The Topological Aspects of Phthalocyanines and Porphyrins Dendrimers

Topological descriptors are the numerical quantities of a graph that characterize various structural properties of it. In environmental sciences, pharmacy and in mathematical chemistry such descriptors are used for the quantitative structure-activity and property relationships (QSAR/QSPR) studies in which physicochemical properties of compounds are correlated with their molecular structures. A large spectrum of topological descriptors is available, among which the distance-based and bond-additive indices are frequently used in QSPR/QSAR studies. In this paper, the different versions of Szeged, Padmakar-Iven (PI) and Mostar indices for the molecular graphs of two types of dendrimers having phthalocyanines and porphyrins as their cores are illustrated through the numerical way. We have obtained exact analytical expressions of these indices by using the cut method.


I. INTRODUCTION
Dendrimers are tree-like polymeric macromolecules with distinct and homogeneous sizes and shapes. The structure of dendrimers consists of three distinct parts, a central core, the interior structure having repeated branches, and the outer structure with functional groups. The increase in the number of repeated branching units identify the generation of the dendrimer and is answerable for the formation of a globular structure. Dendrimers attracted great attention in gene and drug delivery applications due to having highly controllable architecture [1]- [6]. By electrostatic or hydrophobic attraction, the drugs and oligonucleotides are either bound to the surface or encapsulated in their inner cavities. They can also be covalently linked by interacting with terminal functional groups.
In reticular chemistry, quantitative structure-property / quantitative structure-activity relationship (QSPR/QSAR) schemes are correlation/regression models used to associate with various biological and physicochemical activities. The Harold Wiener initiated this study; in 1947, he introduced The associate editor coordinating the review of this manuscript and approving it for publication was Sun-Yuan Hsieh . a distance-based structure invariant to find out the boiling point of alkanes. In this process, the topologically invariant quantities change the chemical structure into a specific mathematical number named as topological descriptor or index. In chemical graph theory, the molecular structure descriptors establish, which models organic compounds into hydrogen-suppressed graphs with a relationship of vertices (resp. edges) to atoms (resp. bonds). The structure descriptors give efficient regression models, which correspond to different biological and physicochemical properties, including the critical temperature, critical volume, critical pressure, the boiling point, the heat of formation, enthalpy of vaporization of organic compounds. There are several classes of topological descriptors, including counting-based descriptors, valency-based descriptors, spectrum-based descriptors and distance-based descriptors. When distance-based descriptors correlate with physicochemical characteristics of organics structures, they provide the regression model with high prediction power. We refer to the books of Gutman and Furtula, and Todeschini and Consonni for readers for theory and applications of topological descriptors respectively.
A graph S is an ordered pair S(V (S), E(S)), where V (S) (resp. E(S)) is called the vertex (resp. edge) set of the graph S. A chemical graph is obtained from any given chemical structure in which atoms related to vertices whereas bonds related to edges. The degree of a vertex is the total number of edges attached with it. For example, the degree of a vertex is at most 4 in the chemical graph that is constructed from an organic compound. In this manuscript, every graph is assumed to be an undirected, simple and finite molecular graph.
A map from the set of simple connected graphs to the set of real numbers is called a topological descriptor which is topologically invariant under graph automorphisms. In literature till present, several different types of topological descriptors are introduced and studied. The distance-based topological descriptors are defined on distances in any graph including additive/multiplicative Harary indices, the Gutman index, the dd index, the (hyper) Wiener index. The degree-based topological descriptors are stated on degrees of vertices of any given graph such as the Randic connectivity index, the atombond connectivity (ABC) index, and the Zagreb index.
The peripheral edge in S is an edge of it if there is the extra number of vertices close to its one end-vertex as compare to its other end-vertex. From the other side, for uy ∈ E(S), the largest value of the absolute difference between the number of closer-vertices to u than to y represented as n u (e|S) and the number of closer-vertices to y than to u (signified as n y (e|S) tells us about a peripheral position of uy in S). The bond-additive topological indices are mainly used to specify the characteristics of chemical graphs. The Wiener index [7] is a significant bond-additive index, in which any bond gives a contribution that is equal to the product of the number of atoms on each side of the bond. It is famous that the Wiener index is used to describe the boiling point of the family of Alkanes. In relation to this, Gutman introduced the topological descriptor named as Szeged index in [8]. For any given graph S, the Szeged index is stated as follows: This Szeged index correlates the boiling points of not only acyclic hydrocarbons but also cyclic hydrocarbons. In 2009, Gutman has shown that this index can also be used to correlate various physicochemical and biological properties [9]. For a comprehensive study on the success of this index, the interested reader is referred to [10]. The other versions of Szeged index named as edge-Szeged and total-Szeged indices were introduced in [11] and [12] respectively. These indices are described for S as follows: (2) Inspired from the progress of the Szeged index, Khadikar et al. defined Padmakar-Iven (PI) index [13] which showed a significance in QSAR. For any graph S, the PI index is stated as under: The other versions of PI index named as edge-PI and total-PI indices are described in [14] for S as follows: Recently, Wang et al. [15] and Mogharrab et al. [16] examined the PI index for extremal cacti graphs and triangular-free graphs respectively. Motivated by the various productive indices such as PI [13], Szeged [8], Zagreb [17], irregularity [18], and revised-Szeged [19]- [21], recently, a novel bond-additive index introduced by Doslic et al. in [22] and it is famous as the Mostar index. For any connected graph S, it was specified as: Recently, the other versions of Mostar index named as edge-Mostar and total-Mostar indices were introduced by Arockiaraj et al. [23]. These indices are described for S as follows: For further results related to different versions of Mostar index, the interested readers are referred to [24], [25], [27]- [31]. The evaluation and analysis of topological indices of molecular structures are modern trends of research, which are of the significant importance in nanotechnology and theoretical chemistry. We refer the interested reader to [28], [32]- [41] for an introduction to this subject. We use the cut method that was suggested in [42] and they used it to find the Wiener index of graphs. This method was extended for general graphs by setting up a correlation between the canonical metric representation of the graph and its Wiener index in [43]. The standard cut method was also established for other topological descriptors of chemical graphs. For a graph S, e = uy and f = u 1 y 1 edges of S belong to the Djokovic-Winkler relation if d S (u, u 1 ) + d S (y , y 1 ) = d S (u, y 1 ) + d S (y , u 1 ) and it is always reflexive and symmetric, and transitive on partial cubes. The relation will construct equivalence classes F 1 , F 2 , · · · , F k of the edge set E(S) of a partial cube S and named as cuts or classes. For a comprehensive study of the cut method, see [44].

II. THE TOPOLOGICAL ASPECTS OF CARBOXYLATE-TERMINATED ZINC PHTHALOCYANINE DENDRIMERS
The structural motif of phthalocyanines has invited important research works during the last couple of decades [45], [46]. This class of molecules has often been used as pigments and dyes in the early days, this interest has transferred in present-days for their usage as building blocks for the formation of new molecular materials for dye-sensitized solar cells [47]- [49], optoelectronics and electronics [45], [46]. Recently, Setaro et al. prepared highly water-soluble dendrimers consisting of a central zinc phthalocyanine moiety and dendritic wedges with terminal carboxylate groups. In this section, we present the results related to this dendrimer. Let G 1 (s) be the molecular graph of this dendrimer, where s represents the generation stage of the dendrimer. Now, we find the vertex PI, vertex Szeged and vertex Mostar indices of G 1 (s).
Theorem 1: For the molecular graph G 1 (s), we have It is an easy exercise to observe that |V (G 1 (s))| = 9(2 s+2 + 1) and |E(G 1 (s))| = 8(5 × 2 s + 2). We define the pendant cuts {P t : 1 ≤ t ≤ 2 s+2 }, which take place on the boundaries of the dendrimer. The edges between two hexagons can be described in three sets {T l : 1 ≤ l ≤ s}, {T l : 1 ≤ l ≤ s} and {T l : 1 ≤ l ≤ s}. In the Table 1, we give the values of n G 1 (s) (y) and n G 1 (s) (w) in P t , T l , T l and T l .
The cuts of a hexagon that is not in the core of G 1 (s) are shown in Figure 1 and represented by {C l : 1 ≤ l ≤ s},   Table 2, we give the values of n G 1 (s) (y) and n G 1 (s) (w) in C l , C l , C l , A l , A l and A l .
Also, we represent the remaining vertices in the core of G 1 (s) by Therefore, by using the Tables 1-3, we get the following closed expressions for the vertex PI index, vertex Szeged TABLE 1. The values of n G 1 (s) (y ) and n G 1 (s) (w ) in P t , T l , T l and T l .
The values of n G 1 (s) (y ) and n G 1 (s) (w ) in C l , C l , C l , A l , A l and A l .

TABLE 3.
The values of n G 1 (s) (y ) and n G 1 (s) (w ) in O, Q and F , and also the edges w 1 w k , 2 ≤ k ≤ 5 in the core. VOLUME 8, 2020 index and vertex Mostar index.
Now, we find the edge PI, edge Szeged and edge Mostar indices of G 1 (s).
The cuts of a hexagon that is not in the core of G 1 (s) are shown in Figure 1 and represented by {C l : 1 ≤ l ≤ s}, {C l : 1 ≤ l ≤ s} and {C l : 1 ≤ l ≤ s} and the cuts of a hexagon in the core are represented as {A l : 1 ≤ l ≤ 4}, Table 5, we give the values of m G 1 (s) (y) and Also, we represent the remaining vertices in the core of G 1 (s) by Therefore, by using the Tables 4-6, we get the following closed expressions for the edge PI index, edge Szeged index and edge Mostar index.
The values of m G 1 (s) (y ) and m G 1 (s) (w ) in P t , T l , T l and T l .
Now, we find the total PI, total Szeged and total Mostar indices of G 1 (s).
Theorem 3: For the molecular graph G 1 (s), we have 1) PI t (G 1 (s)) = 3040 × 2 2s }, which take place on the boundaries of the dendrimer. The edges between two hexagons can be described in three In the Table 7, we give the values of t G 1 (s) (y) and t G 1 (s) (w) in P t , T l , T l and T l .
The cuts of a hexagon that is not in the core of G 1 (s) are shown in Figure 1 Table 8, we give the values of t G 1 (s) (y) and t G 1 (s) (w) in C l , C l , C l , A l , A l and A l .
Also, we represent the remaining vertices in the core of G 1 (s) by . The values of t G 1 (s) (y ) and t G 1 (s) (w ) in P t , T l , T l and T l .
The values of t G 1 (s) (y ) and t G 1 (s) (w ) in C l , C l , C l , A l , A l and A l .
. The values of t G 1 (s) (y ) and t G 1 (s) (w ) in O, Q and F , and also the edges w 1 w k , 2 ≤ k ≤ 5 in the core.
Therefore, by using the Tables 7-9, we get the following closed expressions for the total PI index, total Szeged index and total Mostar index.

III. THE TOPOLOGICAL ASPECTS OF PAMAM DENDRIMERS WITH PORPHYRIN CORE
Since the 1980s various types of dendrimers have been developed, but among all the one obtained from polyamidoamine (PAMAM) are apparently most useful in drug delivery, because they are biocompatible, hydrophilic and non-immunogenic systems. The PAMAM is mostly ethylenediamine, however the diaminobutane, diaminoexane and diaminododecane can also be practice. They comprise on ethylenediamine and methyl acrylate branching units, and have a full generation of amines and in half generations of the carboxyl terminated groups [50]- [53]. Raquel et al. synthesized the PAMAM dendrimers with a porphyrin core via the microwave method in [54]. In this section, we present the results related to this dendrimer. Let G 2 (s) be the molecular graph of this dendrimer, where s represents the generation stage of the dendrimer. Now, we find the vertex PI, vertex Szeged and vertex Mostar indices of G 2 (s). Theorem 4: For the molecular graph G 2 (s), we have Proof 4: It is an easy exercise to observe that |V (G 2 (s))| = 16(2 s+1 + 1) and |E(G 2 (s))| = 8(2 s+3 + 3). We define the pendant cuts {P t : 1 ≤ t ≤ 2 s+2 }, which take place on the boundaries of the dendrimer. The cuts of a hexagon of G 2 (s) are shown in Figure 2 and represented by   10. The values of n G 2 (s) (y ) and n G 2 (s) (w ) in P t , C l , C l , C l and A l .

TABLE 11
. The values of n G 2 (s) (y ) and n G 2 (s) (w ) in T l , U l , W l , X l , Y l , Z l and V l . we give the values of n G 2 (s) (y) and n G 2 (s) (w) in P t , C l , C l , C l and A l .
The edges after the hexagons can be described in the following sets {T l : 1 ≤ l ≤ s}, {U l : 1 ≤ l ≤ s}, Table 11, we give the values of n G 2 (s) (y) and n G 2 (s) (w) in T l , U l , W l , X l , Y l , Z l and V l .
Therefore, by using the Tables 10-12, we get the following closed expressions for the vertex PI index, vertex Szeged index and vertex Mostar index.
The edges after the hexagons can be described in the sets In the Table 14, we give the values of m G 2 (s) (y) and m G 2 (s) (w) in T l , U l , W l , X l , Y l , Z l and V l . Also we present the remaining vertices in the core of G 2 (s) by 13. The values of m G 2 (s) (y ) and m G 2 (s) (w ) in P t , C l , C l , C l and A l .
Now, we find the total PI, total Szeged and total Mostar indices of G 2 (s).
The edges after the hexagons can be described in the following sets such that {T l : 1 ≤ l ≤ s}, {U l : 1 ≤ l ≤ s}, In the Table 17, we give the values of t G 2 (s) (y) and t G 2 (s) (w) in T l , U l , W l , X l , Y l , Z l and V l .
Therefore, by using the Tables 16-18, we get the following  closed expressions for the total PI index, total Szeged index  TABLE 17. The values of t G 2 (s) (y ) and t G 2 (s) (w ) in T l , U l , W l , X l , Y l , Z l and V l .

IV. CONCLUSION
An ongoing direction in mathematical and computational chemistry is the characterization of molecular structures with the help of graphical descriptors. These invariants have been also established the applicable implementations in QSAR/QSPR studies appropriate for hazard assessment of chemicals, molecular design and novel drug discovery. Dendrimers have an eye-catching place in the research VOLUME 8, 2020 considering that of their remarkable physical and chemical properties and the comprehensive range of promising applications in various areas such as physics, chemistry, biology, medicine, and engineering. The Szeged indices have been proven that they can associate highly with various physicochemical and biological properties, for instance, see [9]. Different chemical applications of the PI indices were yielded, and it has proven that the PI indices associates well with the various indices as well as with the biological activities and physico-chemical properties of the underline structure, for instance, see [13]. Mostar index is a new suggested quantity; it has not been so far applied and suggested to be used in its physicochemical or biological researches. Recently, one such work has been performed in this direction for the acentric factor of some octane isomers, and it is confirmed that there is a linear relation between Mostar index and acentric factors of these isomers. In this paper, we have obtained exact analytical expressions of different versions of Szeged, PI and Mostar indices for the molecular graphs of two types of dendrimers by using the cut method. Our results could be beneficial in the models of QSPR/QSAR relationships for these molecular structures for assessing their properties.