Valency-based molecular descriptors of Bakelite network B N m n $B\text N_{m}^{n}$

Abstract Bakelite network BNmn $BN_{m}^{n}$is a molecular graph of bakelite, a pioneering and revolutionary synthetic polymer (Thermosetting Plastic) and regarded as the material of a thousand uses. In this paper, we aim to compute various degree-based topological indices of a molecular graph of bakelite network BNmn $BN_{m}^{n}$. These molecular descriptors play a fundamental role in QSPR/QSAR studies in describing the chemical and physical properties of Bakelite network BNmn $BN_{m}^{n}$. We computed atom-bond connectivity ABC its fourth version ABC4 geometric arithmetic GA its fifth version GA5 Narumi-Katayama, sum-connectivity and Sanskruti indices, first, second, modified and augmented Zagreb indices, inverse and general Randic’ indices, symmetric division, harmonic and inverse sum indices of BNmn $BN_{m}^{n}$.


Introduction
In Chemical graph theory, chemical compounds are represented by graphs and mathematical techniques are used to solve problems arising in chemistry. A molecular graph is a simple connected graph in which atoms are taken as vertices and chemical bonds are taken as the edges of the graph. Topological indices are numerical numbers associated with molecular graphs of chemical compounds and help us to predict the properties of chemical compounds without performing experiments and save money and time [1,2]. Some applications related to topological indices of molecular graphs are given in [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17].
Throughout this paper, Γ is an ordered pair Γ=(V,E) where V is the set of vertices, and E is the set of edges. The order and size of a graph Γ is the cardinality of vertex set V and edge set E denoted by |V| and |E| respectively. Two vertices u and v are adjacent if e=uv is an edge in graph and e is said to be incident with u and v The valency of a vertex v∈Γ is the number of incident edges with v and is denoted by d v . The minimum and maximum valency of Γ is denoted by δ and Δ, respectively. The first topological index was introduced by Harold Wiener [18], when he was investigating the boiling point of alkane. Initially, this index was named as path number and now it is known as the Wiener index/number [19]. In 1975, after Wiener index, Milan Randic [20], introduced a prominent index known as Randić index and is defined as: This simple graph invariant has found many applications in chemistry. Böllöbás and Erdös [21], extended the idea of Randić and proposed the general Randic index: and inverse Randić index: The additive version of Randic index is known as the sumconnectivity index (SCI) [22] and it gives a high correlation coefficient (0.99) for alkanes. The formula for SCI is given by: Ivan Gutman and Trinajstić [23] proposed two topological indices named as the first and the second Zagreb indices which were denoted by M 1 and M 2 and got an immense attraction of researchers due to the applications in chemistry. First and second and Zagreb indices are defined as follows: The modified second Zagreb index is defined as: In 1998, Estrada et al. [24] established an important index named as the ABC(Γ) index, which is a good model to test stability of linear and branched alkanes and is defined below: In 2009, et al. [25] introduced another remarkable index known as the geometric arithmetic index GA(Γ) and is given by the formula: Three prominent and recently developed indices, denoted by ABC 4 (Γ), GA 5 (Γ) and S(Γ) are proposed by Ghorbani et al. [26], Graovac et al. [27], and Hosamani [28], respectively. These indices are different as compared to other vertex valency-based indices in the sense that they require partitioning of the edges of the network on the basis of neighbors' valency-sum of end vertices for every edge in an interconnected network. Their formulae are given as: where and . Motivated by ABC index, Furtula et al. [29] offered the so called Augmented Zagreb index (AZI) which proved to have a better correlation coefficient as compared to ABC [30] and is defined as: A few more topological indices that have key importance are defined below which include harmonic index (HI), inverse sum index (ISI), and symmetric division index (SDD): To compute topological indices, a huge mount of calculution is required. In 2015, Deutsch and [31] introduced M-polynomial to reduce this calculution where m ij (Γ) represent number of edges vw∈E(Γ) such that {d v (Γ), d w (Γ)}={i,j}. The other polynomials are Hosoya polynomial [32], matching polynomial [33], the Zhang-Zhang polynomial [34], the Schultz polynomial [35] and the Tutte polynomial [36]. Some promising topological indices are worked out in [31] with the help of M-polynomial and are depicted in the table 1 below.
In this paper, we computed all the above defined topological indices for Bakelite Network .  [37]. This revolutionary thermosetting plastic has enormous engineering applications and is used as an electric insulator, propellers, automotive accessories, and medical as well as sports equipment [38,39].

Methodology
To compute our results, we count the number of vertices and number of edges of the graph of the Bakelite Network . The total number of vertices and edges in the graph of Bakelite network are 8mn-n+m and 10mn-2n , respectively. In the graph of , figure 3, n represent number of hexagons in one row and m represent number of hexagons in one column. The vertex set and edge set partition of any graph Γ can generally be defined as [3]    We divide the edge set of graph of Bakelite Network into classes depending on the degrees of end vertices of each edge. Then using this edge partition, we compute our desired results.

Computational Results
In this section, we compute various topological indices of bakelite network In subsequent theorems, we provide the M-polynomial of a bakelite network which will be a core component of the forthcoming theorems. such that Now, the fourth version of ABC can be calculated as follows:

Conclusions
QSARs represent predictive models derived from the application of statistical tools correlating to the biological activity (including desirable therapeutic effect and undesirable side effects) of chemicals (drugs/ toxicants/environmental pollutants) with descriptors representative of molecular structure and/or properties. QSARs are being applied in many disciplines like risk assessment, toxicity prediction, and regulatory decisions in addition to drug discovery and lead optimization. In this article, we obtained numerous molecular descriptors for a molecular graph of a pioneer synthetic polymer called bakelite. We employed edge partitioning procedures on the molecular graph on the basis of valency of vertices d v and sum of valency s v of vertices at unit distance from each other. In addition, we computed M-polynomial of bakelite network along with closed form formulae of various valency-based topological descriptors of substantial significance. Moreover, we took advantage of Matlab and Maple for the simplification and plotting of results. For a future prospect, we propose to develop molecular graphs of certain synthetic polymers (plastics) like polyvinyl chloride (PVC) and Polyethylene terephthalate (PETE), and work out their topological descriptors which will eventually take part in determining a relation as well as comparison between the physical and chemical properties of these synthetic polymers with bakelite. Data Availability Statement: All data required for this research is available in this paper.
Author Contribution: All authors contributed equally in this paper.
Funding Statement: This research is partially funded by University of Management and Technology, Lahore, Pakistan.