International Journal of Analytical Chemistry
Volume 2018 (2018), Article ID 1709073, 11 pages
https://doi.org/10.1155/2018/1709073

Research Article

Computations of the M-Polynomials and Degree-Based Topological Indices for Dendrimers and Polyomino Chains

Young Chel Kwun,¹ Adeel Farooq,² Waqas Nazeer,³ Zohaib Zahid,⁴ Saba Noreen,⁵ and Shin Min Kang6,7

¹Department of Mathematics, Dong-A University, Busan 49315, Republic of Korea
²Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, Pakistan
³Division of Science and Technology, University of Education, Lahore 54000, Pakistan
⁴Department of Mathematics, University of Management and Technology, Lahore 54000, Pakistan
⁵Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan
⁶Department of Mathematics and RINS, Gyeongsang National University, Jinju 52828, Republic of Korea
⁷Center 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 16 July 2018; Accepted 18 October 2018; Published 1 November 2018

Academic Editor: Charles L. Wilkins

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.

Topological indices correlate certain physicochemical properties like boiling point, stability, and strain energy of chemical compounds. In this report, we compute M-polynomials for PAMAM dendrimers and polyomino chains. Moreover, by applying calculus, we compute nine important topological indices of under-study dendrimers and chains.

1. Introduction

The polyomino chains constitute a finite 2-connected floor plan, where each inner face (or a unit) is surrounded by a square of length one. We can say that it is a union of cells connected by edges in a planar square lattice. For the origin of dominoes, we quote [1]. The polyomino chains have a long history dating back to the beginning of the 20th century, but they were originally promoted by Golomb [2, 3]. Dendrimers [4] are repetitively branched molecules. The name comes from the Greek word, which translates to “trees.” Synonymous terms for dendrimers include arborols and cascade molecules. The first dendrimer was made by Fritz Vögtle in [5]. For detailed study about dendrimer structures we refer the reader to [6–9].

Many studies have shown that there is a strong intrinsic link between the chemical properties of chemical compounds and drugs (such as melting point and boiling point) and their molecular structure [10, 11]. The topological index defined on the structure of these chemical molecules can help researchers better understand the physical characteristics, chemical reactivity, and biological activity [12]. Therefore, the study of topological indices of chemical substances and chemical structures of drugs can make up for the lack of chemical experiments and provide theoretical basis for the preparation of drugs and chemical substances. In the previous two decades, a number of topological indices have been characterized and utilized for correlation analysis in pharmacology, environmental chemistry, toxicology, and theoretical chemistry [13]. Hosoya polynomial (Wiener polynomial) [14] plays a pivotal role in finding topological indices that depend on distances. From this polynomial, a long list of distance-based topological indices can be easily evaluated. A similar breakthrough was obtained recently by Klavžar et al. [15], in the context of degree-based indices. In the year 2015, authors in [15] introduced the M-polynomial, which plays similar role “to what Hosoya polynomial does” to determine many topological indices depending on the degree of end vertices [16–20].

In the present paper, we compute M-polynomials for different dendrimer structures and polyomino chains. By applying fundamental calculus, we recover nine degree-based topological indices for these dendrimers and chains.

2. Basic Definitions and Literature Review

In this paper, we fixed G as a connected graph, V (G) is the set of vertices, E (G) is the set of edges, and is the degree of any vertex v. Most of the definitions presented in this section can be found in [17].

Definition 1 (see [15]). The M-polynomial of G is defined as

where , , and is the edge that is  .

The very first topological index was the Wiener index, defined by Wiener in 1945, when he was studying boiling point of alkane [21]. For comprehensive details about the applications of Wiener index, see [22, 23]. After that, in 1975, Milan Randić [24] introduced the first degree-based topological index, which is now known as Randić index and is defined as

The generalized Randić index is defined as

please see [25–29].

The inverse generalized Randić index is defined as

It can be seen easily that the Randić index is particular case of the generalized Randić index and the inverse generalized Randić index. Other oldest degree-based topological indices are Zagreb indices. The first Zagreb index is defined as

and the second Zagreb index is defined as

The second modified Zagreb index is defined as

For detailed study about Zagreb indices, we refer the reader to [30–32]. There are many other degree-based topological indices, for example, symmetric division index:

harmonic index:

inverse sum index:

augmented Zagreb index:

We refer to [33–45] for detailed survey about the above defined indices and applications. Tables exhibited in [15–19] relate some notable degree-based topological indices with M-polynomial with the following notations [17]:

3. Computational Results

In this section we give our computational results.

3.1. M-Polynomials and Degree-Based Indices for PAMAM Dendrimers

Polyamidoamine (PAMAM) dendrimers are hyperbranched polymers with unparalleled molecular uniformity, narrow molecular weight distribution, defined size and shape characteristics, and a multifunctional terminal surface. These nanoscale polymers consist of an ethylenediamine core, a repetitive branching amidoamine internal structure, and a primary amine terminal surface. Dendrimers are “grown” off a central core in an iterative manufacturing process, with each subsequent step representing a new “generation” of dendrimer. Increasing generations (molecular weight) produce larger molecular diameters, twice the number of reactive surface sites and approximately double the molecular weight of the preceding generation. PAMAM dendrimers also assume a spheroidal, globular shape at generation 4 and above (see molecular simulation below). Their functionality is readily tailored, and their uniformity, size, and highly reactive “molecular Velcro” surfaces are the functional keys to their use. Here we consider , which denote PAMAM dendrimers with trifunctional core unit generated by dendrimer generations with n growth stages, and , the PAMAM dendrimers with different core generated by dendrimer generators with n growth stages. is kinds of PAMAM dendrimers with n growth stages

Theorem 2. For the PAMAM dendrimers , we have

Proof. Let denote PAMAM dendrimers with trifunctional core unit generated by dendrimer generations with n growth stages.
The edge set of has following four partitions:

Now

and

Theorem 3. For the PAMAM dendrimers , we have

1.2.3.4.5.6.7.8.9.

Proof. Let denote PAMAM dendrimers with trifunctional core unit generated by dendrimer generations with n growth stages. Let

Then

(1) First Zagreb Index

(2) Second Zagreb Index

(3) Modified Second Zagreb Index

(4) Generalized Randić Index

(5) Inverse Randić Index

(6) Symmetric Division Index

(7) Harmonic Index

(8) Inverse Sum Index

(9) Augmented Zagreb Index

Theorem 4. For the PAMAM dendrimers , we have

Proof. Let be the PAMAM dendrimers with different core generated by dendrimer generators with n growth stages. Then the edge set of has following four partitions:

Now

and

Theorem 5. For the PAMAM dendrimers , we have

1..2..3..4..5..6..7..8..9..

Theorem 6. For the PAMAM dendrimers , we have

Proof. Let be kinds of PAMAM dendrimers with n growth stages.
The edge set of has the following three partitions:

Now

and

Theorem 7. For the PAMAM dendrimers , we have

1..2..3..4..5..6.7.8..9..

3.2. M-Polynomials and Degree-Based Indices for Polyomino Chains

From the geometric point of view, a polyomino system is a finite 2-connected plane graph in which each interior cell is encircled by a regular square. In other words, it is an edge-connected union of cells in the planar square lattice. Polyomino chain is a particular polyomino system such that the joining of the centers (set ci as the center of the ith square) of its adjacent regular composes a path .

Let be the set of polyomino chains with n squares. There are 2n+1 edges in every , where is named as a linear chain and denoted by if the subgraph of induced by the vertices with d(v)=3 is a molecular graph with exactly n-2 squares. Also, can be called a zigzag chain and labelled as if the subgraph of is induced by the vertices with d(v)>2 is .

The angularly connected, or branched, squares constitute a link of a polyomino chain. A maximal linear chain (containing the terminal squares and kinks at its end) in the polyomino chains is called a segment of polyomino chain. Let l(S) be the length of S which is calculated by the number of squares in S. For any segment S of a polyomino chain, we get . Furthermore, we deduce and m=1 for a linear chain with n squares and and m=n-1 for a zigzag chain with n squares.

In what follows, we always assume that a polyomino chain consists of a sequence of segments and , where and . We derive that .

Theorem 8. For a linear polyomino chain , we have

Proof. Let be the polyomino chain with n squares where and m=1. is called the linear chain.
The edge set of has the following three partitions:

Now

and

Theorem 9. For a linear polyomino chain , we have the following:

1..2..3..4..5..6..7..8..9..

Theorem 10. Let be zigzag polyomino chain with n squares such that and . Then

Proof. Let be zigzag polyomino chain with n squares such that and . Polyomino chain consists of a sequence of segments and where and .
The edge set of has the following five partitions:

Now

and

Theorem 11. For the Zigzag polyomino chain for , we have the following:

1..2..3.4..5..6.7..8..9..

Theorem 12. For the polyomino chain with n squares and of m segments and satisfying and , , we have

Proof. Let be the polyomino chain with n squares and of m segments and satisfying and . The edge set of has the following five partitions:

Now

and

Theorem 13. For the polyomino chain with n squares and of m segments and satisfying and , , we have the following:

1..2..3..4..5..6..7..8..9..

Theorem 14. For polyomino chain with squares and segments satisfying and , , we have

Proof. Let be a polyomino chain with n squares and m segments satisfying and . Then the edge set of has the following five partitions:

Now

and

Theorem 15. For polyomino chain with n squares and m segments satisfying and , , we have the following:

1..2..3.4..5..6.7.8..9..

4. Conclusions

Topological indices calculated in this paper help us to guess biological activities, chemical reactivity, and physical features of under-study dendrimers and polyomino chains. For example, Randić index is useful for determining physiochemical properties of alkanes as noticed by chemist Milan Randić in 1975. He noticed the correlation between the Randić index and several physicochemical properties of alkanes like boiling point, vapor pressure, enthalpies of formation, surface area, and chromatographic retention times. Hence our results are helpful in determination of the significance of PAMAM dendrimers and polyomino chains in pharmacy and industry.

Data Availability

All data required for this research is included within this paper.

Conflicts of Interest

The authors do not have any conflicts of interest.

Authors’ Contributions

All authors contributed equally to this paper.

Acknowledgments

The first author is funded by Dong-A University, Korea.

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