M-Polynomials and Topological Indices for Line Graphs of Chain Silicate Network and H-Naphtalenic Nanotubes

Graph theory has provided a very useful tool, called topological index, which is a number from the graphMwith the property that every graph N isomorphic toM value of a topological index must be same for bothM and N. Topological index is a descriptor in graph theory which is used to quantify the physio-chemical properties of the chemical graph. In this paper, we computed closed forms ofM-polynomials for line graphs of H-naphtalenic nanotubes and chain silicate network. FromM-polynomial, we obtained some topological indices based on degrees.


Introduction
In a graph M � (V(M), E(M)), |V| represents the total number of vertices called order of M denoted as m while |E| represents the total number of lines or edges called size of M denoted as n. Fundamental theorem of graph theory is stated as the sum of all the degrees of vertices in a graph M which is equal to two times the total edges in a graph. Mathematically, it can be represented as For other undefined notations, see [1][2][3], a label graph whose vertices and edges correspond to the atoms and chemical bonds of the compound, respectively, this type of graph is said to be a chemical graph. Its vertices and edges are labeled with the kinds of the corresponding atoms and types of bonds. e numerical parameters of a graph which describe its topology are said to be the topological indices or graph invariants. e simplest topological indices do not recognize double bonds and types of atoms (C, N, O, etc.) and ignore hydrogen atoms. e topological indices are defined for undirected connected molecular graphs only. e topological indices of the chemical or molecular graphs also help us to investigate the physio-chemical properties and boiling activities, i.e., heat of formation, surface area, surface tension, vapors pressure, and boiling points of the involved molecular or chemical compound, see [4,5]. Cheminformatics is another emerging field in which quantitative structure property relationship (QSPR) and quantitative structure activity relationship (QSAR) predict the biological activities' properties of the nanomaterials. e study of topological indices is used in (QSPR) and (QSAR) [2]. In graph theory, there are various types of topological indices such as counting-related indices, based on distance and based on degrees. e most studied topological indices are topological indices based on degrees. ese indices are most important and have many useful applications. In this paper, we discussed M-polynomial and some degree-based topological indices. M-Polynomial is rich in producing source of many topological indices based on degrees which correspond to chemical properties of the material under investigation.

Basic Definitions and Literature Review
In this section, we study some basic definitions and literature review. Let a graph M and its line graph L(M) be a graph such that each vertex of L(M) represents an edge of M and two vertices of L(M) are connected if their respective edges share a common vertex. e terminologies and notations of the graphs which are used in this paper are standard. In this paper, we discussed topological indices based on degrees. In 1972, Trinajstic et al. [6] defined a pair of topological indices based on degrees which are first and Zagreb second indices for Π − energy of conjugated molecules. For more detail about topological indices, see [7][8][9]. Let M � (V, E) be a molecular graph with d(k) and d(l) which are degrees of vertices k and l, and then the first and second Zagreb indices are represented by M 1 (M) and M 2 (M), respectively, and stated as (2) After these indices, a new form of second Zagreb index was introduced by Nikolic et al. [6] which is known as modified second Zagreb index. Modified second Zagreb index is another degree-based topological index denoted by MM 2 (M) and defined as ird Zagreb index was first used in 2011 by Fath-Tabar [2], denoted by M 3 (M) and defined as is topological index which computed surface area of polychlorobiphenyls is called symmetric division index [3] and defined as In 2010, augmented Zagreb index was introduced by Furtula et al. [3] which is stated as follows: is remarkable predictive index is used to compute the heat of formation in heptane and octanes.
In 2009, Trinajstic and Zhou derived a index, called sum connectivity index. is index (S(M)) is stated as Inverse sum index, is the graph invariant which was studied as a significant descriptor of surface area for isomers of octane and stated as In 1975, Randic [10] introduced the very first and oldest topological index based on degree stated as In 1998, Bollobas and Erdos [11] and Amic et al. [12] independently defined general Randic index for a simple connected graph M and for a real number α as In 2005, both Hu and Li, mathematician and theoretical chemist, have extensively studied it. Caporossi et al. [13] in 2003 have established very interesting and important mathematical properties. e other harmonic index is defined as Various graph polynomials were introduced in the literature, some of them are Hosoya polynomial, matching polynomial, Zhang-Zhang polynomial, Schultz polynomial, Tutte polynomial, etc. In this section, we study a polynomial called the M-polynomial and show that its role for indices based on degree is similar to the role of the Hosoya polynomial for indices based on distance. See for detail [1,3,14]. In 2015, Munir et al. [15] introduced the concept of M-polynomial. e edge of the M-polynomial is the source of knowledge which contains information about degreebased graph invariants. For M-Polynomial of different graphs, see [16][17][18][19]. Let M be molecular graph and m p,q , where p, q greater than or equal to 1 be the total number of edges of M such that (d(k), d(l)) is equal to p, q.
where α ≠ 0. To know in details for these operators, see [5,12]. e following lemmas [15] are important for our results. Here, we give derivation of all above-discussed indices from M(M, r 1 , s 1 ), as shown in Table 1.

Results and Discussion
In this section, we discuss construction of line graphs and some topological indices based on degrees of chain silicate network [20] and H-naphtalenic nanotubes [21].
In the first line graph of chain silicate network, silicates are the most complicated, largest, and interesting class of minerals. Tetrahrdron (SiO 4 ) are found nearly in all the silicate. We get silicate from metal carbonates with sand or from fusing metal oxides. In chemistry, in SiO 4 tetrahedron, the center vertex represents the silicon icon and the corner vertices of SiO 4 tetrahedron represent oxygen we get chain silicate network ions. ere are many methods to construct the silicate network of the tetrahedron. If tetrahedron silicate joins in a row, then we get chain silicate network, see [20] for details. Chain Silicate CS (1) Table 2. We discuss the results for only p > 3.
Now, the required partial derivatives and integrals are obtained as Now, we obtain

Journal of Mathematics
Consequently,