of the key reasons for the development of the mathematical chemistry. In the chemical graph theory, a molecular structure can be represented in the form of a graph , where vertices and edges of a graph show the atoms and the bonds of a molecular structure, respectively.
A topological index (TI) is an invariant that is assigned to a molecular structure (graph) and is used to characterize the molecule. It may be thought as a convenient device which converts a chemical constitution into a unique number, which is independent of the way in which the corresponding graph has been drawn or labeled. TIs were employed in developing a suitable correlation between the chemical structure and chemical or biological activities and physical properties. Several researchers working in the area of chemical and mathematical sciences have introduced TIs, such as the Wiener index, Randić indices, Zagreb indices, PI index, eccentric index, atombond connectivity index, and forgotten index, which have been used to predict the characteristics of the nanomaterials, drugs, and other chemical compounds. There are several papers to calculate the topological indices of some special molecular graphs [3–12].
The notions of a planar graph, its faces, and an infinite face are well known in the literature. Let be a finite simple connected planar graph, where , , and represent the vertex, edge, and face sets, respectively. A face is incident to an edge if is one of those which surrounds the face. Similarly, a face is incident to a vertex in if is at the end of one of those incident edges; the incidency of to the face is represented by . The face degree in is given as . For the notions and notations not given here, we refer [13] to the readers.
Recently, Jamil et al. [14] introduced a novel topological index named as the face index. The face index helped to predict the energy and the boiling points of selected benzenoid hydrocarbons with the correlation coefficient . For a graph , the face index (FI) can be defined as
In this paper, we calculate the face index of some special molecular graphs which have been widely used in drugs.
2. Main Results
In this section, we investigate the exact formulas of the face index for the molecular structures of vastly studied nanotubes with wide range of applications: , and . To find the face indices of the molecular graphs of these nanotubes, we partitioned the face set depending on the degrees of each face.
2.1. Face Index of and Nanotubes
The 2dimensional lattice of is constructed by the alternatingly positioned squares and octagons (see Figure 1(a)), where , and represent the number of rows, octagons in each row, and squares in each row. A nanotube can be constructed by rolling the 2D lattice of carbon atoms and can be seen in Figure 1(b).
(a)
(b)
Firstly, we prove the following formula which provides the exact values of the face index for .
Theorem 1. Let , where , be the 2dimensional lattice of nanotube; then, the face index of is given as
Proof. Let be the 2dimensional lattice of nanotube with number of rows and let and be the number of octagons and number of squares in each row, respectively. In , the total number of faces in one row is . Let denote the faces having and denote the number of faces with degree . From Figure 1(a), it can be noticed that 2D lattice of contains four types of internal faces , , , and and an external face, . When has rows, then sum of vertex degrees of external face is . The number of internal faces in each row is given in Table 1.
The face index of isThis completes the proof.

Theorem 2. Let and be the graph of 2dimensional lattice of nanotube. Then, the face index of the graph is given as
Proof. Consider a nanotube with number of rows, number of octagons, and number of squares in each row as shown in Figure 2(a). The 2dimensional lattice () of is shown in Figure 2(b). In , the total number of faces in one row is . Let and denote the face with degree and the number of faces with degree , respectively. From the structure of , one can notice that there are three types of internal faces , , and and an external face . The external face has degree . Table 2 illustrates the number of internal faces in based on the degree of each face.
The face index of the graph isThis completes the proof.
(a)
(b)

2.2. Face Index of Nanotube
For and , the 2dimensional lattice of nanotube is obtained by the Cartesian product of the path and the cycle . For and , the example of is shown in Figure 3(b).
(a)
(b)
Theorem 3. Let be the graph of nanotube structure, where and . Then, the face index of is equal to
Proof. We will prove the result for . Let denote the graph of nanotube structure. From Figure 3, we can notice that the graph contains three types of internal faces, namely, , and , and an external face of degree . By applying the definition and using the values from Table 3, the face index of can be computed asThis completes the proof.

2.3. ZigZag Nanotube
Consider the graph of zigzag polyhex nanotube structure, where denotes the number of rows and the number of hexagons in each row is represented by . Figure 4 illustrates the nanotube and its 2dimensional structure.
(a)
(b)
Theorem 4. For , let represent the 2dimensional graph of structure. The face index of is
Proof. Let be a polyhex nanotube with number of rows and number of hexagons in each row and be the 2dimensional graph of structure. The molecular graph of is shown in Figure 4. Let denote the face having degree , i.e., , and let denote the number of . The molecular graph of contains two types of internal faces and and an external face . When has rows, then the face degree of is . The number of internal faces with the given number of rows is listed in Table 4.
The face index of the graph iswhich is the required result.

2.4. Nanotube
The 2dimensional graph of is shown in Figure 5(b), and the nanotube can be constructed by rolling this lattice of carbon atoms (Figure 5(a)), where represents the number of rows and is the hexagons in each row.
(a)
(b)
Theorem 5. Let , where , be the graph of 2dimensional lattice of armchair polyhex nanotube. The face index of is
Proof. Let represent the 2dimensional molecular graph of with number of rows and number of hexagons in each row. From Figure 5(b), we can easily notice that contains 2 types of internal faces and and an external face, and the degree of external face is . The cardinalities of internal faces with given degree and given number of rows are explained in Table 5.
The face index of isand the proof is complete.

3. Conclusion
In [14], using multiple linear regression, it has been shown that the novel face index can predict the electron energy and boiling point of benzenoid hydrocarbon with a correlation coefficient greater than 0.99. Therefore, this index can be useful in QSPR/QSAR studies. In this paper, we have computed the novel face index of some nanotubes.
Data Availability
No data were used to support the study.
Disclosure
This research was carried out as a part of the employment of the authors.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
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Copyright
Copyright © 2020 Ansheng Ye et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.