# On Topological Indices of Fractal and Cayley Tree Type Dendrimers

Authors: Muhammad Imran; Abdul Qudair Baig; Waqas Khalid
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## 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 forα={-1,1,1/2,-1/2}. Furthermore, we give closed analytical results of these indices for fractal trees and Cayley trees.


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## Body

## 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, Randic´ index, Zagreb indices, and GA and ABC index, are used to predict the chemical and physical properties and bioactivities of different compounds and structures.The graphG 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 (BID) 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 S(G) of line graphs of subdivision graphs of 2D-lattice, nanotube, nanotorus of TUC4C8[p,q] and polycyclic aromatic hydrocarbons PAHk 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.(1)R-1/2G=∑uv∈EG1du×dv.Bolloba´s et al. [25] and Amic et al. [26] introduced the general Randic´ index independently. The properties and useful results of general Randic´ are referred to in [27–29]. It is defined as follows: (2)RαG=∑uv∈EGdu×dvα, where α∈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 (3)ABCG=∑uv∈EGdu+dv-2du×dv, where du is the degree of vertex u.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(4)M1G=∑uv∈EGdu+dv.M2G=∑uv∈EGdu×dv. Furtula et al. [33] proposed the modified version of ABC index, which is called augmented Zagreb index. It is defined as (5)AZIG=∑uv∈EGdu×dvdu+dv-23. The geometric arithmetic index GA of a graph G is introduced by Vukic˘evic´ et al. [34] and is defined as (6)GAG=∑uv∈EG2du×dvdu+dv. The fourth version of atom bond connectivity index ABC4 of a graph G is introduced by Ghorbhani et al. [35] and is defined as (7)ABC4G=∑uv∈EGSu+Sv-2Su×Sv, where Su=∑uv∈E(G)dv and Sv=∑uv∈E(G)du.The fifth version of geometric arithmetic indexGA5 of a graph G is introduced by Graovoc et al. [36] and is defined as (8)GA5G=∑uv∈EG2Su×SvSu+Sv. The Sanskruti index S(G) of a molecular graph G is introduced by S. M. Hosamani [24] in 2016. It is defined as (9)SG=∑uv∈EGSu×SvSu+Sv-23. 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 connectivityABC, geometric arithmetic GA, general Randic´α={-1,1,1/2,-1/2} only, first and second Zagreb, augmented Zagreb, GA5, ABC4, and Sanskruti indices for fractal tree. Also, we compute the atom bond connectivity ABC, geometric arithmetic GA, general Randic´α={-1,1,1/2,-1/2} only, augmented Zagreb, GA5, ABC4, 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 aggregateπ-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 (ABC) 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 GA 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 S(G) 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 consideredFp where p≥0 is the iterations. If p=0, then F0 is an edge connecting two vertices. Fp is obtained from Fp-1 by using two steps on each existing edge in Fp-1. The first step is to create a path of three links with the two same end points. The second step is to create k new vertices for each of the two middle vertices in the path. After that, attach them to the middle vertices.Figure1 depicts structure for different values of k. The fractal trees F3 and F4 for k=2 and k=3 are depicted in Figures 2(a) and 2(b), respectively.Figure 1
Construction model for next generations of the fractal trees.Figure 2
(a)The representation of fractal treeF3 for k=2 is illustrated. (b) The representation of fractal tree F4 for k=3 is illustrated.
(a) (b)Firstly, we compute above-mentioned topological indices for fractal tree dendrimer by using edge partition technique and graph theoretic methods. InFp, the pendent vertices are 42pk-28k+14p-8, the 4 degree vertices are 7p-5, and the k+2 degree vertices are 42p-28. The edge set of Fp is divided into three partitions based on the degree of end vertices. The first edge partition consists of 42pk-28k+14p-8 edges uv, where du=1 and dv=k+2. The second edge partition consists of 28p-20 edges uv, where du=4 and dv=k+2. The third edge partition consists of 21p-14 edges uv, where du=dv=k+2. The cardinality of vertices and edges in Fp is 42pk-28k+63p-41 and 42pk-28k+63p-42, respectively. The tabular representation of edge partition technique is depicted in Table 1.Table 1
Edge partition ofFp based on degrees of end vertices of each edge.
(du,dv) Frequency (1,k+2) 42 p k - 28 k + 14 p - 8 (4,k+2) 28 p - 20 (k+2,k+2) 21 p - 14Theorem 1.
Consider the fractal tree dendrimerFp with p,k≥2. Its atom bond connectivity index is as follows: (10)ABCFp=1k+242pkk+1k+2-28kk+1k+2+7p2k+1k+2+2k+2k+4+32k+1-24k+1k+2+5k+4k+2+72k+1Proof.
Let fractal tree dendrimerFp be a tree graph for p iterations. The atom bond connectivity index of Fp can be computed by using Table 1 in the following formula. (11)ABCFp=∑uv∈EFpdu+dv-2du×dvABCFp=42pk-28k+14p-81+k+2-21×k+2+28p-204+k+2-24×k+2+21p-14k+2+k+2-2k+2×k+2After simplification of the above form, we get our required result of (10).Theorem 2.
Consider the fractal tree dendrimerFp with p,k≥2. Its geometric arithmetic index is as follows: (12)GAFp=21p-14+k+284pkk+3-56kk+3+28pk+3+112pk+6-16k+3-80k+6Proof.
Let fractal tree dendrimersFp be a chemical tree graph for p iterations. The geometric arithmetic index of Fp can be computed by using Table 1 in the following formula: (13)GAFp=∑uv∈EFp2du×dvdu+dv.GAFp=42pk-28k+14p-821×k+21+k+2+28p-2024×k+24+k+2+21p-142k+2×k+2k+2+k+2After simplification of the above form, we got our required versions of (12).Theorem 3.
Consider the fractal tree dendrimerFp with p,k≥2. Its first and second Zagreb index are as follows: (14)M1Fp=42k2p+210pk-28k2-140k+294p-200 and (15)M2Fp=63k2p+294pk-42k2-200k+336p-232.Proof.
Let fractal tree dendrimerFp be a tree graph for p iterations. The first and second Zagreb index of Fp can be computed by using Table 1. First we compute first Zagreb index of Fp. (16)M1Fp=∑uv∈EFpdu+dv.M1Fp=42pk-28k+14p-81+k+2+28p-204+k+2+21p-14k+2+k+2M1Fp=42k2p+210pk-28k2-140k+294p-200The second Zagreb index of Fp can be computed by using the following formula: (17)M2Fp=∑uv∈EGdu×dv.M2Fp=42pk-28k+14p-81×k+2+28p-204×k+2+21p-14k+2×k+2M2Fp=63k2p+294pk-42k2-200k+336p-232Theorem 4.
Consider the fractal tree dendrimerFp with p,k≥2. Its general Randic´ index is as follows:(18)RαFp=63k2p+294pk-42k2-200k+336p-232,ifα=1,42k2p+105pk-28k2-69k+63p-40k+22,ifα=-1,21pk-14k+42p-28+k+2E+7p-6,ifα=12,42pk-28k+28p-18k+2+21p-14k+2,ifα=-12.Proof.
Let fractal tree dendrimerFp be a tree graph for p iterations. The general Randic´ index Rα of Fp can be computed by using Table 1. For α=1, the general Randic´ index takes the following form: (19)R1Fp=∑uv∈EFpdu×dvR1Fp=42pk-28k+14p-81×k+2+28p-204×k+2+21p-14k+2×k+2R1Fp=63k2p+294pk-42k2-200k+336p-232For α=-1, the formula of general Randic´ index takes the following form: (20)R-1Fp=∑uv∈EFp1du×dvR-1Fp=42pk-28k+14p-811×k+2+28p-2014×k+2+21p-141k+2×k+2R-1Fp=42k2p+105pk-28k2-69k+63p-40k+22For α=1/2, the formula of general Randic´ index takes the following form: (21)R1/2Fp=∑uv∈EFpdu×dvR1/2Fp=42pk-28k+14p-81×k+2+28p-204×k+2+21p-14k+2×k+2R1/2Fp=21pk-14k+42p-28+k+2E+7p-6For α=-1/2, the formula of general Randic´ index takes the following form: (22)R-1/2Fp=∑uv∈EFp1du×dvR-1/2Fp=42pk-28k+14p-811×k+2+28p-2014×k+2+21p-141k+2×k+2R-1/2Fp=42pk-28k+28p-18k+2+21p-14k+2.Theorem 5.
Consider the fractal tree dendrimerFp with p,k≥2. Its augmented Zagreb index is as follows: (23)AZIFp=21pk3-14k3+126pk2-84k2+588pk-392k+280p-176k+22k+13+2567p-5k+2k+43Proof.
Let fractal tree dendrimerFp be a tree graph for p iterations. The augmented Zagreb index of Fp can be computed by using Table 1 in the following formula. (24)AZIFp=∑uv∈EFpdu×dvdu+dv-23.AZIFp=42pk-28k+14p-81×k+21+k+2-23+28p-204×k+24+k+2-23+21p-14k+2×k+2k+2+k+2-23After simplification of the above form, we got (23).Table2 shows the edge partition of fractal tree dendrimer Fp 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.Table 2
Edge partition ofFp based on degree sum of end vertices of each edge.
( S u , S v ) frequency (k+2,2k+3) ( 14 p - 8 ) ( k + 1 ) (k+2,2k+6) 28 p k - 20 k (2k+3,2k+6) 14 p - 8 (2k+6,2k+6) 7 p - 6 (2k+6,4k+8) 28 p - 20Theorem 6.
Consider the fractal tree dendrimerFp with p,k≥2. Its fourth atom bond connectivity index is as follows: (25)ABC4Fp=7p-64k+102k+6+14p-82k+33k+13/2k+2+4k+72k+6+1+23/2k314p-10k+3Proof.
Let fractal tree dendrimerFp be a tree graph for p iterations. The fourth atom bond connectivity index of Fp can be computed by using Table 2 in the following formula: (26)ABC4Fp=∑uv∈EFpSu+Sv-2Su×SvABC4Fp=14p-8k+1k+2+2k+3-2k+2×2k+3+28pk-20kk+2+2k+6-2k+2×2k+6+14p-82k+3+2k+6-22k+3×2k+6+7p-62k+6+2k+6-22k+6×2k+6+28p-202k+6+4k+8-22k+6×4k+8After simplification of the above form, we got (25).Theorem 7.
Consider the fractal tree dendrimerFp with p,k≥2. Its fifth geometric arithmetic index is as follows: (27)GA5Fp=7p-6+7p-583k+42k+6k+23/23k+73k+8+47p-42k+3k+23k+5+2k+64k+9Proof.
Let fractal tree dendrimerFp be a tree graph for p iterations. The fifth geometric arithmetic index of Fp can be computed by using Table 2 in the following formula: (28)GA5Fp=∑uv∈EFp2Su×SvSu+Sv.GA5Fp=14p-8k+12k+2×2k+3k+2+2k+3+28pk-20k2k+2×2k+6k+2+2k+6+14p-822k+3×2k+62k+3+2k+6+7p-622k+6×2k+62k+6+2k+6+28p-2022k+6×4k+82k+6+4k+8After simplification of the above form, we got our result of (27).Theorem 8.
Consider the fractal tree dendrimerFp with p,k≥2. Its Sanskruti index is as follows: (29)SFp=1275+2k3-7+4k31105923+k3-7+4k327+27k+98k2+k3-6+7p+25+2k324524+k217112-22596p-33313p+128k4-5+7p+8k3-328+497p+k-49442+71260p+14p-8k+22k+3327k+12Proof.
Let fractal tree dendrimerFp be a tree graph for p iterations. The Sanskruti index of Fp can be computed by using Table 2 in the following formula: (30)SFp=∑uv∈EFpSu×SvSu+Sv-23SFp=14p-8k+1k+2×2k+3k+2+2k+3-23+28pk-20kk+2×2k+6k+2+2k+6-23+14p-82k+3×2k+62k+3+2k+6-23+7p-62k+6×2k+62k+6+2k+6-23+28p-202k+6×4k+82k+6+4k+8-23After simplification of the above form, we got (29).


## 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 consideredFp where p≥0 is the iterations. If p=0, then F0 is an edge connecting two vertices. Fp is obtained from Fp-1 by using two steps on each existing edge in Fp-1. The first step is to create a path of three links with the two same end points. The second step is to create k new vertices for each of the two middle vertices in the path. After that, attach them to the middle vertices.Figure1 depicts structure for different values of k. The fractal trees F3 and F4 for k=2 and k=3 are depicted in Figures 2(a) and 2(b), respectively.Figure 1
Construction model for next generations of the fractal trees.Figure 2
(a)The representation of fractal treeF3 for k=2 is illustrated. (b) The representation of fractal tree F4 for k=3 is illustrated.
(a) (b)Firstly, we compute above-mentioned topological indices for fractal tree dendrimer by using edge partition technique and graph theoretic methods. InFp, the pendent vertices are 42pk-28k+14p-8, the 4 degree vertices are 7p-5, and the k+2 degree vertices are 42p-28. The edge set of Fp is divided into three partitions based on the degree of end vertices. The first edge partition consists of 42pk-28k+14p-8 edges uv, where du=1 and dv=k+2. The second edge partition consists of 28p-20 edges uv, where du=4 and dv=k+2. The third edge partition consists of 21p-14 edges uv, where du=dv=k+2. The cardinality of vertices and edges in Fp is 42pk-28k+63p-41 and 42pk-28k+63p-42, respectively. The tabular representation of edge partition technique is depicted in Table 1.Table 1
Edge partition ofFp based on degrees of end vertices of each edge.
(du,dv) Frequency (1,k+2) 42 p k - 28 k + 14 p - 8 (4,k+2) 28 p - 20 (k+2,k+2) 21 p - 14Theorem 1.
Consider the fractal tree dendrimerFp with p,k≥2. Its atom bond connectivity index is as follows: (10)ABCFp=1k+242pkk+1k+2-28kk+1k+2+7p2k+1k+2+2k+2k+4+32k+1-24k+1k+2+5k+4k+2+72k+1Proof.
Let fractal tree dendrimerFp be a tree graph for p iterations. The atom bond connectivity index of Fp can be computed by using Table 1 in the following formula. (11)ABCFp=∑uv∈EFpdu+dv-2du×dvABCFp=42pk-28k+14p-81+k+2-21×k+2+28p-204+k+2-24×k+2+21p-14k+2+k+2-2k+2×k+2After simplification of the above form, we get our required result of (10).Theorem 2.
Consider the fractal tree dendrimerFp with p,k≥2. Its geometric arithmetic index is as follows: (12)GAFp=21p-14+k+284pkk+3-56kk+3+28pk+3+112pk+6-16k+3-80k+6Proof.
Let fractal tree dendrimersFp be a chemical tree graph for p iterations. The geometric arithmetic index of Fp can be computed by using Table 1 in the following formula: (13)GAFp=∑uv∈EFp2du×dvdu+dv.GAFp=42pk-28k+14p-821×k+21+k+2+28p-2024×k+24+k+2+21p-142k+2×k+2k+2+k+2After simplification of the above form, we got our required versions of (12).Theorem 3.
Consider the fractal tree dendrimerFp with p,k≥2. Its first and second Zagreb index are as follows: (14)M1Fp=42k2p+210pk-28k2-140k+294p-200 and (15)M2Fp=63k2p+294pk-42k2-200k+336p-232.Proof.
Let fractal tree dendrimerFp be a tree graph for p iterations. The first and second Zagreb index of Fp can be computed by using Table 1. First we compute first Zagreb index of Fp. (16)M1Fp=∑uv∈EFpdu+dv.M1Fp=42pk-28k+14p-81+k+2+28p-204+k+2+21p-14k+2+k+2M1Fp=42k2p+210pk-28k2-140k+294p-200The second Zagreb index of Fp can be computed by using the following formula: (17)M2Fp=∑uv∈EGdu×dv.M2Fp=42pk-28k+14p-81×k+2+28p-204×k+2+21p-14k+2×k+2M2Fp=63k2p+294pk-42k2-200k+336p-232Theorem 4.
Consider the fractal tree dendrimerFp with p,k≥2. Its general Randic´ index is as follows:(18)RαFp=63k2p+294pk-42k2-200k+336p-232,ifα=1,42k2p+105pk-28k2-69k+63p-40k+22,ifα=-1,21pk-14k+42p-28+k+2E+7p-6,ifα=12,42pk-28k+28p-18k+2+21p-14k+2,ifα=-12.Proof.
Let fractal tree dendrimerFp be a tree graph for p iterations. The general Randic´ index Rα of Fp can be computed by using Table 1. For α=1, the general Randic´ index takes the following form: (19)R1Fp=∑uv∈EFpdu×dvR1Fp=42pk-28k+14p-81×k+2+28p-204×k+2+21p-14k+2×k+2R1Fp=63k2p+294pk-42k2-200k+336p-232For α=-1, the formula of general Randic´ index takes the following form: (20)R-1Fp=∑uv∈EFp1du×dvR-1Fp=42pk-28k+14p-811×k+2+28p-2014×k+2+21p-141k+2×k+2R-1Fp=42k2p+105pk-28k2-69k+63p-40k+22For α=1/2, the formula of general Randic´ index takes the following form: (21)R1/2Fp=∑uv∈EFpdu×dvR1/2Fp=42pk-28k+14p-81×k+2+28p-204×k+2+21p-14k+2×k+2R1/2Fp=21pk-14k+42p-28+k+2E+7p-6For α=-1/2, the formula of general Randic´ index takes the following form: (22)R-1/2Fp=∑uv∈EFp1du×dvR-1/2Fp=42pk-28k+14p-811×k+2+28p-2014×k+2+21p-141k+2×k+2R-1/2Fp=42pk-28k+28p-18k+2+21p-14k+2.Theorem 5.
Consider the fractal tree dendrimerFp with p,k≥2. Its augmented Zagreb index is as follows: (23)AZIFp=21pk3-14k3+126pk2-84k2+588pk-392k+280p-176k+22k+13+2567p-5k+2k+43Proof.
Let fractal tree dendrimerFp be a tree graph for p iterations. The augmented Zagreb index of Fp can be computed by using Table 1 in the following formula. (24)AZIFp=∑uv∈EFpdu×dvdu+dv-23.AZIFp=42pk-28k+14p-81×k+21+k+2-23+28p-204×k+24+k+2-23+21p-14k+2×k+2k+2+k+2-23After simplification of the above form, we got (23).Table2 shows the edge partition of fractal tree dendrimer Fp 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.Table 2
Edge partition ofFp based on degree sum of end vertices of each edge.
( S u , S v ) frequency (k+2,2k+3) ( 14 p - 8 ) ( k + 1 ) (k+2,2k+6) 28 p k - 20 k (2k+3,2k+6) 14 p - 8 (2k+6,2k+6) 7 p - 6 (2k+6,4k+8) 28 p - 20Theorem 6.
Consider the fractal tree dendrimerFp with p,k≥2. Its fourth atom bond connectivity index is as follows: (25)ABC4Fp=7p-64k+102k+6+14p-82k+33k+13/2k+2+4k+72k+6+1+23/2k314p-10k+3Proof.
Let fractal tree dendrimerFp be a tree graph for p iterations. The fourth atom bond connectivity index of Fp can be computed by using Table 2 in the following formula: (26)ABC4Fp=∑uv∈EFpSu+Sv-2Su×SvABC4Fp=14p-8k+1k+2+2k+3-2k+2×2k+3+28pk-20kk+2+2k+6-2k+2×2k+6+14p-82k+3+2k+6-22k+3×2k+6+7p-62k+6+2k+6-22k+6×2k+6+28p-202k+6+4k+8-22k+6×4k+8After simplification of the above form, we got (25).Theorem 7.
Consider the fractal tree dendrimerFp with p,k≥2. Its fifth geometric arithmetic index is as follows: (27)GA5Fp=7p-6+7p-583k+42k+6k+23/23k+73k+8+47p-42k+3k+23k+5+2k+64k+9Proof.
Let fractal tree dendrimerFp be a tree graph for p iterations. The fifth geometric arithmetic index of Fp can be computed by using Table 2 in the following formula: (28)GA5Fp=∑uv∈EFp2Su×SvSu+Sv.GA5Fp=14p-8k+12k+2×2k+3k+2+2k+3+28pk-20k2k+2×2k+6k+2+2k+6+14p-822k+3×2k+62k+3+2k+6+7p-622k+6×2k+62k+6+2k+6+28p-2022k+6×4k+82k+6+4k+8After simplification of the above form, we got our result of (27).Theorem 8.
Consider the fractal tree dendrimerFp with p,k≥2. Its Sanskruti index is as follows: (29)SFp=1275+2k3-7+4k31105923+k3-7+4k327+27k+98k2+k3-6+7p+25+2k324524+k217112-22596p-33313p+128k4-5+7p+8k3-328+497p+k-49442+71260p+14p-8k+22k+3327k+12Proof.
Let fractal tree dendrimerFp be a tree graph for p iterations. The Sanskruti index of Fp can be computed by using Table 2 in the following formula: (30)SFp=∑uv∈EFpSu×SvSu+Sv-23SFp=14p-8k+1k+2×2k+3k+2+2k+3-23+28pk-20kk+2×2k+6k+2+2k+6-23+14p-82k+3×2k+62k+3+2k+6-23+7p-62k+6×2k+62k+6+2k+6-23+28p-202k+6×4k+82k+6+4k+8-23After simplification of the above form, we got (29).


## 3. Cayley Tree Dendrimer

The Cayley tree is a kind of dendrimers, also called Bethe lattice. The construction procedure of Cayley treeCs,t(s≥3,t≥0) consists of t iterations. s is the number of nodes at first iteration. Cs,0, for t=0, consist of only a central vertex. For t=1, Cs,1 is obtained by creating s nodes and attaching them to the central vertex by an edge. For t>1, the Cayley tree Cs,t is obtained from Cs,t-1 by creating s-1 nodes and attaching them to each of the pendent vertices of Cs,t-1. The Cayley tree network C4,3 is depicted in Figure 3.Figure 3
The representation of Cayley treeC4,3 is illustrated.InCs,t, the pendent vertices are s(s-1)n-1 and the s degree vertices are 2∑i=1ts-1i-1-(s-1)n-1. The edge set of Cs,t is divided into two partitions based on the degree of end vertices. The first edge partition consists of s(s-1)n-1 edges uv, where du=1 and dv=s. The second edge partition consists of s∑i=1ts-1i-1-s(s-1)t-1 edges uv, where du=dv=s. The cardinality of vertices and edges in Cs,t is 2∑i=1ts-1i-1+s-1t and s∑i=1t(s-1)i-1, respectively. The tabular representation of edge partition technique is depicted in Table 3.Table 3
Edge partition ofCs,t based on degrees of end vertices of each edge.
( d u , v v ) Frequency (1,s) s s - 1 t - 1 (s,s) s ∑ i = 1 t ‍ s - 1 i - 1 - s s - 1 t - 1Theorem 9.
Consider the Cayley tree dendrimerCs,t with s,t≥3. Its atom bond connectivity index is as follows:(31)ABCCs,t=ss-1t-1/2+2s-1∑i=1ts-1i-1-s-1t-1Proof.
Let Cayley tree dendrimerCs,t be a tree graph for t iterations. The atom bond connectivity index of Cs,t can be computed by using Table 3 in the following formula: (32)ABCCs,t=∑uv∈ECs,tdu+dv-2du×dvABCCs,t=ss-1t-11+s-21×s+s∑i=1ts-1i-1-ss-1t-1s+s-2s×sAfter simplification of the above form, we got (31).Theorem 10.
Consider the Cayley tree dendrimerCs,t with s,t≥3. Its geometric arithmetic index is as follows: (33)GACs,t=ss-1t-12s-s-1+ss+1∑i=1ts-1i-1s+1Proof.
Let Cayley tree dendrimerCs,t be a tree graph for t iterations. The geometric arithmetic index of Cs,t can be computed by using Table 3 in the following formula: (34)GACs,t=∑uv∈ECs,t2du×dvdu+dv.GACs,t=ss-1t-121×s1+s+s∑i=1ts-1i-1-ss-1t-12s×ss+sAfter simplification of the above form, we got (33).Theorem 11.
Consider the Cayley tree dendrimerCs,t with s,t≥3. Its general Randic´ index is as follows: (35)RαCs,t=s3∑i=1ts-1i-1-s2s-1t,ifα=1,1ss-1t+∑i=1ts-1i-1,ifα=-1,s2∑i=1ts-1i-1+ss-1t-1s-s,ifα=12,∑i=1ts-1i-1-s-1t-11-s,ifα=-12.Proof.
Let Cayley tree dendrimerCs,t be a tree graph for t iterations. The general Randic´ index Rα of Cs,t can be computed by using Table 3. For α=1, the general Randic´ index takes the following form: (36)R1Cs,t=∑uv∈ECs,tdu×dvR1Cs,t=ss-1t-11×s+s∑i=1ts-1i-1-ss-1t-1s×ss3∑i=1ts-1i-1-s2s-1tFor α=-1, the formula of general Randic´ index takes the following form: (37)R-1Cs,t=∑uv∈ECs,t1du×dvR-1Cs,t=ss-1t-111×s+s∑i=1ts-1i-1-ss-1t-11s×sR-1Cs,t=1ss-1t+∑i=1ts-1i-1For α=1/2, the formula of general Randic´ index takes the following form: (38)R1/2Cs,t=∑uv∈ECs,tdu×dvR1/2Cs,t=ss-1t-11×s+s∑i=1ts-1i-1-ss-1t-1s×sR1/2Cs,t=s2∑i=1ts-1i-1+ss-1t-1s-sFor α=-1/2, the formula of general Randic´ index takes the following form: (39)R-1/2Cs,t=∑uv∈ECs,t1du×dvR-1/2Cs,t=ss-1t-111×s+s∑i=1ts-1i-1-ss-1t-11s×sR-1/2Cs,t=∑i=1ts-1i-1-s-1t-11-s.Theorem 12.
Consider the Cayley tree dendrimerCs,t with s,t≥3. Its augmented Zagreb index is as follows: (40)AZICs,t=s488-s3s-1t-4+s3s-13∑i=1ts-1i-1Proof.
Let Cayley tree dendrimerCs,t be a tree graph for t iterations. The augmented Zagreb index of Cs,t can be computed by using Table 3 in the following formula: (41)AZICs,t=∑uv∈ECs,tdu×dvdu+dv-23.AZICs,t=ss-1t-11×s1+s-23+s∑i=1ts-1i-1-ss-1t-1s×ss+s-23After simplification of the above form, we got (40).Table4 shows the edge partition of Cayley tree dendrimer Cs,t 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.Table 4
Edge partition ofCs,t based on degree sum of end vertices of each edge.
( S u , S v ) frequency (s,2s-1) s s - 1 t - 1 (2s-1,s2) s s - 1 t - 2 (s2,s2) s ∑ i = 1 t s - 1 i - 1 - s 2 s - 1 t - 2Theorem 13.
Consider the Cayley tree dendrimerCs,t with s,t≥3. Its fourth atom bond connectivity index is as follows: (42)ABC4Cs,t=s-1t-3/2s+32s-1-2s+1+ss-1t-1/23s2s-1+2s2-1s∑i=1ts-1i-1Proof.
Let Cayley tree dendrimerCs,t be a tree graph for t iterations. The fourth atom bond connectivity index of Cs,t can be computed by using Table 4 in the following formula: (43)ABC4Cs,t=∑uv∈ECs,tSu+Sv-2Su×SvABC4Cs,t=ss-1t-1s+2s-1-2s×2s-1+ss-1t-22s-1+s2-22s-1×s2+s∑i=1ts-1i-1-s2s-1t-2s2+s2-2s2×s2After simplification of the above form, we got (42).Theorem 14.
Consider the Cayley tree dendrimerCs,t with s,t≥3. Its fifth geometric arithmetic index is as follows: (44)GA5Cs,t=ss-1t-12s2s-13s-1-ss-1+2s2s-1s-1s2+2s-1+s∑i=1ts-1i-1Proof.
Let Cayley tree dendrimerCs,t be a tree graph for t iterations. The fifth geometric arithmetic index of Cs,t can be computed by using Table 4 in the following formula: (45)GA5Cs,t=∑uv∈ECs,t2Su×SvSu+Sv.GA5Cs,t=ss-1t-12s×2s-1s+2s-1+ss-1t-222s-1×s22s-1+s2+s∑i=1ts-1i-1-s2s-1t-22s2×s2s2+s2After simplification of the above form, we got (44).Theorem 15.
Consider the Cayley tree dendrimerCs,t with s,t≥3. Its Sanskruti index is as follows: (46)SCs,t=s4s-1t-42s-1327+s2s-13s-1s+33-s102s+13s-1+s132s2-13∑i=1ts-1i-1Proof.
Let Cayley tree dendrimerCs,t be a tree graph for t iterations. The Sanskruti index of Cs,t can be computed by using Table 4 in the following formula: (47)SCs,t=∑uv∈ECs,tSu×SvSu+Sv-23SCs,t=ss-1t-1s×2s-1s+2s-1-23+ss-1t-22s-1×s22s-1+s2-23+s∑i=1ts-1i-1-s2s-1t-2s2×s2s2+s2-23After simplification of the above form, we got (46).


## 4. Conclusion

In this paper, we have computed some degree based topological indices for fractal tree and Cayley tree dendrimers. We have computed the atom bond connectivityABC, geometric arithmetic GA, general Randic´, first and second Zagreb, augmented Zagreb, GA5, ABC4, and Sanskruti indices for fractal tree. Also, we have computed the atom bond connectivity ABC, geometric arithmetic GA, general Randic´, augmented Zagreb, GA5, ABC4, and Sanskruti index. Furthermore, we have computed closed results of above-mentioned indices for both fractal and Cayley tree dendrimers.



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