Abstract
Graph theory has provided a very useful tool, called topological index, which is a number from the graph M with the property that every graph N isomorphic to M value of a topological index must be same for both M 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 of M-polynomials for line graphs of H-naphtalenic nanotubes and chain silicate network. From M-polynomial, we obtained some topological indices based on degrees.
1. Introduction
A graph is an order pair, , where is the set of vertices or points and is the set of unordered pairs of distinct vertices of . Every ordered pair of points or is a line or edge of if .
In a graph , represents the total number of vertices called order of denoted as while represents the total number of lines or edges called size of denoted as . Fundamental theorem of graph theory is stated as the sum of all the degrees of vertices in a graph which is equal to two times the total edges in a graph. Mathematically, it can be represented as
For other undefined notations, see [1–3], 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. The numerical parameters of a graph which describe its topology are said to be the topological indices or graph invariants. The simplest topological indices do not recognize double bonds and types of atoms (, , , etc.) and ignore hydrogen atoms. The topological indices are defined for undirected connected molecular graphs only.
The 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 and quantitative structure activity relationship (QSAR) predict the biological activities’ properties of the nanomaterials. The study of topological indices is used in and [2]. In graph theory, there are various types of topological indices such as counting-related indices, based on distance and based on degrees. The most studied topological indices are topological indices based on degrees. These 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.
2. Basic Definitions and Literature Review
In this section, we study some basic definitions and literature review. Let a graph and its line graph be a graph such that each vertex of represents an edge of and two vertices of are connected if their respective edges share a common vertex. The 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 of conjugated molecules. For more detail about topological indices, see [7–9]. Let be a molecular graph with and which are degrees of vertices and , and then the first and second Zagreb indices are represented by and , respectively, and stated as
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 and defined as
Third Zagreb index was first used in 2011 by Fath-Tabar [2], denoted by and defined as
This 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:
This 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. This index 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 and for a real number 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.
The 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. The edge of the M-polynomial is the source of knowledge which contains information about degree-based graph invariants. For M-Polynomial of different graphs, see [16–19]. Let be molecular graph and , where greater than or equal to 1 be the total number of edges of such that is equal to . M-Polynomial of is stated as
Relation of topological indices and M-polynomials is defined aswhere . To know in details for these operators, see [5, 12].
The following lemmas [15] are important for our results.
Lemma 1. Let be a graph with and . Then,
Lemma 2. Let be a graph of size and order . Then, the line graph of is a graph of order and size .
Here, we give derivation of all above-discussed indices from , as shown in Table 1.
3. 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 are found nearly in all the silicate. We get silicate from metal carbonates with sand or from fusing metal oxides. In chemistry, in tetrahedron, the center vertex represents the silicon icon and the corner vertices of tetrahedron represent oxygen we get chain silicate network ions. There 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) and its line graphs L(CS(p)) for different values of p are shown in Figures 1 to 4. In graph, total vertices and total edges are and , respectively. There are three types of vertices having degrees 4, 7, and 10.
(a)
(b)
Let represent total edges connecting the vertices of degrees and . The line graph of chain silicate network contains only , , , , and edges. The number of , , , , and edges in different values is mentioned in Table 2. We discuss the results for only .
Theorem 1. Let be the edge version of chain silicate network. Then, M-polynomial of is
Proof. are the required M-polynomials.
Graphical representation of M-polynomial where is shown in Figure 5.
Theorem 2. Let be the line graph of chain silicate network andbe its M-polynomial. Then, first Zagreb index, second Zagreb index, second modified Zagreb, and general Randic, , where , , where and , obtained from M-polynomial are as follows:
Proof. Let be the M-polynomial of the line graph of chain silicate network. Then,Now, the required partial derivatives and integrals are obtained asNow, we obtainConsequently,
Theorem 3. Let be the line graph of chain silicate network andbe M-polynomial. Then, (H(M)), (IS(M)), and (AZI(M)) obtained from M-polynomial are as follows:
Proof. Let be a M-polynomial of the line graph of chain silicate network. Then,Now, the required expressions are obtained asNow, we obtainConsequently,
3.1. H-Naphtalenic Nanotubes
In line graph of H-naphtalenic nanotubes , there are two types of vertices having degrees 3 and 4 and there are three partitions of edges such as , , and . , , and means there are three edges between the vertices of degrees , , and . Graphs of H-naphtalenic nanotubes and its line graphs for x, y = 1, are shown in Figures 6 to 8. Total edges and vertices of line graph of H-naphtalenic nanotubes () ([21] for detail) are and , respectively.
Theorem 4. Let be the line graph of H-naphtalenic nanotubes. Then, the M-polynomial of is
Proof. From Figures 6 to 8, we noted that there are two types of vertices and three types of edges, detail given below:We conclude thatM-Polynomial of isGraphical representation of M-polynomial of line graph of H-naphtalenic nanotubes () is shown in Figure 9.
Theorem 5. Let be the line graph of H-naphtalenic nanotubes and M-polynomial of isthen
Proof. Same as above.
4. Conclusion
In this paper, we derived the results of M-polynomial for line graph of chain silicate network. M-polynomial gives the source of information about the closed form of topological indices based on degrees of molecular graph M. These results can play an essential role to determine the properties and uses of this network in industries, electronics, and pharmacies.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare no conflicts of interest.
Authors’ Contributions
All authors contributed equally in writing this paper. All authors read and approved the final manuscript.
Acknowledgments
The work was supported by the Higher Education Commission of Pakistan.