M-Polynomials and Associated Topological Indices of Sodalite Materials

Natural zeolites are commonly described as macromolecular sieves. Zeolite networks are very trendy chemical networks due to their low-cost implementation. Sodalite network is one of the most studied types of zeolite networks. It helps in the removal of greenhouse gases. To study this rich network, we use an authentic mathematical tool known as M-polynomials of the topological index and show some physical and chemical properties in numerical form, and to understand the structure deeply, we compare different legitimate M-polynomials of topological indices, concluding in the form of graphical comparisons.


Introduction
Macromolecular sieves, commonly known as natural zeolites, are becoming more investigated and researched [1][2][3]. e custom-made molecular capturing of zeolite makes it significant and very applicable. Due to their lower costs, natural zeolites are considered as useful in bulk mineral applications [4]. According to the complexity and varying sizes, zeolites are characterized into different groups. For example, tschernichite, stilbite, mordenite, chabazite, faujasite, sodalite and Linde type A are most useful in terms of commercial applications, molecular sieves, or natural zeolites [5]. Among all the zeolite structures stated above, synthetic compounds and mineral's crystal structure with the frameworks of sodalite structure are the enormously studied compounds [6,7]. Some physical and chemical properties of sodalite and zeolite type structures in terms of topological indices can be found in [8][9][10].
e sodalites with the highest thermodynamic stability are considered as one of the top ranked structures among all the zeolites [11,12]. e sodalites are also considerable due to their crystallographic standpoints [13]. e sodalite structure's unit cell is built up by two cages. e six and four membered rings represent the basic topology of each cage. ese rings are also shared by two parallel cages. To trap the molecules, in the sodalite, cavities are built up by tailor-made composition. Without damaging the crystal structures, tap molecules like water and CO 2 have ability of desorbing and adsorbing molecules. As a result, for the removal of greenhouse gases and water, zeolites can be used [14]. For further studies related to zeolites, sodalite, and its useful applications, we refer the readers to [15,16].
As we know that the study of M-polynomial is a very useful combination of numerical descriptors and algebraic theory, for the study of chemical networks, one can find an intense study on this topic; very particular and selected articles are cited here. Nanotube-related networks or structures were studied in [17][18][19], M-polynomials for different generalized families of graphs were discussed in [20][21][22], convex polytopes were studied in [23], chemical benzenoid structures were discussed in [24], a fine relation of M-polynomial with the probabilistic theory was discussed in [25], the study of this topic on metal organic structure was detailed in [26,27], and some computer-related networks in terms of M-polynomials can be found in [28,29]; there are many types of polynomials, and one of the varieties can be found in [30].
Topological index is a function in numerical form and describes the biological, chemical, and physical properties of given molecular graph in domain by a systematic way. Few very interesting and highly recommended topological indices are given in the following definitions along with their M-polynomials. e researchers in [31] studied polyhex and prism structures in terms of some M-polynomials. e titania nanotubes were studied in terms of some M-polynomials in [32]. e structure of nanostar dendrimers and polyhex nanotubes was studied in [33]. For few interesting studies of chemical structures and their topological properties and different shapes, we refer the readers to [34,35]. ere are many ways and types of computing topological indices for different structures, for example, Shabbir et al. [36] computed the edge-based interpolation with the degree of vertices and topological indices, Nadeem et al. and Hong et al. [37,38] computed the topological indices for the metal organic structure; line graph of some families of graphs and their topological indices are resulted in [39,40], some ev-and ve-degree-based descriptors are studied in [41,42], and the topological descriptors for some computer-related structures are discussed in [43,44]. Some neural networks and their topological descriptors were discussed in [45,46].
To compute our major results, we need to define some basics; for example, let χ be a graph with vertex set χ and edge set E(χ); their cardinalities are represented by n χ and m χ , respectively. Degree of a vertex (say λ a ) is the cardinality of incident edges with a vertex. ere is another way to define edge set, which is called edge partition, usually written as (ξ a , ξ b ), where ξ a , ξ b are the degrees of vertices a and b, respectively, and a, b create an edge say ab.
Following are some useful definitions of M-polynomials which are helpful to find our main results and create formulas for M-polynomials of different topological indices. Definition 1. In 1988, Hosoya polynomial was introduced in [47]; with a relevance to Hosoya polynomial, in 2015, modified polynomial or usually named as M-polynomial was given in [48]. is particular type of polynomial is closely related to degree-based topological indices. Using a particular type of format, one can get the topological indices from M-polynomials of a graph. is M-polynomial can be defined as where m i,j (χ) is the number of edges of graph χ such that i ≤ j.
Definition 2. e first and second Zagreb indices were introduced in 1972 by Gutman [49,50], and their M-polynomials are defined as [48] (2) e second modified Zagreb index was given in [51], and its M-polynomial is defined as [48] Definition 4. In 1988, Bollobas and Erdos and Amic et al. [52,53] proposed the general Randić index independently. Following are general Randić index, general inverse Randić index, and their M-polynomials [48]: where In this research work, we studied sodalite's three-dimensional structure which is an important compound of zeolite network, in terms of different M-polynomials of topological indices defined above. Moreover, we explain the structure more precisely by taking different numerical examples and doing comparative study between computed M-polynomials, and at the end, we conclude the study in the form of graphical representations of derived numerical examples.

Results of M-Polynomials on
Sodalite Network e graph shown in Figure 1 is the three-dimensional sodalite network S p,q,r . It contains 4(pq + pr + qr) + 12pqr total number of vertices and 4(pq + pr + qr) + 5 24pqr total number of edges. As shown in Figure 1, it contains two types of vertices and three types of edges, which are useful for our main results. Figure 1 is studied in [14] in detail to explore some useful structures related to sodalite and zeolites. Now, the first result is the main and general M-polynomial of sodalite network S p,q,r . Theorem 1. Let S p,q,r be a sodalite material, with p, q, r ≥ 1, whose structure can be seen in Figure 1. en, its M-polynomial is M S p,q,r , x, y � 24pqrx 4 y 4 + 4(pq + pr + qr) Proof. From Figure 1, which is the construction of sodalite materials, we can observe that there are two vertex set partitions: with |∨ 3 | � 8(pq + pr + qr) and |∨ 4 | � 12pqr − 4(pq + pr + qr). Also, there are three edge partitions based on degree of end vertices of each edge that is defined as e cardinality of these edge partitions is □ Lemma 1. Let S 4,4,4 be a sodalite material, with p, q, r ≥ 1. en, the differential operator is Proof. Differentiating equation (8) S y � 6pqrx 4 y 4 + 4(pq + pr + qr) Proof. As we know from equation (6), S x � x 0 (M (S p,q,r ; t, y)/t)dt and using the general M-polynomial for the S p,q,r from equation (8) in it, after simplification, we obtain S x . Similarly, from equation (7), S y � y 0 (M(S p,q,r ; x, t)/t)dt and using the general M-polynomial for the S p,q,r from equation (8) in it, after simplification, we obtain S y . Now, we will use these differential and integral operators from Lemmas 1 and 2 to obtain our main results. In the following theorem, we determined the M-polynomial of first and second Zagreb indices. Theorem 2. Let S p,q,r be a sodalite material, with p, q, r ≥ 1, and P M 1 be the M-polynomial of first Zagreb index. en, P M 1 (S p,q,r ) is Mathematical Problems in Engineering 3 Proof. According to Definition 2, the formula for the M-polynomial of first Zagreb index for S p,q,r is defined as P M 1 (S p,q,r ) � (D x + D y )(M(S p,q,r ; x, y)), by using the differential operators that are defined in Lemma 1 for S p,q,r . After some algebraic simplifications, we will get the M-polynomial of first Zagreb index for S p,q,r as follows: P M 1 S p,q,r � 192pqrx 4 y 4 + 8(pq + pr + qr) 6x 3 y 3 + 7x 3 y 4 − 12x 4 y 4 + 8(p + q + r) 3x 3 y 3 − 7x 3 y 4 + 4x 4 y 4 .
Similarly, by using the values of differential operators that are defined in Lemma 1 for S p,q,r in P M 2 (S p,q,r ) � (D x D y )(M(S p,q,r ; x, y)) and after simplification, we will get the M-polynomial of second Zagreb index for S p,q,r : P M 2 S p,q,r � 384pqrx 4 y 4 + 24(pq + pr + qr) 3x 3 y 3 + 4x 3 y 4 − 8x 4 y 4 + 4(p + q + r) 9x 3 y 3 − 24x 3 y 4 + 16x 4 y 4 . (16) □ Theorem 3. Let S p,q,r be a sodalite material, with p, q, r ≥ 1, and Pm M 2 be the M-polynomial of second modified Zagreb index. en, Pm M 2 (S p,q,r ) is Proof. According to Definition 3, the formula for the M-polynomial of second modified Zagreb index for S p,q,r is defined as PM M 2 (S p,q,r ) � (S x S y )(M(S p,q,r ; x, y)), by using the integral operators that are defined in Lemma 2 for S p,q,r . After some algebraic simplifications, we will get the M-polynomial of second modified Zagreb index for S p,q,r as Proof. According to Definition 4, the formula for the M-polynomial of general Randić index for S p,q,r is defined as q,r ; x, y)), by using the differential operators that are defined in Lemma 1 for S p,q,r . After some algebraic simplifications, we will get the M-polynomial of general Randić index for S p,q,r as Proof. According to Definition 4, the formula for the M-polynomial of general inverse Randić index for S p,q,r is defined as P IR α (S p,q,r ) � (S α x S α y )(M(S p,q,r ; x, y)), by using the integral operators that are defined in Lemma 2 for S p,q,r . After some algebraic simplifications, we will get the M-polynomial of general inverse Randić index for S p,q,r as □ P IR α S p,q,r � 24 16 α pqrx 4 y 4 + 4(pq + pr + qr)

Results of Topological Indices from M-Polynomials on Sodalite Materials
Proof. By putting the value of general M-polynomial which is defined in equation (8) in equation (6), after simplification, we get S x . Similarly, by putting the value of general M-polynomial in equation (7), after simplification, we get S y . After simplification, we obtain S y . After that, in both equations, substituting (x, y) � (1, 1) and simplifying, we can easily obtain [S x ] (x,y)�(1,1) and [S y ] (x,y)�(1,1) .

□
Now, we will use these absolute values from the differential and integral operators which are from Lemmas 3 and 4 to obtain our main results. In the following theorem, we determined the topological indices from M-polynomial of first and second Zagreb indices. Theorem 6. Let S p,q,r be a sodalite material, with p, q, r ≥ 1, and M 1 , M 2 be the first and second Zagreb indices. en, M 1 (S p,q,r ) and M 2 (S p,q,r ) are M 1 S p,q,r � 192pqr + 8(pq + pr + qr), M 2 S p,q,r � 384pqr − 24(pq + pr + qr) + 4(p + q + r). (25) Proof. According to Definition 2, the formula for the M-polynomial of first Zagreb index for S p,q,r is defined as P M 1 (S p,q,r ) � (D x + D y )(M(S p,q,r ; x, y)), by using the differential operators that are defined in Lemma 1 for S p,q,r . After some algebraic simplifications, we will get the M-polynomial of first Zagreb index for S p,q,r . Now using Lemma 3 and evaluating it in Definition 2 and after some algebraic simplifications, we will get the first Zagreb index for S p,q,r as Similarly, by using the absolute values of differential operators that are defined in Lemma 3 for S p,q,r in P M 2 (S p,q,r ) � (D x D y )(M(S p,q,r ; x, y)) and after simplification, we will get the second Zagreb index for S p,q,r : M 2 S p,q,r � 384pqr − 24(pq + pr + qr) + 4(p + q + r).
Proof. According to Definition 3, the formula for the M-polynomial of second modified Zagreb index for S p,q,r is defined as m M 2 (S p,q,r ) � (S x S y )(M(S p,q,r ; x, y)), by using the integral operators that are defined in Lemma 2 for S p,q,r . After some algebraic simplifications, we will get the M-polynomial of second modified Zagreb index for S p,q,r . Now using Lemma 4 and evaluating it in Definition 3 and after some algebraic simplifications, we will get the modified Zagreb index for S p,q,r as Proof. According to Definition 4, the formula for the M-polynomial of general Randić index for S p,q,r is defined as R α (S p,q,r ) � (D α x D α y )(M(S p,q,r ; x, y)), by using the differential operators that are defined in Lemma 1 for S p,q,r . After some algebraic simplifications, we will get the M-polynomial of general Randić index for S p,q,r . Now using Lemma 3 and evaluating it in Definition 4 and after some algebraic simplifications, we will get the general Randić index for S p,q,r as R α S p,q,r � 16 α × 24pqr + 4(pq + pr + qr) □ Theorem 9. Let S p,q,r be a sodalite material, with p, q, r ≥ 1, and R α be the general inverse Randić index; then, IR α (S p,q,r ) is IR α S p,q,r � 24 16 α pqr + 4(pq + pr + qr) Proof. According to Definition 4, the formula for the M-polynomial of general inverse Randić index for S p,q,r is defined as IR α (S p,q,r ) � (S α x S α y )(M(S p,q,r ; x, y)), by using the       integral operators that are defined in Lemma 2 for S p,q,r . After some algebraic simplifications, we will get the M-polynomial of general inverse Randić index for S p,q,r . Now using Lemma 4 and evaluating it in Definition 4 and after some algebraic simplifications, we will get the general inverse Randić index for S p,q,r as IR α S p,q,r � 24 16 α pqr + 4(pq + pr + qr)

Conclusion and Discussion
In this study, we discussed the most important structure from zeolite structures which is known as sodalite material network and we symbolized the fetched graph of this network by S p,q,r , defined in Figure 1. e moving parameters for this structure are p, q, r ≥ 1. We study this structure for all these parameters in terms of different M-polynomials and topological indices derived from resulting M-polynomials. Figures 2-7 show the three-dimensional plot of resulting M-polynomials of first, second, and modified Zagreb M-polynomials and general and general inverse Randić M-polynomials, respectively. Figure 8 shows the contour plot of topological index derived from M-polynomials while Figure 9 shows the plot of all the topological indices derived from M-polynomials.

Data Availability
No data were used to support this study.

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