M-Polynomials and Degree-Based Topological Indices of the Molecule Copper(I) Oxide

Topological indices are numerical parameters used to study the physical and chemical residences of compounds. Degree-based topological indices have been studied extensively and can be correlated with many properties of the understudy compounds. In the factors of degree-based topological indices, M-polynomial played an important role. In this paper, we derived closed formulas for some well-known degree-based topological indices like ﬁrst and second Zagreb indices, the modiﬁed Zagreb index, the symmetric division index, the harmonic index, the Randi´c index and inverse Randi´c index, and the augmented Zagreb index using calculus.


Application Background.
A graph that represents the construction of a molecule and also their connectivity is known as a molecular graph, and such a representation is generally known as topological representations of molecule. Molecular graphs are normally characterized by means of exclusive topological basis for parallel of chemicals shape of a molecule with organic, chemical, or bodily homes. Study of graph has some programs of various topological indices in quantitative structure-activity relationship (QSAR) and quantitative structure-property relationship (QSPR), digital screenings, and computational drug designing citations as shown in [1,2]. us far, several exclusive topological indices have been established, and maximum of them are most effective graph descriptors in [3,4]; apart, some indices have proven their parallel with organic, chemical, or physical residences of secure molecules in [5][6][7][8][9][10][11][12][13][14][15][16][17].
In the field of mathematics, any graph has vertices and edges that are represented by the atoms and chemical bonds.
Graph that represents the construction of molecules and their connectivity is known as a molecular graph, and such representation is usually referred as topological representation of molecules. ere are some significant topological indices like distance-based topological indices, degree-based topological indices, and primarily based topological indices. Among these works, distance primarily based topological indices unit works out a crucial task in a chemical graph started, specifically in chemistry [18,19]. Many fields have many features that can be solved with the help of graphs. In the physiochemical compounds or network systems, we have a tendency to abstractly outline exclusive ideas in modeling of mathematics. We have a tendency to refer to as the distinctive names, such as Randić index and national capital index.
A topological index is a numerical parameter of a graph and describes its topology. It describes the molecular shape numerically and is applied within the advancement of qualitative structure-activity relationships (QSARs). e following are the 3 types of topological indices: (1) Degree-based.
Degree-based topological indices were studied extensively and may be correlated with many residences of the understudy molecular compounds. ere is a strong relationship among distance-based and degree-based topological indices in [20]. Most commonly known invariants of such kinds are degree-based topological indices. ese are actually the numerical values that correlate the structure with various physical properties, chemical reactivities, and biological activities. Topological indices are sincerely the numerical values that relate the shape to one of a kind of physical residences, artificial reactivity, and natural biological activities [21,22].
Loads of research has been executed inside the course of M-polynomial, as in the case of Munir et al., processed M-polynomial and related lists of triangular boron nanotubes in [6], polyhex nanotubes in [23], nanostar dendrimers in [4], and titania nanotubes in [5]. M-Polynomials and topological lists of V-phenylenic nanotubes and nanotori. In this paper, the objective is to process the M-polynomial of the crystallographic realistic structure of the atom copper(I) oxide (Cu 2 O) [8,24].

Crystallographic Structure of Cu 2 O(m; n).
Copper oxide is a p-type semiconductor and inorganic compound. Copper oxide is a chemical element with formula Cu 2 O(m; n). Cu 2 O(m; n) is a certainly happening reddish coral that is particularly used in chemical sensors and solar orientated cells in [8,24]. It has many advantages such as photochemical effects, stability, pigment, a fungicide, nontoxicity, and low cost. It has potential applications in new energy, sensing, sterilization, and other fields. It has narrow band gap and is easily excited by visible light.
Cu 2 O(m; n) is additionally responsible for the pink shading in Benedict's test and is the essential cause to select Cu 2 O (see Figures 1 and 2). e promising projects of Cu 2 O(m; n) are mainly on chemical sensors, sunlight-based cells, photocatalysis, lithium particle batteries, and catalysis. Here, we have taken into consideration a monolayer of Cu 2 O(m; n) for satisfaction. To ultimate the basis for Cu 2 O(m; n), we pick out the setting of this graph as Cu 2 O(m; n) be the chemical graph of copper(I) oxide with (m; n) unit cells within the aircraft.  [3,8]. Within the factors of degreebased topological indices, we compete necessary role of M-polynomial. Readers can refer to [9][10][11][12][13][14][15][16][17][27][28][29][30][31][32][33][34][35]. It is the foremost general progressive polynomial and an additionally closed formula alongside 10 distance-based topological indices is given by M-polynomial. It is explained as

Degree-Based Topological Indices.
Any purpose on a graph which does not build upon numbering of its vertices is molecular descriptor. is is also called as topological index. Topological indices are most useful in the field of isomeric discrimination, chemical validation, QSAR, QSPR, and a pharmaceutical drug form. Topological indices are accessed from the system of molecule. ere are some important degree-based topological indices defined, and the first Zagreb index was introduced by Gutman and Trinajstić as follows: Gutman and Trinajstić proposed the second Zagreb index in 1972, which is stated as e second modified Zagreb index is defined as General 1 st and 2 nd multiplicative Zagreb indices are introduced by Kulli, Stone, Wang, and Wei and are stated as e general 1 st and 2 nd Zagreb indices proposed by Kulli, Stone, Wang, and Wei are stated as In 1987, Fajtlowicz in [36] proposed the harmonic index and stated e inverse sum index is defined: Symmetric division index is described as SU and XU recognized general Randić index or general multiplicative Randić index stated as follows (Table 1):   2 Journal of Chemistry such that us, the M-polynomial of □ Theorem 2. Crystallographic structure of the graph of copper(I) oxide G ≈ Cu 2 0[m; n], where n; m ≥ 1. We have We have to find  Journal of Chemistry Multiply a on both sides: Similarly, Now, the first Zagreb index is After solving, the result is e 3D plot of first Zagreb index is given in Figure 3 We have to find D b D a ; first, we take D a : Now, take D b : e second Zagreb index is After solving, the result is e 3D plot of second Zagreb index is given in Figure 4

Topological Indices f(t, s) M(G; t, s) First Zagreb index t + s M 1 (G; t, s) � (D t + D s )M(G; t, s)| t�s�1 Second Zagreb index
Ts Now, we have to find S a S b ; first, we find S a : Taking integration on both sides, Now, take S b and then
e 3D plot of Randić index is given in Figure 6    Journal of Chemistry Now, we have to find S a S b , and first, we find S a : Similarly, take S b : (43) Take α on the above equation: e inverse Randić is

Journal of Chemistry
e 3D plot of inverse Randić index is represented in Figure 7 (f or u � 1 left, v � 1 middle, and w � 1 right), and we see the dependent variables of the inverse Randić index on the involved parameters. Proof. suppose First, we have to find S b : Now, take D a : Similarly, Take D b : Now, the symmetric division index is Put the values After the calculation, the result is e 3D plot of symmetric division index is given in Figure 8 Proof. suppose First, we have to find J f (a;b): Jf(a, b) � Jf(a, a) � 4(m + n − 1)a 3 + 4(mn − m − n + 1)a 4 + 8mna 6 . (56) Take S a : 8 Journal of Chemistry (57) e harmonic index is Now, the result is e 3D plot of harmonic index is given in Figure 9 (f or u � 1 left, v � 1 middle, and w � 1 right), and we see the dependent variables of the harmonic index on the involved parameters.   e 3D plot of inverse sum index is given in Figure 10 (f or u � 1 left, v � 1 middle, and w � 1 right), and we see the dependent variables of the inverse sum index on the involved parameters.

Data Availability
No data were used in this study.

Disclosure
All authors have not any fund, grant, and sponsor for supporting publication charges.