Journal of Chemistry
Volume 2018 (2018), Article ID 8213950, 8 pages
https://doi.org/10.1155/2018/8213950

Research Article

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

Young Chel Kwun,¹ Ashaq Ali,² Waqas Nazeer,³ Maqbool Ahmad Chaudhary,² and Shin Min Kang4,5

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

Correspondence should be addressed to Waqas Nazeer; nazeer.waqas@ue.edu.pk and Shin Min Kang; smkang@gnu.ac.kr

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.

Figure 1: Triangular benzenoid.

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. Let

Then,

(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.

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

Name of index
p = 1
p = 2
p = 3
p = 4
p = 5

M-polynomial

First Zagreb Index
24
72
138
222
324

Second Zagreb Index
46.50000000
132
244.5000000
384
550.5000000

Modified second Zagreb Index
1.500000000
2.833333333
4.500000000
6.500000000
8.833333333

Randić Index for
17.39387692
41.09081537
73.78775383
115.4846923
166.1816308

Inverse Randić Index for
2.898979486
7.348469229
12.79795897
19.24744872
26.69693846

Symmetric Division Index
18
5
90
135
186

Harmonic Index
3
6.400000000
10.80000000
16.20000000
22.60000000

Inverse Sum Index
6
17.70000000
33.90000000
27.30000000
39.90000000

Augmented Zagreb Index
48
466.1718750
918.5156250
1405.031250
1925.718750

Figure 2: 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 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 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.

Table 2: Topological indices of benzenoid hourglass for different values of p.

Name of index
p = 1
p = 2
p = 3
p = 4
p = 5

M-polynomial

First Zagreb Index
36
132
264
432
636

Second Zagreb Index
44
170
350
584
872

Modified second Zagreb Index
1.777777778
4.444444444
7.777777778
11.77777778
16.44444444

Randić Index for
92.81601318
198.9976439
323.1792746
465.3609054
625.5425361

Inverse Randić Index for
3.648129059
8.097618797
14.54710855
22.99659829
33.44608803

Symmetric Division Index
15.33333333
53.33333333
103.3333333
165.3333333
239.3333333

Harmonic Index
3.733333333
10.53333333
19.33333333
30.13333333
44.26666667

Inverse Sum Index
9.200000000
32.60000000
65
106.4000000
156.8000000

Augmented Zagreb Index
77.56250000
241.9062500
474.5937500
775.6250000
1145

Now, we study benzenoid jagged-rectangle shown in Figure 3.

Figure 3: Benzenoid jagged-rectangle system.

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.

Table 3: Topological indices of jagged-rectangle benzenoid system for different values of p and q.

Name of index
p = 1, q = 1
p = 2, q = 11
p = 3, q = 11
p = 4, q = 11
p = 5, q = 11

M-polynomial

First Zagreb Index
12
54
96
138
180

Second Zagreb Index
−24
9
42
75
108

Modified second Zagreb Index
1.944444444
3.388888889
4.833333333
6.277777778
7.722222222

Randic Index for
15.79795897
46.59591794
77.39387692
108.1918359
138.9897949

Inverse Randic Index for
3.966326495
7.932652990
11.89897949
15.86530598
19.83163248

Symmetric Division Index
16.66666667
39.33333333
62
84.66666667
107.3333333

Harmonic Index
3.933333333
7.866666667
11.80000000
15.73333333
19.66666667

Inverse Sum Index
9.200000000
24.50000000
39.80000000
55.10000000
70.40000000

Augmented Zagreb Index
57.21875000
168.9531250
280.6875000
392.4218750
504.1562500

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.

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