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

Key Laboratory of Pattern Recognition and Intelligent Information Processing, Institutions of Higher Education of Sichuan Province, Chengdu University, Chengdu 610106, China Department of Mathematics, COMSATS Institute of Information Technology, Attock Campus, Islamabad, Pakistan Punjab College of Commerce and Science, Attock Campus, Lahore, Pakistan Department of Applied Mathematics, Iran University of Science and Technology (IUST) Narmak, 16844 Tehran, Iran


Introduction
e chemical compounds can be represented by using the mathematical tools of graph theory.e 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.
e topological descriptors are the numerical parameters of the chemical graph which characterize its topology and are usually graph invariants.ey 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.
e properties like boiling point, strain energy, viscosity, fracture toughness, and heat of formation are connected to the chemical structure under study.is fact plays a major role in the field of chemical graph theory .e computation of the general polynomial is formed whose derivatives or integrals or composition of both are evaluated at some particular point.en, 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][24][25][26].e Hosoya polynomial is a polynomial whose derivatives evaluated at 1 give Wiener and hyper Wiener index [27].e Hosoya polynomial and Zagreb polynomials are considered to be of the general form in the determination of distance-based and degree-based indices, respectively.e 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.e 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].
e M-polynomial of a graph G is formulated as [28] e path number was the first distance-based topological index defined by Wiener [31] in 1947. is index is now called as the Wiener index.It has many famous mathematical and chemical applications [31,32].Later on, Milan Randić proposed and formulated the Randić index of a graph GR − (1/2) (G).
e general Randić index was proposed and defined independently by Bollobás et al. [33] and Amić 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][36][37][38].e general Randić index and inverse Randić index are formulated as ( e first and second Zagreb indices are introduced by Gutman and Trinajstić [25,39,40].Both first and second Zagreb indices and the second modified index are formulated as 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 e other version of the Randic index is the harmonic index [43] and is defined as e inverse sum index is formulated as [44] e augmented Zagreb index gives best approximation of heat of formation of alkanes [45,46].It is formulated as [47] Let M(G; x, y) � f(x, y), and then Table 1 relates above described topological indices with M-polynomial [28], where

Main Results and Discussion
OḰeeffe 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. e P-type surface is coordinated to the Cartesian arranges in the Euclidean space.e reader can discover more about this intermittent surface in [49,50]. is structure was required to be combined as 3D carbon solids and no such combination was accounted before.is 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 60 , fullerenes have also been studied in depth, see detail in [51,52].e molecular graph of the benzene ring embedded in the P-type surface network is depicted in Figure 1. e cardinality of vertices and edges of the given molecular graph are 24mn and 32mn − 2m − 2n, respectively.e 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  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.

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 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 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.
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.
such that Now, applying the definition of M-polynomial to the graph of the benzene ring embedded in the P-type network, we have Table 2: Vertex partition of the benzene ring embedded in the Ptype surface network based on degrees of each vertex.
Table 3: Edge partition of the benzene ring embedded in the P-type surface network based on degrees of end vertices of each edge.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. is is plotted by using Maple 13. e 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.
e 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.e same behavior occurs for negative values of y.

Topological Indices Derived from M-Polynomial of BR (m,n)
e 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:
( 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: In order to prove the above nine results, we use the following formulas: Now, we have the following computations:  Journal of Chemistry Now, by using all the aforementioned values from equations ( 19)- (30) in Table 1, the topological indices defined in Table 1 are obtained. ( )) e symmetric division, harmonic, inverse sum, and augmented Zagreb indices are plotted by using Maple 13. e graphical representation depicts different behavior of indices by changing the parameters m and n. e blue, green, red, and black colors show the symmetric division, harmonic, inverse sum, and augmented Zagreb indices, respectively, as depicted in Figure 4    In future, we will sketch and design some new chemical graphs/networks and compute their M-polynomial and examine their underlying topological properties.

Figure 1 :
Figure 1: Chemical graph of the benzene ring embedded in a Ptype surface network in 2D.

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

Figure 3 :Figure 4 :Figure 5 Figure 6 :
Figure 3: (a) e 2D plot of M-polynomial of the benzene ring embedded in the P-type surface network by fixing the parameter x.(b) e 2D plot of M-polynomial of the benzene ring embedded in the P-type surface network by fixing the parameter y.

□ 5 .
ConclusionsWe 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.e 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).

Figure 7 :Figure 8 :
Figure 7: (a) e 2D plot of the Randić index for BR(m, n).(b) e 2D plot of the inverse Randić index for BR(m, n).