On Computation of Face Index of Certain Nanotubes

School of Information Science and Engineering, Chengdu University, Chengdu 610106, China Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan Department of Mathematics, Riphah International University, 14 Ali Road, Lahore, Pakistan Department of Natural Sciences and Humanities, University of Engineering and Technology, Lahore (RCET), Lahore, Pakistan National University of Science and Technology, Islamabad, Pakistan Department of Mathematics and Statistics, Institute of Southern Punjab, Multan, Pakistan Department of Mathematics, COMSATS University Islamabad, Vehari Campus, Vehari 61110, Pakistan


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.
is 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]). is 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 G � (V(G), E(G)), where vertices V � V(G) and edges E � E(G) of a graph G 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, atom-bond connectivity index, and forgotten index, which have been used to predict the characteristics of the nanomaterials, drugs, and other chemical compounds.
ere are several papers to calculate the topological indices of some special molecular graphs [3][4][5][6][7][8][9][10][11][12]. e notions of a planar graph, its faces, and an infinite face are well known in the literature. Let G � (V(G), E(G), F(G)) be a finite simple connected planar graph, where V(G), E(G), and F(G) represent the vertex, edge, and face sets, respectively. A face f ∈ F(G) is incident to an edge e ∈ E(G) if e is one of those which surrounds the face. Similarly, a face f ∈ F(G) is incident to a vertex v in G if v is at the end of one of those incident edges; the incidency of v to the face f is represented by v∼f. e face degree f in G is given as d(f) � v∼f d (v). 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. e face index helped to predict the energy and the boiling points of selected benzenoid hydrocarbons with the correlation coefficient r > 0.99. For a graph G, the face index (FI) can be defined as (1) In this paper, we calculate the face index of some special molecular graphs which have been widely used in drugs.

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: 6 , TUC 4 , and TUVC 6 . To find the face indices of the molecular graphs of these nanotubes, we partitioned the face set depending on the degrees of each face.

Face Index of TUC
e 2-dimensional lattice of TUC 4 C 8 (S) [n, q, r] is constructed by the alternatingly positioned squares C 4 and octagons C 8 (see Figure 1(a)), where n, q, and r represent the number of rows, octagons in each row, and squares in each row. A TUC 4 C 8 (S)[n, q, r] nanotube can be constructed by rolling the 2D lattice of carbon atoms and can be seen in Figure 1 Firstly, we prove the following formula which provides the exact values of the face index for TUC 4 Proof.
Let G be the 2-dimensional lattice of TUC 4 C 8 (S)[n, q, r] nanotube with n number of rows and let q and r be the number of octagons and number of squares in each row, respectively. In TUC 4 C 8 (S)[n, q, r], the total number of faces in one row is q + r. Let f j denote the faces having w∼f j d w � j and |f j | denote the number of faces with degree j. From Figure 1(a), it can be noticed that 2D lattice of TUC 4 C 8 (S) [7,4,4] contains four types of internal faces f 12 , f 20 , f 22 , and f 24 and an external face, f ∞ . When TUC 4 C 8 (S)[n, q, r] has n rows, then sum of vertex degrees of external face is 8q + 12r. e number of internal faces in each row is given in Table 1.
e face index of TUC 4 is completes the proof.
Proof. Consider a TUC 4 C 8 (R)[n, q, r] nanotube with n number of rows, q number of octagons, and r number of squares in each row as shown in Figure 2(a). e 2-dimensional lattice (H) of TUC 4 C 8 (R)[n, q, r] is shown in Figure 2(b). In H, the total number of faces in one row is q + r. Let f j and |f j | denote the face with degree j and the number of faces with degree j, respectively. From the structure of H, one can notice that there are three types of internal faces f 11 , f 12 , and f 24 and an external face f ∞ . e external face has degree 12q + 2r.  Figure 3(b).
Proof. We will prove the result for p > 3. Let K denote the graph of TUC 4 C[p, q] nanotube structure. From Figure 3, we can notice that the graph K contains three types of internal faces, namely, f 3q , f 14 , and f 16 , and an external face of degree 3q. By applying the definition and using the values from Table 3, the face index of K can be computed as is completes the proof. 6 [n, q] Nanotube. Consider the graph K of TUHC 6 [n, q] zig-zag polyhex nanotube structure, where n denotes the number of rows and the number of hexagons in each row is represented by q. Figure 4 illustrates the nanotube TUHC 6 [n, q] and its 2-dimensional structure.

Zig-Zag TUHC
Theorem 4. For n, q ≥ 1, let K represent the 2-dimensional graph of TUHC 4 [n, q] structure. e face index of K is Proof. Let TUHC 6 [n, q] be a polyhex nanotube with n number of rows and q number of hexagons in each row and K be the 2-dimensional graph of TUHC 6 [n, q] structure. e molecular graph of TUHC 6 [n, q] is shown in Figure 4. Let f j denote the face having degree j, i.e., w∼f j d w � j, and let |f j | denote the number of f j . e molecular graph of TUHC 6 [n, q] contains two types of internal faces f 17 and f 18 and an external face f ∞ . When TUHC 6 [n, q] has n rows, then the face degree of f ∞ is 10q. e number of internal faces with the given number of rows is listed in Table 4.
which is the required result.
Proof. Let L represent the 2-dimensional molecular graph of TUVC 6 [n, q] with n number of rows and q number of hexagons in each row. From Figure 5(b), we can easily notice that L contains 2 types of internal faces f 16 and f 18 and an external face, and the degree of external face is 14q. e cardinalities of internal faces with given degree and given number of rows are explained in Table 5. e face index of L � TUVC 6 [n, q] is Table 3: Numbers of f 11 , f 12 , and f 24 with given number of rows.
and the proof is complete.

Conclusion
In [14], using multiple linear regression, it has been shown that the novel face index can predict the π electron energy and boiling point of benzenoid hydrocarbon with a correlation coefficient greater than 0.99. erefore, 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
is research was carried out as a part of the employment of the authors.

Conflicts of Interest
e authors declare that they have no conflicts of interest.