Topological indices are numerical numbers associated to molecular graphs and are invariant of a graph. In QSAR/QSPR study, Zagreb indices are used to explain the different properties of chemical compounds at the molecular level mathematically. They have been studied extensively due to their ease of calculation and numerous applications in place of the existing chemical methods which needed more time and increased the costs. In this paper, we compute precise values of new versions of Zagreb indices for two classes of dendrimers.
Dendrimers are discrete nanostructures with the welldefined, homogeneous and monodisperse structure having treelike arms with low polydispersity and high functionality. The structure of these materials has a great impact on their physical and chemical properties. Due to their singular behaviour, these are acceptable for a wide range of potential applications in several areas of research, technology and treatment. With bettered synthesis, additional understandings of their unique characteristics and recognition of new applications, dendrimers will become hopeful candidates for further exploitation in drug discovery and clinical applications. Developing of commercial implementations of dendrimer technology will provide strength for its functionality in the future (Abbasi et al., 2014; Boad et al., 2006; Klajnert and Bryszewska, 2001).
Throughout the paper we consider G to be a finite, simple and connected molecular graph. The vertex set and the edge set of G are denoted by V(G) and E(G), respectively. In a molecular graph, the vertices correspond to atoms and the edges correspond to chemical bonds between the atoms. For an element in a molecular graph G = (V, E), we mean either a vertex or an edge. Two vertices u and v are called adjacent if there is an edge between them and we write it as e = uv or e = vu. Similarly, two edges e and f are called adjacent if they have a common vertex. The degree of a vertex u ∈ V(G) is the cardinality of the set of edges incident to u and is denoted by d_{u}. For a given edge e = uvE(G), the degree of e is defined as d_{uv} = d_{u} + d_{v} – 2.
The topological index is a numerical number linked with chemical constitutions. This number claims to correlate the chemical structures with its many physical/ chemical properties, biological activity or chemical reactivity. Many topological indices have been introduced based on the transformation of a molecular graph into a number that examines the relationship between the structure, properties, and activity of chemical compounds in molecular modelling. These topological indices are invariant under graph isomorphism, If A and B are two molecular graphs such that A ≅ B, then we have Top(A) = Top(B), where Top(A) and Top(B) denote the topological indices of A and B, respectively. In the field of nanotechnology, biochemistry, and chemistry, different topological indices are observed to be useful in structureproperty relationship, isomer discrimination, and structureactivity relationship. In recent decades, many topological indices have been defined and utilized for chirality, similarity/dissimilarity, study of molecular complexity, chemical documentation, isomer discrimination, structureproperty relationship and structureactivity relationship, lead optimization, drug design, and database selection, etc. There are three main types of topological indices: distancebased, degreebased, and countingrelated. For further studies of numerous kinds of topological indices of graphs and chemical structures (Aslam et al., 2017, 2018; Gao et al., 2018; Iqbal et al., 2017, 2019; Kang et al., 2018).
Among the degreebased Topological indices the Zagreb indices are the oldest and most studied molecular structure descriptors and they found significant applications in chemistry. Nowadays, there exist hundreds of papers on Zagreb indices and related matters. Gutman and Trinajstić (1972) introduced the first and second Zagreb index based on the degree of vertices of a graph G. The first and second Zagreb index of a molecular graph G are denoted and defined as:
These formulas were obtained analyzing the structural dependency of total π electron energy. It was observed that these terms increase with the increase extent of branching of carbonatom Skelton. This shows that these formulas provide the quantitative measures of molecular branching.
After this, many new extended and reformulated versions of Zagreb indices have been introduced, e.g. see: Ali et al. (2018), Ashrafi et al. (2010), Borovicanin et al. (2017), Braun et al. (2005), Das and Ali (2019), Gutman and Trinajstić (1972), Gutman and Das (2004), Javaid et al. (2019), Khalifeh et al. (2009), Kok et al. (2017), Liu et al. (2020), Zhou (2004), and Zhou and Gutman (2004, 2005). Recently, a new version of Zagreb indices has been introduced by Alwardi et al. (2018) and they named it as entire Zagreb indices. In addition these indices take into account the relations between the vertices and edges between vertices. The first and second entire Zagreb indices are defined by Alwardi et al. (2018).
This article is organized as follows: in Section 2, we compute the entire Zagreb indices for a class of Trizane based dendrimer, whereas Section 3 contains computation of these indices for water soluble PDI cored sendrimers. Conclusion and references will close this article.
In this section, we will compute the entire Zagreb indices for the molecular graph of Triazine based dendrimer (Gajjar et al., 2015). Let D_{1}(n) be the molecular graph of this dendrimer, where n represents the generation stage of D_{1}(n). The number of vertices and edges in D_{1}(n) is
Representatives  Degree  Frequency  Representatives  Degree  Frequency  Representatives  Degree  Frequency 

β_{1}  3  2  a_{j}  3  2^{2j}^{−1}  f_{j}  2  2^{2j}^{+1} 
β_{2}  2  4  b_{j}  2  2^{2j}  g_{n}  2  2^{2n}^{+1} 
β_{3}  2  4  c_{j}  3  2^{2j}  g_{j}, j ≠ n  2  2^{2j}^{+1} 
β_{4}  3  2  d_{j}  2  2^{2j}^{−1}  h_{n}  1  2^{2n}^{+1} 
β_{5}  2  2  e_{j}  3  2^{2j}  h_{j} , j ≠ n  2  2^{2j}^{+1} 
In Table 2, we find the edge partition with respect to the pairs of end vertices of sets A and B, degree of each edge and their frequencies of occurrence.
Edge  Degree  Frequency  Edge  Degree  Frequency  Edge  Degree  Frequency 

[β_{1},β_{1}]  4  1  [β_{5},a_{1}]  3  2  [e_{j},f_{j}]  3  2^{2j}^{+1} 
[β_{1},β_{2}]  3  4  [a_{j},b_{j}]  3  2^{2j}  [f_{j},g_{j}]  2  2^{2j}^{+1} 
[β_{2},β_{3}]  2  4  [b_{j},c_{j}]  3  2^{2j}  [g_{n},h_{n}]  1  2^{2n}^{+1} 
[β_{3},β_{4}]  3  4  [c_{j},d_{j}]  3  2^{2j}  [g_{j},h_{j}] j ≠ n  2  2^{2j}^{+1} 
[β_{4},β_{5}]  3  2  [c_{j},e_{j}]  4  2^{2j}  [h_{j},a_{j}_{+1}] j ≠ n  3  2^{2j}^{+1} 
For D_{1}(n), the first entire Zagreb index is given by:
Proof. Followed by values depicted in Tables 1 and 2, and the expression of first entire Zagreb index,
After several calculational steps, we obtain the following:
To compute the second entire Zagreb index, at first, we compute the nonrepeated collection of representatives with corresponding adjacent vertices and their frequencies of occurrence as shown in Table 3. Secondly, we find the edge partition with respect to the pairs of end vertices of sets A and B, their corresponding adjacent edges and frequencies of occurrence. These calculations are shown in Table 4. Finally, we find all the edges on which a specific vertex is incident and their frequencies of occurrence for n ≥ 1. These computations are shown in Table 5.
Representatives  Adjacent vertices  Frequency  Representatives  Adjacent vertices  Frequency 

β_{1}  β_{1}, β_{2}  1,4  b_{j}  c_{j}  2^{2j} 
β_{2}  β_{3}  4  c_{j}  d_{j},e_{j}  2^{2j},2^{2j} 
β_{3}  β_{4}  4  e_{j}  f_{j}  2^{2j}^{+1} 
β_{4}  β_{5}  2  f_{j}, j ≠ n  g_{j}  2^{2j}^{+1} 
β_{5}  a_{1}  2  g_{n}  f_{n},h_{n}  2^{2n}^{+1},2^{2n}^{+1} 
a_{j}  b_{j}  2^{2j}  g, j ≠ n _{j}  h_{j}  2^{2j}^{+1} 
h_{j} , j ≠ n  a_{j}_{+1}  2^{2j}^{+1} 
Edge  Adjacent edges  Frequency  Edge  Adjacent edges  Frequency 



4 


2^{2j},2^{2j} 
[β_{1},β_{2}]  [β_{1},β_{2}], [β_{2},β_{3}]  2,4  [c_{j},d_{j}]  [c_{j},d_{j}], [c_{j},e_{j}]  2^{2j}^{−1},2^{2j} 


4 


2^{2j}^{+1} 


2,4 


2^{2j},2^{2j}^{+1} 


2 


2^{2j}^{+1} 
[β_{5},a_{1}]  [a_{1},b_{1}]  4  [g_{j},h_{j}] j ≠ n  [h_{j},a_{j}_{+1}]  2^{2j}^{+1} 
[a_{j},b_{j}]  [a_{j},b_{j}], [b_{j},c_{j}]  2^{2j}^{−1},2^{2j}  [h_{j},a_{j}_{−1}] j ≠ n  [a_{j}_{+1},b_{j}_{+1}]  2^{2j}^{+2} 
Representatives  Edges on which representatives are incident  Frequency  Representatives  Edges on which representatives are incident  Frequency 

β_{1}  [β_{1},β_{1}], [β_{1},β_{2}]  1,4  c_{j}  [b_{j},c_{j}], [c_{j},d_{j}], [c_{j},e_{j}]  2^{2j},2^{2j}, 2^{2j} 
β_{2} 

4,4  d_{j} 

2^{2j} 
β_{3}  [β_{2},β_{3}], [β_{3},β_{4}]  4,4  e_{j}  [c_{j},e_{j}], [e_{j},f_{j}]  2^{2j}, 2^{2j}^{+1} 
β_{4} 

4,2  f_{j} 

2^{2j}^{+1},2^{2j}^{+1} 
β_{5} 

2,2  g_{n} 

2^{2n}^{+1},2^{2n}^{+1} 
a_{j}  [β_{5},a_{1}], [a_{j},b_{j}]  2,2^{2j}  g_{j} j ≠ n  [f_{j},g_{j}], [g_{j},h_{j}]  2^{2j}^{+1},2^{2j}^{+1} 
b_{j}  [a_{j},b_{j}], [b_{j},c_{j}]  2^{2j},2^{2j}  h_{j} j ≠ n  [g_{j},h_{j}], [h_{j},a_{j}_{+1}]  2^{2j}^{+1},2^{2j}^{+1} 
h_{n}  [g_{n},h_{n}]  2^{2n}^{+1} 
Now, we are ready to compute the second entire Zagreb index.
For D_{1}(n) the second entire Zagreb index is given by:
Proof. By using the values of Tables 15 and the definition of second entire Zagreb index, we calculate
By means of simple calculations, we derive:
The watersoluble perylenediimide (PDI)cored dendrimers have an important place among the other dendrimers due to their wide range of potential applications and have many advantages include low cytotoxicity, excellent photo stability, versatile surface modification, high quantum yield, strong red fluorescence, and biological applications including fluorescence livecell imaging, gene delivery, and fluorescent labelling (Kok et al., 2017). Here we will calculate the entire Zagreb indices for the molecular graph of one class of watersoluble PDIcored.
Let D_{2}(n) be the molecular graph of this dendrimer, where n represents the generation stage of D_{2}(n). It is easy to see that the number of vertices and edges in D_{2}(n) are 20(2^{n} + 1) and 20 × 2^{n} + 26, respectively. To compute the entire Zagreb indices of D_{2}(n), we will compute the required information for the sets of representatives of V(D_{2}(n)). We will use computational arguments for this computation. First we partition the molecular graph D_{2}(n) into two sets C and D. For the set C, we label the representatives by γ_{l}, where 1 ≤ l ≤ 9, and for the set D, these representatives are labelled by a_{i}, b_{i}, c_{i}, d_{n}, e_{n}, f_{n}, g_{n}, h_{n}, where 1 ≤ i ≤ n. Table 6 shows these representatives, their degrees and frequencies of their occurrence. The labelled molecular graphs with n = 1 and n = 2 are shown in Figure 2.
Representatives  Degree  Frequency  Representatives  Degree  Frequency  Representatives  Degree  Frequency 

γ_{1}  3  4  γ_{7}  3  4  d_{n}  2  2^{n}^{+1} 
γ_{2}  3  2  γ_{8}  1  4  e_{n}  3  2^{n}^{+1} 
γ_{3}  2  4  γ_{9}  3  2  f_{n}  1  2^{n}^{+1} 
γ_{4}  3  2  a_{i}  3  2^{i}  g_{n}  2  2^{n}^{+1} 
γ_{5}  2  4  b_{i}  2  2^{i}^{+1}  h_{n}  1  2^{n}^{+1} 
γ_{6}  3  4  c_{i}  2  2^{i}^{+1} 
In Table 7, we find theedge partition with respect to the pairs of end vertices of sets C and D, degree of each edge and their frequencies of occurrence.
Edge  Degree  Frequency  Edge  Degree  Frequency  Edge  Degree  Frequency 


4  2 

4  4 

3  2^{i}^{+1} 

4  4 

2  4 

2  2^{n}^{+1} 

3  4  [γ_{7},γ_{9}]  4  4  [d_{n},e_{n}]  3  2^{n}^{+1} 

4  2 

4  2 

2  2^{n}^{+1} 
[γ_{3},γ_{5}]  2  4  [a_{i},b_{i}]  3  2^{i}^{+1}  [e_{n},g_{n}]  3  2^{n}^{+1} 
[γ_{4},γ_{6}]  4  4  [b_{i},c_{i}]  2  2^{i}^{+1}  [g_{n},h_{n}]  1  2^{n}^{+1} 
[γ_{5},γ_{6}]  3  4 
Now, in the following theorem we compute the first entire Zagreb index.
For D_{2}(n), the first entire Zagreb index is given by:
Proof. According to values in Tables 6 and 7, and the definition of first entire Zagreb index the value of
After some calculations, we obtain the following:
To compute the second entire Zagreb index, we will have to find three things, firstly in Table 8, we compute the nonrepeated collection of representatives with corresponding adjacent vertices and their frequencies of occurrence. Secondly, we find the edge partition with respect to the pairs of end vertices of sets C and D, their corresponding adjacent edges and frequencies of occurrence. These calculations are shown in Table 9. Finally, we find all the edges on which a specific vertex is incident and their frequencies of occurrence for n ≥ 1. These computations are shown in Table 10.
Representatives  Adjacent vertices  Frequency  Representatives  Adjacent vertices  Frequency 

γ_{1}  γ_{1}, γ_{2}, γ_{3}  2,4,4  γ_{9}  a_{1}  2 
γ_{2}  γ_{4}  2  a_{i}  b_{i}  2^{i}^{+1} 
γ_{3}  γ_{5}  4 

c_{i}  2^{i}^{+1} 
γ_{4}  γ_{6}  4  c_{n}  d_{n}  2^{n}^{+1} 
γ_{5}  γ_{6}  4  d_{n}  e_{n}  2^{n}^{+1} 
γ_{6}  γ_{7}  4  e_{n} 

2^{n}^{+1,}2^{n}^{+1} 
γ_{7}  γ_{8},γ_{9}  4,4  g_{n}  h_{n}  2^{n}^{+1} 
Edge  Adjacent edges  Frequency  Edge  Adjacent edges  Frequency 

[γ_{1},γ_{1}]  [γ_{1},γ_{2}], [γ_{1},γ_{3}]  4,4  [γ_{7},γ_{9}]  [γ_{7},γ_{9}], [γ_{9},a_{1}]  2,4 


2,4,4 


4 


4 

[a_{i},b_{i}], [b_{i},c_{i}]  2^{i}, 2^{i}^{+1} 


4 


2^{i}^{+1} 


4 


2^{i}^{+2} 


2,4,4 


2^{n}^{+1} 


4 


2^{n}^{+1} 


4,4 


2^{n}^{+1}, 2^{n}^{+1} 


4 


2^{n}^{+1}, 2^{n}^{+1} 
Representatives  Edges on which representatives are incident  Frequency  Representatives  Edges on which representatives are incident  Frequency 

γ_{1}  [γ_{1},γ_{1}], [γ_{1},γ_{2}], [γ_{1},γ_{3}]  2,4,4  a_{i}  [γ_{9},a_{1}], [a_{i},b_{i}]  2,2^{i}^{+1} 
γ_{2} 

4,2,4  b_{i} 

2^{i}^{+1}, 2^{i}^{+1} 
γ_{3} 

4,4  c_{i} 

2^{i}^{+1} 
γ_{4}  [γ_{2},γ_{4}], [γ_{4},γ_{6}]  2,4  c_{i}, i ≠ n  [b_{i},c_{i}], [c_{i},a_{i}_{+1}]  2^{i}^{+1}, 2^{i}^{+1} 
γ_{5} 

4,4  f_{n} 

2^{n}^{+1} 
γ_{6}  [γ_{4},γ_{6}], [γ_{5},γ_{6}], [γ_{6},γ_{7}]  2,4,4  e_{n}  [d_{n},e_{n}], [e_{n},g_{n}], [e_{n},f_{n}]  2^{n}^{+1}, 2^{n}^{+1}, 2^{n}^{+1} 
γ_{7} 

4,4,4  g_{n} 

2^{n}^{+1}, 2^{n}^{+1} 
γ_{8} 

4  h_{n} 

2^{n}^{+1} 
γ_{9}  [γ_{7},γ_{9}], [γ_{9},a_{1}]  4,2  d_{n}  [c_{n},d_{n}], [d_{n},e_{n}]  2^{n}^{+1}, 2^{n}^{+1} 
Now, we are ready to compute the second entire Zagreb index.
For D_{2}(n), the second entire Zagreb index is given by:
Proof. Using Tables 610 and the definition of second entire Zagreb index, the value of
After some simplifications, we obtain the following:
In this rapid era of technological improvement, a large number of new chemical structures emerge every year. To find out the chemical properties of such a large number of compounds and drugs requires a large amount of chemical experiments. In this regard, computing different types of topological indices has supplied the evidence of such medicinal behaviour of several compounds and drugs.We considered two classes of dendrimers and studied entire Zagreb indices for their molecular graphs. It will be interesting to compute these indices for other chemical structures, which may be helpful to understand their underlying topologies.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11761083
Author Contribution
Conceptualization: Wei Gao; Investigation: Zahid Iqbal and Muhammad Ishaq; Methodology: Zahid Iqbal, Muhammad Ishaq, and Adnan Aslam; Software: Muhammad Aamir; Validation: Abdul Jaleel and Muhammad Aamir; Writing – original draft: Zahid Iqbal and Adnan Aslam; Writing – review and editing: Wei Gao and Abdul Jaleel.
Conflict of interest: Authors state no conflict of interest.
Data availability statement: No data was used to support this study.
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