Molecular Properties of Symmetrical Networks Using Topological Polynomials

Abstract A numeric quantity that comprehend characteristics of molecular graph Γ of chemical compound is known as topological index. This number is, in fact, invariant with respect to symmetry properties of molecular graph Γ. Many researchers have established, after diverse studies, a parallel between the physico chemical properties like boiling point, stability, similarity, chirality and melting point of chemical species and corresponding chemical graph. These descriptors defined on chemical graphs are extremely helpful for researchers to conduct regression model like QSAR/QSPR and better understand the physical features, complexity of molecules, chemical and biological properties of underlying compound. In this paper, several structure descriptors of vital importance, namely, first, second, modified and augmented Zagreb indices, inverse and general Randic indices, symmetric division, harmonic, inverse sum and forgotten indices of Hex-derived Meshes (networks) of two kinds, namely, HDN1(n) and HDN2(n) are computed and recovered using general approach of topological polynomials.


Introduction
Graph theory is a standout amongst the most extraordinary and one of a kind branch of mathematics by which the showing of any structure is made possible. As of late, it achieves much consideration among scientists on account of its extensive variety of utilizations in Computer science, electrical systems, interconnected systems, biological networks, and in chemistry, and so forth. The chemical graph theory is the quickly developing zone among scientists. It helps in comprehension about the basic properties of a molecular graph. There is a considerable measure of molecular compounds which have assortment of utilizations in the elds of business, commercial, industrial, pharmaceutical chemistry, every day life, and in research facility. In recent years researchers got immense attraction toward an emerging eld Cheminformatics, an interplay between Chemistry, Mathematics, Statistics and Information Science. In fact, one of the main reasons behind the signi cance of Cheminformatics is the interlacing of these core areas of science. Graph Theory attained exceptional place in Mathematical Chemistry and this novel branch got the name Chemical graph theory which became increasingly common among researchers and deals with molecular graph of a chemical compound to calculate various topological indices to understand and predict the physicochemical properties of chemical compounds [1]. During QSAR/ QSPR study, the regression analysis relies upon molecular descriptors and is responsible to understand and predict the chemical, biological and physical characteristics of compounds. This provides basis in designing new chemical compounds and drugs having features of our own interest. In literature, numerous types of degree based, distance based as well as topological polynomials [2] related topo-logical indices of molecular graphs have been introduced and many of them turned out to be applicable in mathematical chemistry. For instance, among the most wellknown and well read graph invariants are the Wiener index, Szeged index [3], the Randić indices, Zagreb indices [4,5], atom-bond connectivity, geometric arithmetic and the Hosoya Z indices [6]. Although all above mentioned classes of indices have their own signi cance, however, degree based indices are well read and nd real importance in chemical graph theory and therein biochemistry [7][8][9][10][11]. A graph Γ(V , E) with vertex set V and edge set E is connected, if there exists a connection between any pair of vertices in Γ. A network is simply a connected graph having no multiple edges and loops. For a graph Γ, the degree of a vertex v is the number of edges incident with v and denoted by dv. Paul Manuel et al., [12] conjectured that the minimum metric dimension of hex-derived networks HDN (n) and HDN (n) lies between 3 and 5 and this open problem has partially been answered by Dacheng Xu and Jianxi Fan [13]. Furthermore, Imran et al., [14] discussed some topological properties such as atom-bond connectivity, geometric arithmetic and Randić indices of the network under discussion. We have studied and computed some new indices as well as recovered some indices presented in [15][16][17][18][19] by using entirely di erent and general approach. In this article, throughout, Γ is considered to be connected, nite, undirected and simple network with V(Γ) = Vertex set, E(Γ) = Edge set and dv = degree of vertex where v ∈ V(Γ).

. Preliminaries
De nition 1. Deutsch and Klavžar [20] introduced Mpolynomial for graph Γ = (V , E) as follows: De nition 2. The forgotten polynomial of Γ is given by: In 1947, the idea of topological index was rst conceived and originated by Harold Wiener [21] during the study on boiling point of para n (bi-product of petroleum) and he referred it as path number but afterward path number was assigned the name of its inventor and entitled as Wiener index/number [22]. This pioneering, eminent and well studied index of chemical graph Γ is distance-based index. It deserves high rank in theoretical Chemistry and Chemical graph theory due to its theoretical as well as applicable nature. In 1975, Milan Randić [23] introduced a topological index with the name branching index and is de ned as: Randić observed and established the fact that there exists a relationship between Randić index and various properties (boiling point, enthalpy of formation, surface area) of alkanes.
Later on in 1998, two distinguished researchers, Böllöbás and Erdös [24], extended the idea to all real numbers and the new index received the name general Randić index and is de ned below: Moreover, inverse Randić index is de ned by formula, In 1972, Ivan Gutman and Trinajstic [25] proposed two topological indices named as the rst and the second Zagreb indices and soon after, these indices were used to analyze the structure-dependency of total π − electron of molecular graph. First, second and modi ed Zagreb indices are de ned as follows:

De nition 1. Generalized Zagreb Index
The concept of generalized Zagreb index was established by Azari and Iranmanesh [26] and de ned as where a, b ∈ Z + . Few more topological indices of our interest having utmost importance are de ned below which include harmonic index (HI), inverse sum index (ISI), augmented Zagreb index (AZI) [27] and forgotten index (FI): (13) .

Applications of Topological polynomials
Several topological polynomials were established in the literature and played vital role in mathematical chemistry. Among other graph polynomials like matching polynomial [28], the Zhang-Zhang polynomial (Clar covering polynomial) [29], the Schultz polynomial [30], the Tutte polynomial [31], the most signi cant polynomials are Hosoya polynomial [32] introduced in 1988 and Mpolynomial established in 2015. We can e ciently determine exact formulae for various degree and distancebased topological indices with the help of M-polynomial and Hosoya polynomial, respectively. M-polynomial is the tool which conceals lot of facts regarding degree based graph invariants. Moreover, the M-polynomial provides a very good correlation for the stability of linear alkanes as well as the branched alkanes and for computing the strain energy of cyclo alkanes [33]. For a certain family of networks, we normally use di erent formula to calculate each individual topological index. M-polynomial got advantage over this approach as we only need to operate speci c di erential, integral or both operators on corresponding polynomial to get various vertex-based indices. Many closed form degree-based topological indices of Triangular Boron Nanotubes and Jahangir graph Jn,m are computed using M-polynomial [34,35]. The Zhang-Zhang polynomial (Clar covering polynomial), were found to occur for computation of the total π-electron energy of the molecules within speci c approximate expressions. The Randic index is a topological descriptor that has correlated with a lot of chemical characteristics of the molecules and has been found to be parallel to compute the boiling point and Kovats constants of the molecules.

Materials and Methods
Our main results includes the formulation of algebraic structures of M-polynomial and F-polynomial of HDN (n) and HDN (n), respectively. Then, we computed as well as recovered various topological indices of vital importance. In particular, rst, second, modi ed and augmented Zagreb indices, general and inverse Randić indices, symmetric division, harmonic, inverse sum and forgotten indices of these networks via topological polynomials. To compute our results, we used the method of combinatorial computing, vertex partition method, edge partition method, graph theoretical tools, analytic techniques, degree counting method, and sum of degrees of neighbor's method. Moreover, we used Maple for mathematical calculations, veri cations, and plotting these mathematical results.

. Methodology and Construction of Hex-Derived Network HDN (n) Formulas
The construction of rst kind of hex-derived network HDN (n) is fairly simple and is achieved by placing additional node in each triangular face of hexagonal mesh HX(n) [36] and then joining this extra nodes with all nodes of triangular face. For an alternate version of construction of HDN (n) from HX(n). There are n − n + number of nodes (vertices) and n − n + number of edges in HDN (n). This new network has many advantage over the one from which it is obtained, for instance, represent con guration similar to molecular lattice structures in chemistry. This network is also called mesh network and mostly used in networking of computers to minimize cost, to achieve high performance and reliability. Moreover, HDN (n) is planar and this property gets advantage over non-planar network as far as cost is concerned. Figure  1 depicts a hex-derived network of rst kind with dimension 4.
For the sake of simplicity as well as with out loss of generality, we assume HDN (n) = Γ . As we know total number of vertices of Γ are given by |V(Γ )| = n − n+ and total number of edges are |E(Γ )| = n − n + . In Γ , we observe eight categories of edges on the basis of degree of the vertices of each edge which lead to edge partition of graph and is depicted in the table below. Now, using de nition 1 and de nition 2 to compute M-Polynomial and Forgotten Polynomial, respectively of Γ as follows: • M-Polynomial of hex-derive network HDN (n) Harmonic Index (HI) Sx Jf (x, y)  Now we compute the toplogical indices for hex-derive network Γ , namely rst, second, modi ed and augmented Zagreb indices, Randić indices, SSD index, harmonic index, ISI index and forgotten index. By applying the operators given in derivation of Table 1 on M-polynomial and Forgotten polynomials as follows: Again using derivation formulae of topological indices over M-polynomial from Table , we get

. Methodology and Construction of Hex-Derived Network HDN (n) Formulas
The architecture of second kind of hex-derived network HDN (n) is bit sophisticated as it is obtained from the merger of hexagonal network HX(n) of dimension n with honeycomb network HC(n − ) of dimension n − . The construction of HDN (n) can be accomplished by taking union of HX(n) with its bounded dual HC(n − ) and then by joining each honeycomb vertex with the three vertices of the corresponding face of HX(n). There are n − n + number of nodes (vertices) and n − n + number of edges in HDN (n). Figure 4 depicts a hex-derived network of second kind with dimension 4.
Again for the sake of simplicity, suppose HDN (n) = Γ . We know total number of vertices of Γ are given by |V(Γ )| = n − n + and total number of edges are |E(Γ )| = n − n + . In Γ , we observe eight categories of edges on the basis of degree of the vertices of each edge which lead to edge partition of graph and is depicted in the table below. Now, using de nition 1 and de nition 2 to compute M-Polynomial and Forgotten Polynomial, respectively of Γ as follows:  Now we compute the toplogical indices for hex-derive network Γ , namely rst, second, modi ed and augmented Zagreb indices, Randić indices, SSD index, harmonic index, ISI index and forgotten index. By applying the operators given in derivation of Table 1 Sx JDx Dy f (x, y) = x + (n − )x + (n − )x + ( n − n + )x ( + x ) + (n − )x + nx Again using derivation formulae of topological indices over M-polynomial from table 1, we get  In addition, ISI index and forgotten index has much better predictive power than the predictive power of the Randić index, so the ISI index and forgotten index is more useful than the Randić index for α ∈ {− , − / } as compared to the Randić index for α ∈ { , / } in the case of HDN (n) and HDN (n) . Since the rst and second Zagreb indexes were found to occur for computation of the total π-electron energy of the molecules, in the case of HDN (n) and HDN (n), their values provide total π-electron energy in increasing order for higher values of n. For future work we propose investigation of some new type of chemical graphs and networks to compute certain degree based topological indices using polynomials.
Funding This research is supported by Quality Engineering Projects of Education Department of Anhui province (Grant No.2018jyxm1074) and Natural Science Fund of Education Department of Anhui province (Grant No.KJ2018A0598).