On Topological Indices of Fractal and Cayley Tree Type Dendrimers

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 inmathematical 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 andCayley 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 α = {−1, 1, 1/2, −1/2}. Furthermore, we give closed analytical results of these indices for fractal trees and Cayley trees.


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 ć 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][4][5][6][7][8][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][18][19][20][21][22].The Sanskruti index () of line graphs of subdivision graphs of 2D-lattice, nanotube, nanotorus of  4  8 [, ] 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.
Bollob á s et al. [25] and Amic et al. [26] introduced the general Randi ć index independently.The properties and useful results of general Randi ć are referred to in [27][28][29].It is defined as follows: where  ∈ R.
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 Vuki c evi ć et al. [34] and is defined as The fourth version of atom bond connectivity index  4 of a graph  is introduced by Ghorbhani et al. [35] and is defined as where   = ∑ V∈()  V and  V = ∑ V∈()   .
The fifth version of geometric arithmetic index  5 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 2016.It is defined as Moreover the paper is organized as follows.

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.

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  ≥ 0 is the iterations.If  = 0, then  0 is an edge connecting two vertices.  is obtained from  −1 by using two steps on each existing edge in  −1 .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.
Proof.Let fractal tree dendrimer   be a tree graph for  iterations.The atom bond connectivity index of   can be computed by using Table 1 in the following formula.

𝐴𝐵𝐶 (𝐹
After simplification of the above form, we get our required result of (10).
Theorem 2. Consider the fractal tree dendrimer   with ,  ≥ 2. Its geometric arithmetic index is as follows: Proof.Let fractal tree dendrimers   be a chemical tree graph for  iterations.The geometric arithmetic index of   can be computed by using Table 1 in the following formula: After simplification of the above form, we got our required versions of (12).
Theorem 3. Consider the fractal tree dendrimer   with ,  ≥ 2. Its first and second Zagreb index are as follows: and Proof.Let fractal tree dendrimer   be a tree graph for  iterations.The first and second Zagreb index of   can be computed by using Table 1.First we compute first Zagreb index of   .
The second Zagreb index of   can be computed by using the following formula: Theorem 4. Consider the fractal tree dendrimer   with ,  ≥ 2. Its general Randi ć index is as follows: Proof.Let fractal tree dendrimer   be a tree graph for  iterations.The general Randi ć index   of   can be computed by using Table 1.For  = 1, the general Randi ć index takes the following form: For  = −1, the formula of general Randi ć index takes the following form: For  = 1/2, the formula of general Randi ć index takes the following form: For  = −1/2, the formula of general Randi ć index takes the following form: Theorem 5. Consider the fractal tree dendrimer   with ,  ≥ 2. Its augmented Zagreb index is as follows: Proof.Let fractal tree dendrimer   be a tree graph for  iterations.The augmented Zagreb index of   can be computed by using Table 1 in the following formula.
After simplification of the above form, we got (23).
Table 2 shows the edge partition of fractal tree dendrimer   based on the sum of degrees of end vertices of each edge.This table is used in computing the fourth atom bond connectivity and fifth geometric arithmetic index.Theorem 6.Consider the fractal tree dendrimer   with ,  ≥ 2. Its fourth atom bond connectivity index is as follows: Proof.Let fractal tree dendrimer   be a tree graph for  iterations.The fourth atom bond connectivity index of   can be computed by using Table 2 in the following formula: Proof.Let fractal tree dendrimer   be a tree graph for  iterations.The fifth geometric arithmetic index of   can be computed by using Table 2 in the following formula: After simplification of the above form, we got our result of (27).
Proof.Let fractal tree dendrimer   be a tree graph for  iterations.The Sanskruti index of   can be computed by using Table 2 in the following formula: After simplification of the above form, we got (29).

Cayley Tree Dendrimer
The Cayley tree is a kind of dendrimers, also called Bethe lattice.The construction procedure of Cayley tree  , ( ≥  Proof.Let Cayley tree dendrimer  , be a tree graph for  iterations.The atom bond connectivity index of  , can be computed by using Table 3 in the following formula: After simplification of the above form, we got (31).Proof.Let Cayley tree dendrimer  , be a tree graph for  iterations.The geometric arithmetic index of  , can be computed by using Table 3 in the following formula: After simplification of the above form, we got (33).
Theorem 11.Consider the Cayley tree dendrimer  , with ,  ≥ 3. Its general Randi ć index is as follows: Proof.Let Cayley tree dendrimer  , be a tree graph for  iterations.The general Randi ć index   of  , can be computed by using Table 3.For  = 1, the general Randi ć index takes the following form: For  = −1, the formula of general Randi ć index takes the following form: For  = 1/2, the formula of general Randi ć index takes the following form: For  = −1/2, the formula of general Randi ć index takes the following form: Theorem 12. Consider the Cayley tree dendrimer  , with ,  ≥ 3. Its augmented Zagreb index is as follows: Proof.Let Cayley tree dendrimer  , be a tree graph for  iterations.The augmented Zagreb index of  , can be computed by using Table 3 in the following formula: After simplification of the above form, we got (40).
Table 4 shows the edge partition of Cayley tree dendrimer  , based on the sum of degrees of end vertices of each edge.This table is used in computing the fourth atom bond connectivity and fifth geometric arithmetic index.Proof.Let Cayley tree dendrimer  , be a tree graph for  iterations.The fourth atom bond connectivity index of  , can be computed by using Table 4 in the following formula: Proof.Let Cayley tree dendrimer  , be a tree graph for  iterations.The fifth geometric arithmetic index of  , can be computed by using Table 4 in the following formula: ) +  13   (2 ( 2 − 1))

Figure 1 :
Figure 1: Construction model for next generations of the fractal trees.

Figure 1
depicts structure for different values of .The fractal trees  3 and  4 for  = 2 and  = 3 are depicted in Figures 2(a) and 2(b), respectively.Firstly, we compute above-mentioned topological indices for fractal tree dendrimer by using edge partition technique and graph theoretic methods.In   , the pendent vertices are 42 − 28 + 14 − 8, the 4 degree vertices are 7 − 5, and the +2 degree vertices are 42−28.The edge set of   is divided into three partitions based on the degree of end vertices.The first edge partition consists of 42 − 28 + 14 − 8 edges V, where   = 1 and  V =  + 2. The second edge partition consists of 28−20 edges V, where   = 4 and  V = +2.The third edge partition consists of 21 − 14 edges V, where   =  V = +2.The cardinality of vertices and edges in   is 42− 28 + 63 − 41 and 42 − 28 + 63 − 42, respectively.The

Figure 2 :
Figure 2: (a)The representation of fractal tree  3 for  = 2 is illustrated.(b) The representation of fractal tree  4 for  = 3 is illustrated.

Table 1 :
Edge partition of   based on degrees of end vertices of each edge.
tabular representation of edge partition technique is depicted in Table1.Theorem 1.Consider the fractal tree dendrimer   with ,  ≥ 2. Its atom bond connectivity index is as follows:

Table 2 :
Edge partition of   based on degree sum of end vertices of each edge.
Consider the fractal tree dendrimer   with ,  ≥ 2. Its fifth geometric arithmetic index is as follows:

Table 3 :
Edge partition of  , based on degrees of end vertices of each edge.

Table 4 :
Edge partition of  , based on degree sum of end vertices of each edge.