M-Polynomial and Topological Indices of Benzene Ring Embedded in P-Type Surface Network.

1. Introduction

The chemical compounds can be represented by using the mathematical tools of graph theory. The mathematical models that are based on the polynomials of the chemical compounds and crystal structures can be used in order to predict and forecast their chemical properties and bioactivities. Mathematical chemistry is rich in tools like functions and polynomials which predict the properties of molecular graphs and crystal structures. The topological descriptors are the numerical parameters of the chemical graph which characterize its topology and are usually graph invariants. They explain the structure of chemical compounds mathematically and are utilized in the study of quantitative structure property and activity relationships (QSPR/QSAR).

A topological index is a numerical value which describes and explains an important information about the chemical structure. A great variety of such indices are studied and used in theoretical chemistry, pharmaceutical research, drugs, and different areas of science. The properties like boiling point, strain energy, viscosity, fracture toughness, and heat of formation are connected to the chemical structure under study. This fact plays a major role in the field of chemical graph theory [1-22].

The computation of the general polynomial is formed whose derivatives or integrals or composition of both are evaluated at some particular point. Then, the simplified form yields the molecular descriptor. For instance, there are polynomials like forgotten polynomials, Zagreb polynomials, and Hosoya polynomials, but these polynomials give rise to one or two topological indices [23-26]. The Hosoya polynomial is a polynomial whose derivatives evaluated at 1 give Wiener and hyper Wiener index [27]. The Hosoya polynomial and Zagreb polynomials are considered to be of the general form in the determination of distance-based and degree-based indices, respectively. The M-polynomial is a new and recent polynomial. It will open up new results of chemical graphs and insights in the study of topological descriptors based on degrees. The main importance of this polynomial is that it can give exact forms of more than ten degree-based molecular descriptors [28, 29]. Rapid development and advancements are being made in this new polynomial. Recently, Kwun et al. computed M-polynomial and topological indices of V-phenylenic nanotube and nanotori [30].

The M-polynomial of a graph G is formulated as [28]

M(G; x,y)= [summation over ([delta][less than or equal to]j[less than or equal to]j)] [m.sub.ij](G)[x.sup.i][y.sup.j], (1)

where [m.sub.ij](G) is the number of edges uv e [member of](G) such that ([d.sub.u], [d.sub.v]) = (i,j), [delta] = min [d.sub.v] | v [member of] V(G), and [DELTA] = max [d.sub.v] | v [member of] V(G).

The path number was the first distance-based topological index defined by Wiener [31] in 1947. This index is now called as the Wiener index. It has many famous mathematical and chemical applications [31, 32]. Later on, Milan Randic proposed and formulated the Randic index of a graph [GR.sub.-(i/2)](G).

[R.sub.-(1/2)](G) = [summation over (uv[member of]E(G))] 1/[square root of [d.sub.u][d.sub.v]]. (2)

The general Randic index was proposed and defined independently by Bollobas et al. [33] and Amic et al. [34]. Due to its useful and important results in the field of mathematical chemistry, it has been widely used by both mathematicians and chemists. For a survey of these results, see references [35-38]. The general Randic index and inverse Randic index are formulated as

[R.sub.[alpha]](G) = [summation over (uv[member of]E(G))] [([d.sub.u][d.sub.v]).sup.[alpha]], R[R.sub.[alpha]](G) = [summation over (uv[member of]E(G))] 1/[([d.sub.u][d.sub.v]).sup.[alpha]]. (3)

The first and second Zagreb indices are introduced by Gutman and Trinajstic [25,39,40]. Both first and second Zagreb indices and the second modified index are formulated as

[mathematical expression not reproducible]. (4)

Recently, the symmetric division deg index of a graph G is introduced [41]. It is the significant index which is used to determine the total surface area of polychlorobiphenyls [42] and is defined as

[mathematical expression not reproducible]. (5)

The other version of the Randic index is the harmonic index [43] and is defined as

H(G) = [summation over (uv[member of]E(G))] 2/[d.sub.u] + [d.sub.v]. (6)

The inverse sum index is formulated as [44]

I(G) = [summation over (uv[member of]E(G))] [d.sub.u][d.sub.v]/[d.sub.u] + [d.sub.v]. (7)

The augmented Zagreb index gives best approximation of heat of formation of alkanes [45, 46]. It is formulated as [47]

A(G) = [summation over (uv[member of]E(G))] [([d.sub.u][d.sub.v]/[d.sub.u] + [d.sub.v] - 2).sup.3]. (8)

Let M (G; x, y)= f (x, y), and then Table 1 relates above described topological indices with M-polynomial [28], where

[mathematical expression not reproducible]. (9)

2. Main Results and Discussion

OKeeffe et al. have distributed around a quarter century a letter managing two 3D systems of benzene, and one of the structures was known as 6.82P (or additionally polybenzene) and has a place with the space gather Im3m, compared with the P-type surface [48]. Actually, this is insertion of the hexagon fix in the surface of negative ebb and flow P. The P-type surface is coordinated to the Cartesian arranges in the Euclidean space. The reader can discover more about this intermittent surface in [49, 50]. This structure was required to be combined as 3D carbon solids and no such combination was accounted before. This has aroused a lot of research enthusiasm of researchers to carbon nanoscience. As much as the graphenes were picked up a moment Nobel prize after [C.sub.60], fullerenes have also been studied in depth, see detail in [51, 52].

The molecular graph of the benzene ring embedded in the P-type surface network is depicted in Figure 1. The cardinality of vertices and edges of the given molecular graph are 24mn and 32mn - 2m - 2n, respectively. The vertex set consists of two vertex partitions in the benzene ring embedded in the P-type surface network, as shown in Table 2. Furthermore, the edge set consists of three edge partitions. The first edge partition contains 4m + 4n edges uv, where deg(u) = deg(v) = 2. The second edge partition contains 16mn edges uv, where deg (u) = 2 and deg(v) = 3. The third edge partition contains 16mn - 2m - 2n edges uv, where deg(u) = deg(v) = 3. Table 3 shows the edge partition in the benzene ring embedded in the P-type surface network. We compute the M-polynomial of the benzene ring embedded in the P-type surface network. Also, we present the graphical representation of this graph in 2D and 3D by using Maple 13. In the end, we compute and simplify the topological indices by using the M-polynomial of the benzene ring embedded in the P-type surface network.

3. M-Polynomial of Benzene Ring Embedded in P-Type Network

Theorem 1. Consider the graph of a benzene ring embedded in the P-type surface network BR (m, n) with m, n > 1, and then the M-polynomial of this graph is given by

M(BR(m, n); x, y) = (4m + 4n)[x.sup.2][y.sup.2] + (16mn)[x.sup.2][y.sup.3] + (16mn - 6m - 6n)[x.sup.3][y.sup.3].

Proof. Let the graph of a benzene ring embedded in the P-type surface network with m and n being the number of unit cells in the columns and rows, respectively. It consists of two vertices and three edge partitions. From Figure 1, it is easy to observe that

[absolute value of V (BR (m, n))] = 24mn, [absolute value of E(BR(m,n))] = 32mn - 2m - 2n. (11)

From Table 2, it can be seen that there are two partitions of the vertex set of the benzene ring embedded in the P-type surface network.

[V.sub.1] (BR (m, n)) = {u [member of] V (BR (m, n)) | [d.sub.u] = 2}, [V.sub.2] (BR (m, n))= {u [member of] V (BR (m, n)) | [d.sub.u] = 3}, (12)

such that

[absolute value of [V.sub.1] (BR(m, n))] = 8mn + 4m + 4n, [absolute value of [V.sub.2] (BR(m, n))] = 16mn - 4m - 4n. (13)

From Table 3, it can be seen that there are three partitions of the edge set of the benzene ring embedded in the P-type surface network.

[mathematical expression not reproducible], (14)

such that

[mathematical expression not reproducible]. (15)

Now, applying the definition of M-polynomial to the graph of the benzene ring embedded in the P-type network, we have

[mathematical expression not reproducible]. (16)

The 3D graphical representation of M-polynomial of the benzene ring embedded in the P-type surface network BR (m, n) is depicted in Figure 2. This is plotted by using Maple 13. The graph shows different behavior by fixing the values of m and n and changing the parameters x and y. If the 2D graphical representation of M-polynomial of BR (m, n) can be formed by considering the parameter x to be the positive value, then the graph increases by increasing the values of x, and the graph lies in the first and third quadrant. The same behavior occurs for positive values of y, as depicted in Figures 3(a) and 3(b). If the parameter x is taken to be the negative value, then the graph increases by increasing the values of x, and the graph lies in the second and fourth quadrant. The same behavior occurs for negative values of y.

4. Topological Indices Derived from M-Polynomial of BR (m, n)

The following proposition computes the degree-based topological indices that are derived from the M-polynomial of the molecular graph of the benzene ring embedded in the P-type surface network.

Proposition 1. Consider the graph G be a benzene ring embedded in the P-type surface network with m, n > 1; then, we have the following degree-based topological indices:

(1) [M.sub.1](G) = 176mn - 20m - 20n

(2) [M.sub.2](G) = 240mn - 38m - 38n

(3) [sup.m][M.sub.2](G) = (40mn + 3m + 3n)/9

(4) SDD(G) = (200mn - 12m - 12n)/3

(5) H(G) = 176mn/15

(6) I(G) = (24mn - 25m - 25n)/5

(7) A(G) = (9928mn - 1163m - 1163n)/32

(8) R[R.sub.[alpha]](G) = (16mn)(([2.sup.[alpha]] + [3.sup.[alpha]])/[2.sup.[alpha]] [3.sup.2[alpha]]) + (m + n) ((4([3.sup.2[alpha]]) - 6([2.sup.2[alpha]]))/[6.sup.2[alpha]])

(9) [R.sub.[alpha]](G)= (16mn)([3.sup.2[alpha]] + [6.sup.[alpha]]) + (m + n)(4([2.sup.2[alpha]]) - 6([3.sup.2[alpha]]))

Proof. Consider the molecular graph of G be a benzene ring embedded in the P-type surface network with m, n > 1; its M-polynomial is simplified in the first theorem. Now, consider the following:

[mathematical expression not reproducible]. (17)

In order to prove the above nine results, we use the following formulas:

[mathematical expression not reproducible]. (18)

Now, we have the following computations:

[D.sub.x](f(x, y)) = 2(4m + 4n)[x.sup.2][y.sup.2] + 2(16mn)[x.sup.2][y.sup.3] + 3(16mn - 6m - 6n)[x.sup.3][y.sup.3], (19)

[D.sub.y](f(x, y)) = 2(4m + 4n)[x.sup.2][y.sup.2] + 3 (16mn)[x.sup.2][y.sup.3] + 3 (16mn - 6m - 6n)[x.sup.3][y.sup.3], (20)

[D.sub.x][D.sub.y](f(x, y)) = 4 (4m + 4n) [x.sup.2][y.sup.2] + 6(16mn)[x.sup.2][y.sup.3] + 9(16mn - 6m - 6n)[x.sup.3][y.sup.3], (21)

[S.sub.x](f(x, y)) = (4m + 4n)/2 [x.sup.2][y.sup.2] + (16mn)/2 [x.sup.2][y.sup.3] + (16mn - 6m - 6n)/3 [x.sup.3][y.sup.3], (22)

[S.sub.x][S.sub.y](f(x,y)) = (4m + 4n)/4[x.sup.2][y.sup.2] + (16mn)/6 [x.sup.2][y.sup.3] (16mn - 6m - 6n)/9 [x.sup.3][y.sup.3] (23)

[S.sub.x][D.sub.y] (f(x, y)) = (4m + 4n)[x.sup.2][y.sup.2] + 3(16mn)/2 [x.sup.2][y.sup.3] + (16mn - 6m - 6n) [x.sup.3][y.sup.3], (24)

[D.sub.x][S.sub.y] (f(x, y)) = (4m + 4n) [x.sup.2][y.sup.2] 2(16mn)/3 [x.sup.2][y.sup.3] + (16mn - 6m - 6n) [x.sup.3][y.sup.3], (25)

[D.sup.[alpha].sub.x][D.sup.[alpha].sub.y](f(x, y)) = [2.sup.2[alpha]] (4m + 4n) [x.sup.2][y.sup.2] + [6.sup.[alpha]] (16mn) [x.sup.2][y.sup.3] + [3.sup.2[alpha]] (16mn - 6m - 6n) [x.sup.3][y.sup.3], (26)

[S.sup.[alpha].sub.x][S.sup.[alpha].sub.y](f(x,y)) = (4m + 4n)/[2.sup.2[alpha]] - (16mn)/[6.sup.[alpha]][x.sup.2][y.sup.3] + (16mn - 6m - 6n)/[3.sup.2[alpha]] [x.sup.3][y.sup.3], (27)

[S.sub.x]J (f(x, y)) = (4m + 4n)/4 [x.sup.4] + (16mn)/5 [x.sup.5] + (16mn - 6m - 6n)/[3.sup.2[alpha]] [x.sup.3][y.sup.3], (28)

[S.sub.x]J[D.sub.x][D.sub.y](f(x, y)) = 8(4m + 4n)[x.sup.2] + 8(16mn)[x.sup.3] + 729(16mn - 6m - 6n)/64 [x.sup.4]. (29)

[S.sup.3.sub.x][Q.sub.-2]J[D.sup.3.sub.x][D.sup.3.sub.y](f(x, y)) = 8(4m + 4n)[x.sup.2] + 8(16mn)[x.sup.3] + 729(16mn - 6m - 6n)/64 [x.sup.4]. (30)

Now, by using all the aforementioned values from equations (19)-(30) in Table 1, the topological indices defined in Table 1 are obtained.

(1) [M.sub.1](G) = 176mn - 20m - 20n

(2) [M.sub.2](G) = 240mn - 38m - 38n

(3) [sup.m][M.sub.2](G) = (40mn + 3m + 3n)/9

(4) SDD(G) = (200mn - 12m - 12n)/3

(5) H(G) = 176mn/15

(6) I(G) = (24mn - 25m - 25n)/5

(7) A(G) = (9928mn - 1163m - 1163n)/32

(8) R[R.sub.[alpha]](G) = (16mn)(([2.sup.[alpha]] + [3.sup.[alpha]])/[2.sup.[alpha]][3.sup.2[alpha]]) + (m + n) ((4([3.sup.2[alpha]]) - 6([2.sup.2[alpha]]))/[6.sup.2[alpha]])

(9) [R.sub.[alpha]](G)= (16mn)([3.sup.2[alpha]] + [6.sup.[alpha]]) + (m + n)(4 ([2.sup.2[alpha]]) - 6([3.sup.2[alpha]]))

The symmetric division, harmonic, inverse sum, and augmented Zagreb indices are plotted by using Maple 13. The graphical representation depicts different behavior of indices by changing the parameters m and n. The blue, green, red, and black colors show the symmetric division, harmonic, inverse sum, and augmented Zagreb indices, respectively, as depicted in Figure 4(a). Figure 4(b) illustrates the first Zagreb index in blue color, second Zagreb index in green color, and modified Zagreb index in red color.

The 3D plot of the Randic index and inverse Randic index is illustrated in Figures 5 and 6, respectively. It is clearly seen from the graphs that by increasing the values of the parameters m and n, the graph of 5 increases faster than the graph of 6. It can be concluded that the Randic index increases faster than the inverse Randic index.

The 2D plot of the inverse Randic index is depicted in Figure 7(a). This is achieved by using Maple 13 and fixing the value of the parameter m or n. In both cases, if values of the parameter increases then the graph increases gradually and shows different behavior. The 2D plot of the Randic index is depicted in Figure 7(b). By increasing the values of the parameters, the graph increases and depicts different behavior.

Figure 8 illustrates the 3D plot of the augmented Zagreb index for the molecular graph BR (m, n). By increasing the values of the given parameters, the value of indices increases. The value indices of BR (m, n) increase by changing the values of parameters m and n. The indices derived here are the functions that depend on the values of parameters, where m and n are the independent parameters and the index is the dependent parameter.

5. Conclusions

We have computed the general form of M-polynomial for the molecular graph of the benzene ring embedded in the P-type surface network BR (m, n) for the first time. The graphical representation of M-polynomial of BR (m, n) and some of its indices have plotted for different values of the given parameters. Furthermore, we have derived and simplified the exact results for degree-based topological indices of BR (m, n) from the M-polynomial of BR (m, n).

In future, we will sketch and design some new chemical graphs/networks and compute their M-polynomial and examine their underlying topological properties.

https://doi.org/10.1155/2019/7297253

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the Key Project of Sichuan Provincial Department of Education (grant nos. 17ZA0079 and 18ZA0118); Soft Scientific Research Foundation of Sichuan Provincial Science and Technology Department (grant no. 2018ZR0265); and COMSATS Attock and National University of Sciences and Technology, Islamabad, Pakistan.

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Hong Yang, (1) A. Q. Baig, (2) W. Khalid, (3) Mohammad Reza Farahani [ID], (4) and Xiujun Zhang [ID] (1)

(1) Key Laboratory of Pattern Recognition and Intelligent Information Processing, Institutions of Higher Education of Sichuan Province, Chengdu University, Chengdu 610106, China

(2) Department of Mathematics, COMSATS Institute of Information Technology, Attock Campus, Islamabad, Pakistan

(3) Punjab College of Commerce and Science, Attock Campus, Lahore, Pakistan

(4) Department of Applied Mathematics, Iran University of Science and Technology (IUST) Narmak, 16844 Tehran, Iran

Correspondence should be addressed to Xiujun Zhang; woodszhang@cdu.edu.cn

Received 3 July 2019; Accepted 8 October 2019; Published 3 November 2019

Academic Editor: Robert Zalesny

Caption: Figure 1: Chemical graph of the benzene ring embedded in a P-type surface network in 2D.

Caption: Figure 2: The 3D plot of M-polynomial of the benzene ring embedded in the P-type surface network.

Caption: Figure 3: (a) The 2D plot of M-polynomial of the benzene ring embedded in the P-type surface network by fixing the parameter x. (b) The 2D plot of M-polynomial of the benzene ring embedded in the P-type surface network by fixing the parameter y.

Caption: Figure 4: (a) Plot of symmetric division, harmonic, inverse sum, and augmented Zagreb index for fix n parameter. (b) Plot of first Zagreb, second Zagreb, and modified Zagreb index for fix m parameter.

Caption: Figure 5: The 3D plot of RandiC index for BR(m, n).

Caption: Figure 6: The 3D plot of the inverse RandiC index for BR(m, n).

Caption: Figure 7: (a) The 2D plot of the Randic index for BR(m,n). (b) The 2D plot of the inverse Randic index for BR(m,n).

Caption: Figure 8: The 3D plot of the augmented Zagreb index for BR (m, n).
```Table 1: The relationship of topological indices with M-
polynomial.

Topological descriptor   Derivation from f(x,y)

[R.sub.[alpha]] (G)      ([D.sup.[alpha].sub.x][D.sup.[alpha].sub.y])
[alpha] [member of] R    (f(x, y))|[sub.x=y=1]
R[R.sub.[alpha]] (G),    ([S.sup.[alpha].sub.x][S.sup.[alpha].sub.y])
[alpha] [member of] R    (f(x, y))|[sub.x=y=1]
[M.sub.1](G)             ([D.sub.x] + [D.sub.y])(f(x, y))\|[sub.x=y=1]
[M.sub.2](G)             ([D.sub.x][D.sub.y])(f(x, y))|[sub.x=y=1]
[sup.m][M.sub.2](G)      ([S.sub.x][S.sub.y])(f(x, y))|[sub.x=y=1]
H(G)                     2[S.sub.x]J(f(x,y))|[sub.x=1]
SDD(G)                   ([D.sub.x][S.sub.y] + [S.sub.x][D.sub.y])
(f(x, y))|[sub.x=y=1]
I(G)                     [S.sub.x]J[D.sub.x][D.sub.y]
(f(x, y))|[sub.x=1]
A(G)                     [S.sup.3.sub.x][Q.sub.-2]J[D.sup.3.sub.x]
[D.sup.3.sub.y](f(x, y))|[sub.x=1]

Table 2: Vertex partition of the benzene ring embedded in the
P-type surface network based on degrees of each vertex.

[d.sub.v]         2              3

Frequency   8mn + 4m + 4n    16mn-4m-4n

Table 3: Edge partition of the benzene ring embedded in the P-type
surface network based on degrees of end vertices of each edge.

([d.sub.u], [d.sub.v])     (2,2)     (2,3)       (3, 3)

Frequency                 4m + 4n    16mn      16mn-6m-6n
```
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