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.
Theorem 1. Let be a Triangular benzenoid system where p shows the number of hexagons in the base graph and total no. of hexagons in is . Then,
Proof. Let be a triangular benzenoid. Then from Figure 1, we have
The edge set of has the following three partitions:
Now,
Thus, the M-polynomial of is
Now, we derive formulas for many degree-based topological indices using M-polynomial.
Proposition 2. Let be a triangular Benzenoid.
Then,(1)(2)(3)(4)(5)(6)(7)(8)(9)
Proof. LetThen,(1)First Zagreb Index:(2)Second Zagreb Index:(3)Modified second Zagreb Index:(4)Generalized Randic Index:(5)Inverse Randic Index:(6)Symmetric Division Index:(7)Harmonic Index:(8)Inverse Sum Index:(9)Augmented Zagreb Index:Topological indices of Tp for specific values of p are given in Table 1.
Our next target is the benzenoid hourglass system which is obtained from two copies of a triangular benzenoid by overlapping their external hexagons and shown in Figure 2. In Theorem 3, we compute M-polynomial of the benzenoid hourglass system.
Theorem 3. Let denotes the Benzenoid Hourglass. Then, its M-polynomial is
Proof. Let denotes the benzenoid hourglass which is obtained from two copies of a triangular benzenoid by overlapping their external hexagons. Then, we haveThe edge set of has the following three partitions:Now,Thus, the M-polynomial of isNow, we derive formulas for many degree-based topological indices using M-polynomial.
Proposition 4. Let be a Benzenoid Hourglass.
Then,(1)(2)(3)(4)(5)(6)(7)(8)(9)
Topological indices of Xp for specific values of p are given in Table 2.
Now, we study benzenoid jagged-rectangle shown in Figure 3.
Theorem 5. Let denotes a Jagged-rectangle Benzenoid system for all .
Then,
Proof. Let denotes a benzenoid system jagged-rectangle for all . A benzenoid jagged-rectangle forms a rectangle and the number of benzenoid called in each chain alternate p and p − 1.
The edge set of has the following three partitions:Now,Thus, the M-polynomial of is
Proposition 6. Let denotes a Jagged-rectangle Benzenoid system for all .
Then,(1)(2)(3)(4)(5)(6)(7)(8)(9)
Topological indices of Bp,q for specific values of p and q are given in Table 3.
4. Conclusion
In this paper, we computed M-polynomials for triangular, hourglass, and jagged-rectangle benzenoid systems. From these M-polynomials, we recover first Zagreb, second Zagreb, modified second Zagreb, Randić, inverse Randić, symmetric division, inverse sum, and harmonic and augmented Zagreb indices of triangular, hourglass, and jagged-rectangle benzenoid systems. Note that there are no benzenoid molecules having the triangular graphs and hourglass graphs as their skeleton. It is important to mention here that some of these topological indices are calculated directly by using formulas in the literature.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was supported by research funds from Dong-A University.