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On Topological Indices of Fractal and Cayley Tree Type Dendrimers.

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 graph G 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 [TUC.sub.4][C.sub.8][p, q] and polycyclic aromatic hydrocarbons [PAH.sub.k] 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.

[R.sub.-1/2](G) = [summation over (uv[member of]E(G))] 1/[square root of ([d.sub.u] x [d.sub.v])]. (1)

Bollobas 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:

[R.sub.[alpha]](G) = [summation over (uv[member of]E(G))][([d.sub.u] x [d.sub.v]).sup.[alpha]], (2)

where [alpha] [member of] 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

ABC(G) = [summation over (uv[member of]E(G))] [square root of ([[d.sub.u] + [d.sub.v] - 2]/[[d.sub.u] x [d.sub.v]])], (3)

where [d.sub.u] 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

[mathematical expression not reproducible]. (4)

Furtula et al. [33] proposed the modified version of ABC index, which is called augmented Zagreb index. It is defined as

AZI(G) = [summation over (uv[member of]E(G))] [([[d.sub.u] x [d.sub.v]]/[[d.sub.u] + [d.sub.v] - 2]).sup.3]. (5)

The geometric arithmetic index GA of a graph G is introduced by Vukicevic et al. [34] and is defined as

GA(G) = [summation over (uv[member of]E(G))] 2[square root of ([d.sub.u] x [d.sub.v])]/[[d.sub.u] + [d.sub.v]]. (6)

The fourth version of atom bond connectivity index [ABC.sub.4] of a graph G is introduced by Ghorbhani et al. [35] and is defined as

[ABC.sub.4](G) = [summation over (uv[member of]E(G))] [square root of ([[S.sub.u] + [S.sub.v] - 2]/[[S.sub.u] x [S.sub.v]])], (7)

where [S.sub.u] = [[summation].sub.uv[member of]E(G)][d.sub.v] and [S.sub.v] = [[summation].sub.uv[member of]E(G)][d.sub.u].

The fifth version of geometric arithmetic index [GA.sub.5] of a graph G is introduced by Graovoc et al. [36] and is defined as

[GA.sub.5](G) = [summation over (uv[member of]E(G))] 2[square root of ([S.sub.u] x [S.sub.v])]/[[S.sub.u] + [S.sub.v]]. (8)

The Sanskruti index S(G) of a molecular graph G is introduced by S. M. Hosamani [24] in 2016. It is defined as

S(G) = [summation over (uv[member of]E(G))] ([[S.sub.u] x [S.sub.v]]/[[S.sub.u] + [S.sub.v] - 2]). (9)

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 ABC, geometric arithmetic GA, general Randic [alpha] = {-1, 1, 1/2, -1/2} only, first and second Zagreb, augmented Zagreb, [GA.sub.5], [ABC.sub.4], and Sanskruti indices for fractal tree. Also, we compute the atom bond connectivity ABC, geometric arithmetic GA, general Randic [alpha] = {-1, 1, 1/2, -1/2} only, augmented Zagreb, [GA.sub.5], [ABC.sub.4], 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 [pi]-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 considered [F.sub.p] where p [greater than or equal to] 0 is the iterations. If p = 0, then [F.sub.0] is an edge connecting two vertices. [F.sub.p] is obtained from [F.sub.p-1] by using two steps on each existing edge in [F.sub.p-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.

Figure 1 depicts structure for different values of k. The fractal trees [F.sub.3] and [F.sub.4] for k = 2 and k = 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 [F.sub.p], the pendent vertices are 42pk - 28k +1 4p - 8, the 4 degree vertices are 7p - 5, and the k + 2 degree vertices are 42p - 28. The edge set of [F.sub.p] 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 [d.sub.u] = 1 and [d.sub.v] = k + 2. The second edge partition consists of 28p-20 edges uv, where [d.sub.u] = 4 and [d.sub.v] = k+2. The third edge partition consists of 21p- 14 edges uv, where [d.sub.u] = [d.sub.v] = k+2. The cardinality ofvertices and edges in [F.sub.p] is 42pk-28k + 63p - 41 and 42pk - 28k + 63p - 42, respectively. The tabular representation of edge partition technique is depicted in Table 1.

Theorem 1. Consider the fractal tree dendrimer [F.sub.p] with p, k [greater than or equal to] 2. Its atom bond connectivity index is as follows:

[mathematical expression not reproducible] (10)

Proof. Let fractal tree dendrimer [F.sub.p] be a tree graph for p iterations. The atom bond connectivity index of [F.sub.p] can be computed by using Table 1 in the following formula.

[mathematical expression not reproducible] (11)

After simplification of the above form, we get our required result of (10).

Theorem 2. Consider the fractal tree dendrimer [F.sub.p] with p, k [greater than or equal to] 2. Its geometric arithmetic index is as follows:

[mathematical expression not reproducible] (12)

Proof. Let fractal tree dendrimers [F.sub.p] be a chemical tree graph for p iterations. The geometric arithmetic index of [F.sub.p] can be computed by using Table 1 in the following formula:

[mathematical expression not reproducible] (13)

After simplification of the above form, we got our required versions of (12).

Theorem 3. Consider the fractal tree dendrimer [F.sub.p] with p, k [greater than or equal to] 2. Its first and second Zagreb index are as follows:

[M.sub.1]([F.sub.p]) = 42[k.sup.2]p + 210pk - 28[k.sup.2] - 140k + 294p - 200 (14)

and

[M.sub.2]([F.sub.p]) = 63[k.sup.2]p + 294pk - 42[k.sup.2] - 200k + 336p - 232. (15)

Proof. Let fractal tree dendrimer [F.sub.p] be a tree graph for p iterations. The first and second Zagreb index of [F.sub.p] can be computed by using Table 1. First we compute first Zagreb index of [F.sub.p].

[mathematical expression not reproducible] (16)

The second Zagreb index of [F.sub.p] can be computed by using the following formula:

[mathematical expression not reproducible] (17)

Theorem 4. Consider the fractal tree dendrimer [F.sub.p] with p, k [greater than or equal to] 2. Its general Randic index is as follows:

[mathematical expression not reproducible]. (18)

Proof. Let fractal tree dendrimer [F.sub.p] be a tree graph for p iterations. The general Randic index [R.sub.[alpha]] of [F.sub.p] can be computed by using Table 1. For [alpha] =1, the general Randic index takes the following form:

[mathematical expression not reproducible] (19)

For [alpha] = -1, the formula of general Randic index takes the following form:

[mathematical expression not reproducible] (20)

For [alpha] = 1/2, the formula of general Randic index takes the following form:

[mathematical expression not reproducible] (21)

For [alpha] = -1/2, the formula of general Randic index takes the following form:

[mathematical expression not reproducible]. (22)

Theorem 5. Consider the fractal tree dendrimer [F.sub.p] with p, k [greater than or equal to] 2. Its augmented Zagreb index is as follows:

[mathematical expression not reproducible] (23)

Proof. Let fractal tree dendrimer [F.sub.p] be a tree graph for p iterations. The augmented Zagreb index of [F.sub.p] can be computed by using Table 1 in the following formula.

[mathematical expression not reproducible] (24)

After simplification of the above form, we got (23).

Table 2 shows the edge partition of fractal tree dendrimer [F.sub.p] 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 [F.sub.p] with p, k [greater than or equal to] 2. Its fourth atom bond connectivity index is as follows:

[mathematical expression not reproducible] (25)

Proof. Let fractal tree dendrimer [F.sub.p] be a tree graph for p iterations. The fourth atom bond connectivity index of [F.sub.p] can be computed by using Table 2 in the following formula:

[mathematical expression not reproducible] (26)

After simplification of the above form, we got (25).

Theorem 7. Consider the fractal tree dendrimer [F.sub.p] with p, k [greater than or equal to] 2. Its fifth geometric arithmetic index is as follows:

[mathematical expression not reproducible] (27)

Proof. Let fractal tree dendrimer [F.sub.p] be a tree graph for p iterations. The fifth geometric arithmetic index of [F.sub.p] can be computed by using Table 2 in the following formula:

[mathematical expression not reproducible] (28)

After simplification of the above form, we got our result of (27).

Theorem 8. Consider the fractal tree dendrimer [F.sub.p] with p, k [greater than or equal to] 2. Its Sanskruti index is as follows:

[mathematical expression not reproducible] (29)

Proof. Let fractal tree dendrimer [F.sub.p] be a tree graph for p iterations. The Sanskruti index of [F.sub.p] can be computed by using Table 2 in the following formula:

[mathematical expression not reproducible] (30)

After 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 tree [C.sub.s,t] (s [greater than or equal to] 3, t [greater than or equal to] 0) consists of t iterations. s is the number of nodes at first iteration. [C.sub.s,0] for t = 0, consist of only a central vertex. For t = 1, [C.sub.s,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 [C.sub.s,t-1] by creating s - 1 nodes and attaching them to each of the pendent vertices of [C.sub.s,t-1]. The Cayley tree network C433 is depicted in Figure 3.

In [C.sub.s,t], the pendent vertices are s[(s - 1).sup.n-1] and the s degree vertices are 2 [[summation].sup.t.sub.i=1][(s - 1).sup.i-1] - [(s - 1).sup.n-1]. The edge set of [C.sub.s,t] is divided into two partitions based on the degree of end vertices. The first edge partition consists of s[(s - 1).sup.n-1] edges uv, where [d.sub.u] = 1 and [d.sub.v] = s. The second edge partition consists of s[[summation].sup.t.sub.i=1][(s - 1).sup.i-1] - s[(s-1).sup.t-1] edges uv, where [d.sub.u] = [d.sub.v] = s. The cardinality of vertices and edges in [C.sub.s,t] is 2[[summation].sup.t.sub.i=1][(s - 1).sup.i-1] + [(s - 1).sup.t] and s[[summation].sup.t.sub.i=1][(s - 1).sup.i-1], respectively. The tabular representation of edge partition technique is depicted in Table 3.

Theorem 9. Consider the Cayley tree dendrimer [C.sub.s,t] with s, t [greater than or equal to] 3. Its atom bond connectivity index is as follows:

[mathematical expression not reproducible] (31)

Proof. Let Cayley tree dendrimer [C.sub.s,t] be a tree graph for t iterations. The atom bond connectivity index of [C.sub.s,t] can be computed by using Table 3 in the following formula:

[mathematical expression not reproducible] (32)

After simplification of the above form, we got (31).

Theorem 10. Consider the Cayley tree dendrimer [C.sub.s,t] with s, t [greater than or equal to] 3. Its geometric arithmetic index is as follows:

GA([C.sub.s,t]) = [s[(s - 1).sup.t-1](2[square root of s] - s - 1) + s(s + 1)[[summation].sup.t.sub.i=1][(s - 1).sup.i-1]]/[s + 1] (33)

Proof. Let Cayley tree dendrimer [C.sub.s,t] be a tree graph for t iterations. The geometric arithmetic index of [C.sub.s,t] can be computed by using Table 3 in the following formula:

[mathematical expression not reproducible] (34)

After simplification of the above form, we got (33).

Theorem 11. Consider the Cayley tree dendrimer [C.sub.s,t] with s, t [greater than or equal to] 3. Its general Randic index is as follows:

[mathematical expression not reproducible]. (35)

Proof. Let Cayley tree dendrimer [C.sub.s,t] be a tree graph for t iterations. The general Randic index [R.sub.[alpha]] of [C.sub.s,t] can be computed by using Table 3. For [alpha] =1, the general Randic index takes the following form:

[mathematical expression not reproducible] (36)

For [alpha] = -1, the formula of general Randic index takes the following form:

[mathematical expression not reproducible] (37)

For [alpha] = 1/2, the formula of general Randic index takes the following form:

[mathematical expression not reproducible] (38)

For [alpha] = -1/2, the formula of general Randic index takes the following form:

[mathematical expression not reproducible]. (39)

Theorem 12. Consider the Cayley tree dendrimer [C.sub.s,t] with s, t [greater than or equal to] 3. Its augmented Zagreb index is as follows:

[mathematical expression not reproducible] (40)

Proof. Let Cayley tree dendrimer [C.sub.s,t] be a tree graph for t iterations. The augmented Zagreb index of [C.sub.s,t] can be computed by using Table 3 in the following formula:

[mathematical expression not reproducible] (41)

After simplification of the above form, we got (40).

Table 4 shows the edge partition of Cayley tree dendrimer [C.sub.s,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.

Theorem 13. Consider the Cayley tree dendrimer [C.sub.s,t] with s, t [greater than or equal to] 3. Its fourth atom bond connectivity index is as follows:

[mathematical expression not reproducible] (42)

Proof. Let Cayley tree dendrimer [C.sub.s,t] be a tree graph for t iterations. The fourth atom bond connectivity index of [C.sub.s,t] can be computed by using Table 4 in the following formula:

[mathematical expression not reproducible] (43)

After simplification of the above form, we got (42).

Theorem 14. Consider the Cayley tree dendrimer [C.sub.s,t] with s, t [greater than or equal to] 3. Its fifth geometric arithmetic index is as follows:

[mathematical expression not reproducible] (44)

Proof. Let Cayley tree dendrimer [C.sub.s,t] be a tree graph for t iterations. The fifth geometric arithmetic index of [C.sub.s,t] can be computed by using Table 4 in the following formula:

[mathematical expression not reproducible] (45)

After simplification of the above form, we got (44).

Theorem 15. Consider the Cayley tree dendrimer [C.sub.s,t] with s, t [greater than or equal to] > 3. Its Sanskruti index is as follows:

[mathematical expression not reproducible] (46)

Proof. Let Cayley tree dendrimer [C.sub.s,t] be a tree graph for t iterations. The Sanskruti index of [C.sub.s,t] can be computed by using Table 4 in the following formula:

[mathematical expression not reproducible] (47)

After 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 connectivity ABC, geometric arithmetic GA, general Randic, first and second Zagreb, augmented Zagreb, [GA.sub.5], [ABC.sub.4], and Sanskruti indices for fractal tree. Also, we have computed the atom bond connectivity ABC, geometric arithmetic GA, general Randic, augmented Zagreb, [GA.sub.5], [ABC.sub.4], and Sanskruti index. Furthermore, we have computed closed results of above-mentioned indices for both fractal and Cayley tree dendrimers.

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

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

This research is supported by the Start Up Research Grant 2016 of United Arab Emirates University, Al Ain, United Arab Emirates, Grant no. G00002233, and UPAR Research Grant of United Arab Emirates University, Al Ain, United Arab Emirates, Grant no. G00002590.

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Muhammad Imran (iD), (1,2) Abdul Qudair Baig, (3) and Waqas Khalid (4)

(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; imrandhab@gmail.com

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

Academic Editor: Juan L. G. Guirao

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

Caption: Figure 2: (a) The representation of fractal tree [F.sub.3] for k = 2 is illustrated. (b) The representation of fractal tree [F.sub.4] for k = 3 is illustrated.

Caption: Figure 3: The representation of Cayley tree [C.sub.4,3] is illustrated.
Table 1: Edge partition of [F.sub.p] based on degrees of
end vertices of each edge.

([d.sub.u], [d.sub.v])            Frequency

(1, k + 2)                  42pk - 28k + 14p - 8
(4, k + 2)                        28p - 20
(k + 2, k + 2)                    21p - 14

Table 2: Edge partition of [F.sub.p] based on degree sum of
end vertices of each edge.

([S.sub.u], [S.sub.v])          frequency

(k + 2, 2k + 3)              (14p - 8)(k + 1)
(k + 2, 2k + 6)                 28pk - 20k
(2k + 3, 2k + 6)                 14p - 8
(2k + 6, 2k + 6)                  7p - 6
(2k + 6, 4k + 8)                 28p - 20

Table 3: Edge partition of [C.sub.s,t] based on degrees of
end vertices of each edge.

([d.sub.u], [d.sub.v])                 Frequency

(1, s)                             s[(s - 1).sup.t-1]
(s, s)                        s[[summation].sup.t.sub.i=1]
                                  [(s - 1).sup.i-1] -
                                   s[(s - 1).sup.t-1]

Table 4: Edge partition of [C.sub.s,t] based on degree sum
of end vertices of each edge.

([S.sub.u], [S.sub.v])                 frequency

(s, 2s - 1)                        s[(s - 1).sup.t-1]
(2s - 1, [s.sup.2])                s[(s - 1).sup.t-2]
([s.sup.2], [s.sup.2])       s[[summation].sup.t.sub.i=1]
                             [(s - 1).sup.i-1] - [s.sup.2]
                                   [(s - 1).sup.t-2]
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Title Annotation:Research Article
Author:Imran, Muhammad; Baig, Abdul Qudair; Khalid, Waqas
Publication:Discrete Dynamics in Nature and Society
Geographic Code:4EXCR
Date:Jan 1, 2018
Words:5217
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