eu M-POLYNOMIALS AND TOPOLOGICAL INDICES OF SILICATE AND OXIDE NETWORKS

A topological index is a numeric quantity that characterizes the whole structure of a molecular graph of the chemical compound and helps to understand its physical features, chemical reactivities and boiling activities. In 1936, Pólya introduced the concept of a counting polynomial in chemistry and Wiener in 1947 made the use of a topological index working on the boiling point of paraffin. The literature on the counting polynomials and the topological indices of the molecular graphs has grown enormously since those times. In this paper, we study the M -polynomials of the silicate, chain silicate and oxide networks and use these polynomials as a latest developed tool to compute the certain degree-based topological indices such as first Zagreb, second Zagreb, second modified Zagreb, general Randić, reciprocal general Randić, symmetric division deg, harmonic, inverse sum and the augmented Zagreb. we also include a comparison between all the obtained results to show the better one. AMS Subject Classification: 05C07, 92E10


Introduction
For undefined terms, see next section.A number, polynomial or a matrix can uniquely identify a graph.A topological index is a function T op from to the set of real numbers, where is the set of finite simple graphs.In the progressive study of the topological indices for the molecular graphs in the chemical graph theory, many topological indices have been introduced.It is very important to know that all the topological indices are invariant under the graph theoretic operation of graphs isomorphism.The computed topological indices of the molecular graphs help to understand the physical features, chemical reactivities and boiling activities such as heat of evaporation, heat of formation, surface tension, chromatographic retention times, vapor pressure and boiling point of the involved chemical compound [5,16,22].Moreover, in the studies of quantitative structure-activity relationship (QSAR) and quantitative structure-property relationship (QSPR), topological indices are utilized to guess the bioactivity of the chemical compounds [11].
In 1947, for the first time, Wiener [25] introduced the distance-based topological index in chemistry working on the boiling point of paraffin.The most studied indices are degree-based topological indices which are obtained from the degrees of the vertices of the molecular graphs under the certain formulaic conditions.This fact is emphasized in the recent survey [10] that provide a uniform approach to the degree-based topological indices.
In particular, in 1972, Gutman and Trinajsti [12] derived a pair of degreebased molecular descriptors known as the first Zagreb index and the second Zagreb index for the total π-energy of conjugated molecules.Soon after, these indices have been used as the branching indices [7,8].The Zagreb indices are also used in the studies of the quantitative structure-activity relationship and the quantitative structure-property relationship [11].In 1975, Randić [21] defined the degree based topological index which is called by Randić index.In 1998, Bollobás and Erdös [4] and, in 1998, Amić et al. [1] independently defined the generalized Randić index.In 2010, the augmented Zagreb index is defined by Furtula et al. [9].
Many computational results are also obtained related to the aforesaid topological indices on different chemical structures.In particular, Rajan et al. [20] computed the certain topological indices of the silicate, honeycomb and hexagonal networks.Bača et al. [2,3] proved the topological indices for the fullerenes and the carbon nanotube networks.The rhombus type silicate and oxide networks for several topological indices are studied in [15] and the Zagreb indices of the Titania nanotubes can be found in [17].For further study, we refer [7,8,10,11].
In 1935, Pólya [19] defined the concept of counting polynomial in chemistry.Later on, in the literature, numerous polynomials are introduced with remark-able applications in mathematical chemistry.In particular, Hosoya polynomial is a key polynomial in the area of distance-based topological indices.The Wiener index can be obtained as the first derivative of the Hosoya polynomial at numeric value 1.Similarly, the hyper-Wiener index and the Tratch-Stankevich-Zefirov can be computed from the Hosoya polynomial.Recently, in 2015, Deutsch and Klavzar [6] introduced the concept of M -polynomial and showed that the role of this polynomial for the degree-based topological indices is parallel to the role of the Hosoya polynomial for the distance-based topological indices.The M -polynomials and the certain degree-based topological indices of the polyhex nanotubes are studied in [18].
In this paper, we prove the M -polynomials of silicate, chain silicate and oxide networks.The degree-based topological indices such as first Zagreb (M 1 ), second Zagreb (M 2 ), second modified Zagreb (M M 2 ), general Randić (R α ), reciprocal general Randić (RR α ) and symmetric division deg (SDD) are computed with the help of these M -polynomials.In addition, we compute harmonic index (H), inverse sum index (IS) and the augmented Zagreb index (AZI) for all the aforesaid networks.For the better under standing a comparison between all the computed indices is also shown.
The rest of the paper is organized as: Section 2 includes the definitions and formulas which are frequently used in the main results.In Section 3, we compute the main results related to the M -polynomials and the certain degreebased topological indices of the silicate, chain silicate and oxide networks and, finally, Section 4 includes the conclusion between all the obtained results.

Preliminaries
A molecular graph Γ = (V (Γ), E(Γ)) with the vertex set V (Γ) = {v 1 , v 2 , ..., v n } and the edge set E(Γ) is a graph whose vertices (nodes) denote atoms and edges denote bonds between the atoms of any underlying chemical structure.The order and the size of a graph are |V (Γ)| = v and |E(Γ)| = e, respectively.A graph is connected if there exists a connection between any pair of vertices.The distance between two vertices u and v, denoted by d(u, v), is length of the shortest path between u and v in the graph Γ.The degree of a vertex v, denoted by d(v), is the number of the vertices that are connected to v by the edges.A loop is an edge that connects a vertex to itself, and two or more than two edges with the same end points are called multiple edges.In the present study, a molecular graph is simply a finite connected graph without multiple edges and loops.The notions and terminologies of the graphs which are used in this paper are standard.For further study, we refer [14,24].Now, we define some degree-based topological indices and polynomials which are studied in this paper.Definition 2.1.Let Γ be a molecular graph.Then, the first Zagreb and the second Zagreb indices denoted by M 1 (Γ) and M 2 (Γ), respectively are defined as Definition 2.2.Let Γ be a molecular graph.Then, for a real number α, the general Randić index denoted by R α (Γ) is defined as Definition 2.3.Let Γ be a molecular graph.Then, the symmetry division deg index denoted by SDD(Γ) is defined as Definition 2.4.Let Γ be a molecular graph.Then, the harmonic index and the inverse sum index of Γ are defined as follows and Definition 2.5.Let Γ be a molecular graph.Then, augmented Zagreb index of Γ is given by Definition 2.6.Let Γ be a molecular graph and m i,j (Γ); i, j ≥ 1 be the number of edges e = uv of Γ such that {d(u), d(v)} = {i, j}.Then, the M -polynomial of Γ is defined as In the following table, the relations between the aforesaid topological indices and the M -polynomial are defined.x + y In the Table 1, , where α = 0. To know in details for these operators, we refer [6].Now, we discuss the construction of silicate, chain silicate and oxide networks.The class of the silicates is a most interesting class of minerals by far which are obtained by fusing metal oxides or metal carbonates with sand.Essentially all the silicates contain SiO 4 tetrahedra as its basic unit.In chemistry, the corner vertices of SiO 4 tetrahedron represent oxygen ions and the central vertex represents the silicon ion.In graph theory, we call the corner vertices as oxygen nodes and the center vertex as silicon node.By the different arrangements of the tetrahedron silicate, we obtain different silicate structures.Similarly, different silicate networks are constructed by the different silicate structures.
In Figure 1, the silicate network of dimension 2 is presented.In general, the vertices and the edges in a silicate network SL

Main Results
In this section, we present the main results related to the M -polynomials and the certain degree-based topological indices of the silicate, chain silicate and oxide networks.
Proof.From Figure 1, we note that there are two types of vertices in Γ with respect to their degree such as of degree 3 and 6, and three types of edges with respect to degree of end vertices, that is, {3, 3}, {3, 6} and {6, 6}.Thus, we have with |V 1 | = 6n 2 + 6n, and |V 2 | = 9n 2 − 3n, respectively.Consequently, |V (Γ)| = 15n 2 + 3n.Similarly, we have Thus, the partitions of the vertex set and the edge set of the silicate network are given in the Tables 3 and 4.

Conclusions
In this paper, we proved the M -polynomials of the silicate, chain silicate and oxide networks.With the help of these M -polynomials, we also computed the certain degree-based topological indices such as first Zagreb, second Zagreb, second modified Zagreb, general Randić, reciprocal general Randi c, symmetric division deg, harmonic index, inverse sum index and the augmented Zagreb index of these networks.In other words, we can say the M -polynomials are used to compute the certain degree based topological indices as a latest developed tool in the chemical graph theory.Moreover, the obtained M -polynomials and all the computed topological indices are expressed in terms of n, where n shows the dimension of the each network studied in this note.Figures 4,5 and 6 show the AZI as a better one for each network studied in the current paper.Now, we close our discussion with the following lines.These counting polynomials and computed topological indices can help us to understand the physical features, chemical reactivity and biological activities of the silicate, chain (n) of dimension n are |V (SL(n))| = 15n 2 + 3n and |E(SL(n))| = 36n 2 , respectively.A chain silicate network denoted by CS(n) is obtained by arranging n tetrahedra linearly.The Figure 2 presents the chain silicate network for n = 6.The vertices and edges of the chain silicate network CS(n) are |V (CS(n))| = 3n+1 and |E(CS(n))| = 6n.Moreover, If we delete all the silicon ions from the silicate network, then we obtain oxide network as shown in Figure 3 of dimension 2. For dimension n, the vertices and the edges of a oxide network OX(n) are |V (OX(n))| = 9n 2 + 3n and |E(OX(n))| = 18n 2 , respectively.

Figure 5 .
Figure 5.Comparison of the computed indices of CS(n).

Figure 6 .
Figure 6.Comparison of the computed indices of OX(n).

Table 1 .
Derivation of degree-based topological indices from M -polynomial

Table 4 .
The partitions of the edge set of the silicate network SL(n)

Table 5 .
The partitions of the vertex set of the chain silicate network CS(n)

Table 7 .
The partitions of the vertex set of the oxide network OX(n)

Table 8 .
The partitions of the edge set of the oxide network OX(n)