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

Department of Mathematics, Dong-A University, Busan 49315, Republic of Korea Department of Mathematics and Statistics, %e University of Lahore, Lahore, Pakistan Division of Science and Technology, University of Education, Lahore 54000, Pakistan Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Jinju 52828, Republic of Korea Center for General Education, China Medical University, Taichung 40402, Taiwan


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.ese are actually the numerical values that correlate the structure with various physical properties, chemical reactivity, and biological activities [1][2][3][4][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 Mpolynomial in, 2015, to play a role, parallel to Hosoya polynomial to determine closed form of many degree-based topological indices [8][9][10][11].e real power of M-polynomial is its comprehensive nature containing healthy information about degree-based graph invariants.ese invariants are calculated on the basis of symmetries present in the 2dmolecular 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.
is 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.

Basic Definitions and Literature Review
roughout 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 d v denotes the degree of a vertex v.

Definition 1.
e M-polynomial of G is defined as and m ij (G) is the edge vu ∈ E(G) such that where ≤ j [7].

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.e triangular benzenoid system is shown in Figure 1.In the following theorem, we compute M-polynomial of the triangular benzenoid system.Theorem 1.Let T p be a Triangular benzenoid system where p shows the number of hexagons in the base graph and total no. of hexagons in T p is (1/2)p(p + 1).en, ( Proof.Let T p be a triangular benzenoid.en from Figure 1, we have □ e edge set of T p has the following three partitions: Now, us, the M-polynomial of T p is 2 Journal of Chemistry Now, we derive formulas for many degree-based topological indices using M-polynomial.Proposition 2. Let T p be a triangular Benzenoid.en, en, Figure 1: Triangular benzenoid.
Our next target is the benzenoid hourglass system which is obtained from two copies of a triangular benzenoid T p by overlapping their external hexagons and shown in Figure 2. In eorem 3, we compute M-polynomial of the benzenoid hourglass system.□ Theorem 3. Let X p denotes the Benzenoid Hourglass.en, its M-polynomial is Proof.Let X p denotes the benzenoid hourglass which is obtained from two copies of a triangular benzenoid T p by overlapping their external hexagons.en, we have e edge set of X p has the following three partitions: Now,  us, the M-polynomial of X p is Now, we derive formulas for many degree-based topological indices using M-polynomial.□ Proposition 4. Let X p be a Benzenoid Hourglass.en, Topological indices of X p for specific values of p are given in Table 2. Now, we study benzenoid jagged-rectangle shown in Figure 3. Theorem 5. Let B p,q denotes a Jagged-rectangle Benzenoid system for all p, q ∈ N − 1. en, Table 2: Topological indices of benzenoid hourglass for different values of p. Proof.Let B p,q denotes a benzenoid system jagged-rectangle for all p, q ∈ N − 1.A benzenoid jagged-rectangle forms a rectangle and the number of benzenoid called in each chain alternate p and p − 1. e edge set of B p,q has the following three partitions:

Table 1 :
Topological indices of triangular benzenoid for different values of p.