Discrete Dynamics in Nature and Society

Volume 2018, Article ID 2684984, 11 pages

https://doi.org/10.1155/2018/2684984

## On Topological Indices of Fractal and Cayley Tree Type Dendrimers

^{1}Department of Mathematical Sciences, United Arab Emirates University, P.O. Box 15551, Al Ain, UAE^{2}School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Sector H-12, Islamabad, Pakistan^{3}Department of Mathematics, University of Lahore, Pakpattan Campus, Pakistan^{4}Department of Mathematics, COMSATS Institute of Information Technology, Attock Campus, Pakistan

Correspondence should be addressed to Muhammad Imran; moc.liamg@bahdnarmi

Received 5 April 2018; Accepted 11 July 2018; Published 8 August 2018

Academic Editor: Juan L. G. Guirao

Copyright © 2018 Muhammad Imran 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

The topological descriptors are the numerical invariants associated with a chemical graph and are helpful in predicting their bioactivity and physiochemical properties. These descriptors are studied and used in mathematical chemistry, medicines, and drugs designs and in other areas of applied sciences. In this paper, we study the two chemical trees, namely, the fractal tree and Cayley tree. We also compute their topological indices based on degree concept. These indices include atom bond connectivity index, geometric arithmetic index and their fourth and fifth versions, Sanskruti index, augmented Zagreb index, first and second Zagreb indices, and general Randic index for . Furthermore, we give closed analytical results of these indices for fractal trees and Cayley trees.

#### 1. Introduction

Graph theory provides a gateway for chemists and scientist to focus on the topological descriptors of molecular graphs. Molecular compounds can be modeled by using graph theoretic method. These topological descriptors provide a better way to understand and to predict the properties and bioactivities of compounds. Molecular graph is the illustration of a chemical compound or complex structure in which nodes correspond to atoms and edges correspond to chemical bond or physical link between two nodes/vertices. In the QSAR/QSPR study, the topological indices, mainly the Wiener index, Randi index, Zagreb indices, and and index, are used to predict the chemical and physical properties and bioactivities of different compounds and structures.

The graph can be represented by a numerical number, an eigen value, polynomial, and topological index, etc., which represents the structure of a graph. A topological index is a numerical value which indicates some useful information about the chemical compound or structure. It is the arithmetic invariants of a molecular graph and is fruitful and vital to correlate with physiochemical properties and bioactivities of molecular compound/network/dendrimer. Recently, many researchers have found topological descriptor vital for the study of structural properties of molecular graph or network or chemical tree. An acyclic connected graph is called a tree graph. The degree 3 or greater of every vertex of tree is called branching point of tree. A chemical tree is a connected acyclic graph having maximum degree 4. We provide now some literature review of molecular descriptors of molecular graphs/networks, which motivate us to work in this field. The alkanes are the example of chemical trees and the series of saturated hydrocarbons, including methane, ethane, and propane. Motivated by the chemical importance of molecular descriptors of chemical compounds and structures, a lot of research has been done by researchers. The caterpillar trees represent the structure of benzenoid hydrocarbon molecules. The first and second Zagreb index of star-like trees and sun-like graphs and also caterpillar trees containing the hydrocarbons specially ethane, propane, and butane is studied and computed in [1]. The computation of topological indices of degree based for the line graphs of Banana tree graph and Firecracker graph is investigated in [2]. The chemical applications of topological descriptors related to acyclic organic molecules are mentioned in detail in [3–9].

The dendrimers consist of highly branched organic macromolecules with successive generations/iterations of branch units surrounding a central core. There are different types of dendrimers that are discovered so far. These have a wide range of applications in the field of chemistry, nanoscience, biology, etc. The topological indices of famous dendrimers is recently investigated in [10, 11] and the references therein. For instance, the bond incident degree indices of nanostructures and polyomino chains are computed in [12, 13]. The edge version of geometric arithmetic index of polyomino chains of 8 cycles and arbitrary carbon nanocones is computed by [14]. Certain topological indices of honeycomb derived networks are computed in [15]. With the rapid increase of development in medicine manufacturing organizations and industries, a large number of drug products are produced each year. In order to determine the chemical properties of these drugs we focus on theoretical examination of topological indices. Smart polymer family is widely used in anticancer drugs and its some topological indices are computed in [16]. The computation of topological indices and their properties of certain networks, carbon graphite, crystal cubic carbon, copper oxide, and nanotubes are discussed in [17–22]. The Sanskruti index of line graphs of subdivision graphs of 2D-lattice, nanotube, nanotorus of and polycyclic aromatic hydrocarbons is investigated in [23, 24].

There are certain types of topological indices that depend on eccentric based, degree based, and distance based indices, etc. In this article, we compute degree based topological indices only. Bollobs et al. [25] and Amic et al. [26] introduced the general Randi index independently. The properties and useful results of general Randi are referred to in [27–29]. It is defined as follows: where .

The atom bond connectivity index is of vital importance and used in the study of heat formation of alkanes. It is introduced by Estrada et al. [30] and is defined as where is the degree of vertex .

In 1972, the first Zagreb index was introduced by [31]. Later on, the second Zagreb index was introduced by [32]. Both, the first and the second, Zagreb indexes are defined as Furtula et al. [33] proposed the modified version of index, which is called augmented Zagreb index. It is defined as The geometric arithmetic index of a graph is introduced by Vukievi et al. [34] and is defined as The fourth version of atom bond connectivity index of a graph is introduced by Ghorbhani et al. [35] and is defined as where and .

The fifth version of geometric arithmetic index of a graph is introduced by Graovoc et al. [36] and is defined as The Sanskruti index of a molecular graph G is introduced by S. M. Hosamani [24] in . It is defined as Moreover the paper is organized as follows.

We compute additive topological indices based on degree concept of graph for the fractal and Cayley tree dendrimers. We compute the atom bond connectivity , geometric arithmetic , general Randi only, first and second Zagreb, augmented Zagreb, , , and Sanskruti indices for fractal tree. Also, we compute the atom bond connectivity , geometric arithmetic , general Randi only, augmented Zagreb, , , and Sanskruti index. Furthermore, we compute close formulas of these indices for the fractal and Cayley tree dendrimers.

#### 2. Applications of Topological Indices

The first and second Zagreb indices were found to be helpful for calculation of the aggregate -electron energy of the particles inside particular rough articulations [37]. The Randic index is a topological descriptor related to a great deal of synthetic qualities of atoms and was discovered parallel to processing the boiling point and Kovats constants of the particles. The particle bond network () index connects to the security of direct alkanes and stretched alkanes and is used to process the strain vitality of cycloalkanes [38, 39]. In terms of physicoconcoction properties, the index has prescient control superior to the prescient energy of the Randic connectivity index [40]. These are among the graph invariants proposed for the estimation of skeletons of stretching of carbon atoms [41]. The Sanskruti index shows good correlation with entropy of an octane isomers.

##### 2.1. Fractal Tree Dendrimer

The word fractal comes from the Latin word meaning "to break". Fractals are geometric patterns in which every smaller part of the structure is similar to the whole. There are countless examples of fractals like sierpinski triangle, von koch curve, broccoli, ferns, lotus white flower, etc. In this paper, the proposed fractal tree dendrimers are considered where is the iterations. If , then is an edge connecting two vertices. is obtained from by using two steps on each existing edge in . The first step is to create a path of three links with the two same end points. The second step is to create new vertices for each of the two middle vertices in the path. After that, attach them to the middle vertices.

Figure 1 depicts structure for different values of . The fractal trees and for and are depicted in Figures 2(a) and 2(b), respectively.