Journal of Chemistry

Volume 2019, Article ID 7297253, 9 pages

https://doi.org/10.1155/2019/7297253

## M-Polynomial and Topological Indices of Benzene Ring Embedded in P-Type Surface Network

Correspondence should be addressed to Xiujun Zhang; nc.ude.udc@gnahzsdoow

Received 3 July 2019; Accepted 8 October 2019; Published 3 November 2019

Academic Editor: Robert Zaleśny

Copyright © 2019 Hong Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The representation of chemical compounds and chemical networks with the M-polynomials is a new idea, and it gives nice and good results of the topological indices. These results are used to correlate the chemical compounds and chemical networks with their chemical properties and bioactivities. In this article, particular attention will be put on the derivation of M-polynomial for the benzene ring embedded in the P-type surface network in 2D. Furthermore, the topological indices based on the degrees are also derived by using the general form of M-polynomial of the benzene ring embedded in the P-type surface network . In the end, the graphical representation and comparison of the M-polynomial and the topological indices of the benzene ring embedded in the P-type surface network in 2D are described.

#### 1. Introduction

The chemical compounds can be represented by using the mathematical tools of graph theory. The mathematical models that are based on the polynomials of the chemical compounds and crystal structures can be used in order to predict and forecast their chemical properties and bioactivities. Mathematical chemistry is rich in tools like functions and polynomials which predict the properties of molecular graphs and crystal structures. The topological descriptors are the numerical parameters of the chemical graph which characterize its topology and are usually graph invariants. They explain the structure of chemical compounds mathematically and are utilized in the study of quantitative structure property and activity relationships (QSPR/QSAR).

A topological index is a numerical value which describes and explains an important information about the chemical structure. A great variety of such indices are studied and used in theoretical chemistry, pharmaceutical research, drugs, and different areas of science. The properties like boiling point, strain energy, viscosity, fracture toughness, and heat of formation are connected to the chemical structure under study. This fact plays a major role in the field of chemical graph theory [1–22].

The computation of the general polynomial is formed whose derivatives or integrals or composition of both are evaluated at some particular point. Then, the simplified form yields the molecular descriptor. For instance, there are polynomials like forgotten polynomials, Zagreb polynomials, and Hosoya polynomials, but these polynomials give rise to one or two topological indices [23–26]. The Hosoya polynomial is a polynomial whose derivatives evaluated at 1 give Wiener and hyper Wiener index [27]. The Hosoya polynomial and Zagreb polynomials are considered to be of the general form in the determination of distance-based and degree-based indices, respectively. The M-polynomial is a new and recent polynomial. It will open up new results of chemical graphs and insights in the study of topological descriptors based on degrees. The main importance of this polynomial is that it can give exact forms of more than ten degree-based molecular descriptors [28, 29]. Rapid development and advancements are being made in this new polynomial. Recently, Kwun et al. computed M-polynomial and topological indices of V-phenylenic nanotube and nanotori [30].

The M-polynomial of a graph is formulated as [28]where is the number of edges such that , , and .

The path number was the first distance-based topological index defined by Wiener [31] in 1947. This index is now called as the Wiener index. It has many famous mathematical and chemical applications [31, 32]. Later on, Milan Randić proposed and formulated the Randić index of a graph .

The general Randić index was proposed and defined independently by Bollobás et al. [33] and Amić et al. [34]. Due to its useful and important results in the field of mathematical chemistry, it has been widely used by both mathematicians and chemists. For a survey of these results, see references [35–38]. The general Randić index and inverse Randić index are formulated as

The first and second Zagreb indices are introduced by Gutman and Trinajstić [25, 39, 40]. Both first and second Zagreb indices and the second modified index are formulated as

Recently, the symmetric division deg index of a graph is introduced [41]. It is the significant index which is used to determine the total surface area of polychlorobiphenyls [42] and is defined as

The other version of the Randic index is the harmonic index [43] and is defined as

The inverse sum index is formulated as [44]

The augmented Zagreb index gives best approximation of heat of formation of alkanes [45, 46]. It is formulated as [47]

Let , and then Table 1 relates above described topological indices with M-polynomial [28], where