Topological Study of Zeolite Socony Mobil-5 via Degree-Based Topological Indices

Graph theory is a subdivision of discrete mathematics. In graph theory, a graph is made up of vertices connected through edges. Topological indices are numerical parameters or descriptors of graph. Topological index tells the symmetry of compound and helps us to compare those mathematical values, with boiling point, melting point, density, viscosity, hydrophobic surface area, polarity, etc., of that compound. In the present research paper, degree-based topological indices of Zeolite Socony Mobil-5 are calculated. Names of those topological indices are Randić index, first Zagreb index, general sum connectivity index, hyper-Zagreb index, geometric index, ABC index, etc.


Introduction
In graph theory, the term graph was suggested in eighteenth century by Leonhard Euler (1702-1782). He was a Swiss mathematician. He manipulated graphs to solve Konigsberg bridge problem [1][2][3]. Chemical graph theory is a topological division of mathematical chemistry that practices graph theory to model chemical structures mathematically. It studies chemistry and graph theory to view the detailed physical and chemical properties of compounds. A graph G � (V, E) is comprised through a set of vertices V and an edges set E [4].
Topological indices study the properties of graphs that remain constant/unchanged after continuous change in structure. Topological indices explain formation and symmetry of chemical compounds numerically and then help in advancement of QSAR (qualitative structure activity relationship) and QSPR (quantitative structure property relationship). Both QSAR and QSPR are used to build a relation among molecular structure and mathematical tools. ese descriptors are helpful to correlate physio-chemical properties of compounds (enthalpy, boiling and melting point, strain energy, etc.) that is why these descriptors have a large number of applications in chemistry, biotechnology, nanotechnology, etc.
Topological indices are invariants of graph that is why topological indices are independent of pictorial representation of graph. In other words, it is a numerical value that describes the structure of chemical graph [5,6]. Among the three types of topological indices, degree-based indices have great importance. e need to define these indices is to explain physical properties of every chemical structure with a number. Continuous change in shape does not affect the value of topological index. Topological indices are useful in the study of QSAR and QSPR because topological indices show the physical properties and convert the chemical structure into a numerical value.
Distance-based topological indices deal with distances of graph, degree-based topological indices use the concept of degree, and counting-based topological index depends upon counting the edges. Randic explained some characteristics of a topological index. Some of them are explained here.
A topological index should (i) have architectural interpretation (ii) be well-defined (iii) be related with at least one physio-chemical property of compound (iv) be uncomplicated (v) display an appropriate size dependence (vi) modify with modification in structure (vii) locally defined (viii) have related with other indices Topological indices show translations of chemical compounds into distinctive structural descriptors as a numerical value that can be used by QSAR [7,8]. Topological indices are awfully beneficial in describing the properties of given compound. Chemists can use these indices to correlate considerable range of characteristics. Medicine industry is developing new drug designs that are useful for humans, plants, and animals. Many graph theoretical techniques have been established for forecasting of medicinal, environmental, and physio-chemical properties of compounds. It is not astonishing to see such a great victory of graph theory and topological indices in analyzing biological and physical characteristics of chemical compounds.
1.1. ZSM. Zeolites (alumino silicate) are tetrahedrally-linked structures based on silicate and aluminate tetrahedral. Structural chemistry deals with the framework of zeolites; it also works out on the arrangement of cations and other molecules in pore spaces. It belongs to a pentasil class of zeolite. It consists of silica (Si) and alumina (Al). It is named as ZSM-5 due to pore diameter of five angstrom; also, it has Si/Al ratio of five [9]. Size of the molecule depends on the type of structure. It is a crystalline powder. Geometry of pores can be connected in channels in one, two, or three dimensions.

Motivations.
e structure of ZSM-5 has great importance in the field of chemistry, petroleum, and medicine industry. ZSM-5 is useful because of its stability, favorable selectivity, metal tolerance, and flexibility. It is also useful for the treatment of fertilizers. It helps to separate oxygen and nitrogen in the air. is unique structure is useful in petroleum industry as a catalyst. It is generally used in the conversion of methanol to gasoline as well as refining of oil. rough dehydration, it changes alcohol into petrol. Efficiency of LPG can also be increased through ZSM-5 catalyst. It keeps unusual hydrophobicity that is useful to separate hydrocarbons from polar compounds. Basic reason of calculation of topological indices is the industrial uses of ZSM-5 structure.
is index was first presented by Li and Zhao. Its mathematical form is defined in [10][11][12] as follows: First and Second Zagreb Index. ere are two Zagreb groups of indices, denoted by M 1 and M 2 [13][14][15]. Both of these indices are explained in 1970s by Gutman and Tranjistic.
(2) First Zagreb Index. It is defined in [16,17]: (2) (3) Second Zagreb Index. It is defined in [11,16]: Multiple and polynomial Zagreb indices: In 2012, new kinds of Zagreb indices were introduced by Ghorbani and Azimi, named as first and second multiple Zagreb indices represented as PM 1 (G) and PM 2 (G) [11,15,18]. e polynomials are used to find the Zagreb index. First and second Zagreb polynomial indices are written as M 1 (G, j) and M 2 (G, j). (4) First and second multiple Zagreb indices: (5) First and second polynomial Zagreb indices: (6) Hyper-Zagreb Index. Modified Zagreb index is called hyper-Zagreb and that was introduced in 2013 by Shirdil, Rezapour, and Sayadi [19][20][21], mathematically written as (7) Second modified Zagreb index: (8) Reduced second Zagreb index. is index was proposed by Furtula and it is defined as (9) Atom Bond Connectivity Index. It was written in 1998 by Ernesto Estrada and Torres [15,[22][23][24]. It is used to model thermodynamic characteristics of organic compounds (especially alkanes). Mathematically, (11) General Randić Connectivity Index. First degreebased TI was proposed in 1975 by Millan Randić. At that time, it was called as branching index [8,17,18] and used to measure the branching of hydrocarbons. In 1998, Eddrös and Bollobás wrote the general term of this index by changing the factor (−1/2) with αεI R [25]. It is defined as the total sum of weights (d(p)d(q)) α of all the edges pq, d(p) is the degree of p, d(q) is the degree of q, and α ε I R.
(12) Randić index: is index can also be called as first genuine degreebased topological index [15,23]. Randić index is defined as (14) is index was first studied by Favaron, Mahéo, and Saclé [26]. e index is helpful in modeling of boiling points of hydrocarbons. It is defined as It is the analogue of reciprocal Randić index [26,27]. It is defined as follows: (15) Geometric Arithmetic Index. GA index was proposed by Vukicevic̀and Furtula [6,14,15]; it is stated as (17) Forgotten Index. is index was given by Gutman and Furtula in 2015 [16,28,29]. It is denoted by F (G) or F index: (18) General Sum Connectivity Index. e index was proposed by Zhou and Trinajstić [15,23,30]. Mathematically, where α ε I R. (19) Symmetric Division Index. In 2010, Vukicevic̀and Furtula proposed this useful index denoted by SD (G) [28,31,32]: (20) Harmonic Index. Siemion Fajtlowicz wrote a computer program that works for the automatic generation of conjectures in graph theory [11,15]. He also examined the relationship between graph invariants; while doing this work, he found a vertex degree-based quantity. Later on, (in 2012) Zhang rediscovered that unknown quantity and named it as harmonic index. It is written as

Topological Indices of ZSM-5 Graphs
Topological indices remain constant for a given compound; they do not depend on the direction or position of graph. We can predict many physical properties of compounds such as solubility, soil sorption, boiling and melting properties, biodegradability, toxicity, vaporization, and thermodynamic properties.

Description of ZSM-5 Graph.
e graph of ZSM-5 is given in Figure 1 and it is represented by G * .
By using the definition of M 1 (G * , j) in equation (6): From (3), we have which completes the proof.

Theorem 4. First and second multiple Zagreb index of G * of ZSM-5 is given as
Proof. E(G * ) is classified into 3 edge classes based on the degree of end vertices. E 1 (G * ) has 4p edges pq, where where d p � 3, d q � 3. Also consider |E 1 (G * )| � e 2,2 , |E 2 (G * )| � e 2,3 , and |E 3 (G * )| � e 3,3 . We define PM 1 (G * ) as (4): Now, we define PM 2 (G * ) as (5): Proof. E(G * ) is divided into 3 edge divisions based on the degree of end vertices. E 1 (G) holds 4p edges pq, where Since, we have (8), which completes our proof. □ Theorem 6. G * is the graph of ZSM-5. e second modified Zagreb index is given as Proof. Consider G * to be a graph of ZSM-5. E(G * ) is divided into 3 sets based on the degree of end vertices.
Proof. G * is our graph of ZSM-5. E(G * ) is divided into three edge groups.
We define this index in equation (16): Theorem 13. Consider G * to be the graph of ZSM-5. Geometric arithmetic index is described as follows: Proof. e graph G * of zeolite encounters 36pq + 2p − 2q edges and 24pq + 4p vertices. e grouping of the vertices is given as follows: e vertices of degree two are 8p + 4q and of degree three are 24pq − 4p − 4q. Cardinality of E of G * is 36pq + 2p − 2q. e arc group E(G * ) cleaves in 3 disjoint arc groups that rely on the degrees of the end vertices, such as E 1 (G * ) has 4p lines pq, where d p � d q � 2. E 2 (G * ) has 8p + 8q lines pq, where d p � 2 and d q � 3.
research contains the results theoretically not experimentally.

Data Availability
No data were used in this study.

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