Journal of Chemistry

Volume 2018, Article ID 8213950, 8 pages

https://doi.org/10.1155/2018/8213950

*M*-Polynomials and Degree-Based Topological Indices of Triangular, Hourglass, and Jagged-Rectangle Benzenoid Systems

^{1}Department of Mathematics, Dong-A University, Busan 49315, Republic of Korea^{2}Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan^{3}Division of Science and Technology, University of Education, Lahore 54000, Pakistan^{4}Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Jinju 52828, Republic of Korea^{5}Center for General Education, China Medical University, Taichung 40402, Taiwan

Correspondence should be addressed to Waqas Nazeer; kp.ude.eu@saqaw.reezan and Shin Min Kang; rk.ca.ung@gnakms

Received 22 March 2018; Revised 16 July 2018; Accepted 9 August 2018; Published 12 September 2018

Academic Editor: Marjana Novic

Copyright © 2018 Young Chel Kwun 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

Chemical graph theory is a branch of mathematical chemistry which has an important effect on the development of the chemical sciences. The study of topological indices is currently one of the most active research fields in chemical graph theory. Topological indices help to predict many chemical and biological properties of chemical structures under study. The aim of this report is to study the molecular topology of some benzenoid systems. *M*-polynomial has wealth of information about the degree-based topological indices. We compute *M*-polynomials for triangular, hourglass, and jagged-rectangle benzenoid systems, and from these *M*-polynomials, we recover nine degree-based topological indices. Our results play a vital role in pharmacy, drug design, and many other applied areas.

#### 1. Introduction

Mathematical chemistry provides tools such as polynomials and functions to capture information hidden in the symmetry of molecular graphs and thus predict properties of compounds without using quantum mechanics. A topological index is a numerical parameter of a graph and depicts its topology. Topological indices describe the structure of molecules numerically and are used in the development of qualitative structure activity relationships (QSARs). Most commonly known invariants of such kinds are degree-based topological indices. These are actually the numerical values that correlate the structure with various physical properties, chemical reactivity, and biological activities [1–5]. It is an established fact that many properties such as heat of formation, boiling point, strain energy, rigidity, and fracture toughness of a molecule are strongly connected to its graphical structure.

Hosoya polynomial, Wiener polynomial [6], plays a pivotal role in distance-based topological indices. A long list of distance-based indices can be easily evaluated from Hosoya polynomial. A similar breakthrough was obtained recently by Deutsch and Klavžar [7], in the context of degree-based indices. Deutsch and Klavžar [7] introduced *M*-polynomial in, 2015, to play a role, parallel to Hosoya polynomial to determine closed form of many degree-based topological indices [8–11]. The real power of *M*-polynomial is its comprehensive nature containing healthy information about degree-based graph invariants. These invariants are calculated on the basis of symmetries present in the 2d-molecular lattices and collectively determine some properties of the material under observation. Benzenoid hydrocarbons play a vital role in our environment and in the food and chemical industries.

Benzenoid molecular graphs are systems with deleted hydrogens. It is a connected geometric figure obtained by arranging congruent regular hexagons in a plane so that two hexagons are either disjoint or have a common edge.

This figure divides the plane into one infinite (external) region and a number of finite (internal) regions. All internal regions must be regular hexagons. Benzenoid systems are of considerable importance in theoretical chemistry because they are the natural graph representation of benzenoid hydrocarbons. A vertex of a hexagonal system belongs to, at most, three hexagons. A vertex shared by three hexagons is called an internal vertex [12].

In this paper, we study three benzenoid systems, namely, triangular, hourglass, and jagged-rectangle benzenoid systems.

#### 2. Basic Definitions and Literature Review

Throughout this article, we assume *G* to be a connected graph, *V* (*G*) and *E* (*G*) are the vertex set and the edge set, respectively, and denotes the degree of a vertex .

*Definition 1*. The *M*-polynomial of *G* is defined as where and is the edge such that where [7].

Wiener Index and its various applications are discussed in [13–15]. Randić Index, , was introduced by Milan Randić in 1975, defined as . For general details about and its generalized Randić Index, refer [16–20], and the Inverse Randić Index is defined as . Clearly, is a special case of where . This index has many applications in diverse areas. Many papers and books such as [21–23] are written on this topological index as well. Gutman and Trinajstić introduced two indices defined as and . The modified second Zagreb Index is defined as We refer [24–28] to the readers for comprehensive details of these indices. Other famous indices are Symmetric Division Index: SDD(G) = , Harmonic Index: , Inverse Sum Index: , and Augmented Zagreb Index: [29, 30].

Tables presented in [7–10] relate some of these well-known degree-based topological indices with *M*-polynomial with the following reserved notations:

#### 3. Computational Results

In this section, we give our computational results. In terms of chemical graph theory and mathematical chemistry, we associate a graph with the molecular structure where vertices correspond to atoms and edges to bonds. The triangular benzenoid system is shown in Figure 1. In the following theorem, we compute *M*-polynomial of the triangular benzenoid system.