On the Studies of Dendrimers via Connection-Based Molecular Descriptors

Topological indices (TIs) have been utilized widely to characterize and model the chemical structures of various molecular compounds such as dendrimers, neural networks, and nanotubes. Dendrimers are extraordinarily comprehensible, globular, artificially synthesized polymers with a structure of frequently branched units. A mathematical approach to characterize the molecular structures by manipulating the topological techniques, including numerical graphs invariants is the present-day line of research in chemistry. Among all the defined descriptors, the connection-based Zagreb indices are considered to be more effective than the other classical indices. In this manuscript, we find the general results to compute the Zagreb connection indices (ZCIs), namely, first ZCI (1 ZCI), second ZCI (2 ZCI), modified 1 ZCI, modified 2 ZCI, and modified 3 ZCI. Furthermore, we compute the multiplicative ZCI (MZCI), namely, first MZCI (1 MZCI), second MZCI (2 MZCI), third MZCI (3 MZCI), fourthMZCI (4 MZCI), modified 1 MZCI, modified 2MZCI, andmodified 3MZCI. In addition, we compare the calculated values with each other in order to check the superiority.


Introduction
Dendrimers are compartmentalized, versatile, well-defined, synthetic chemical polymers with numerous attributes which make them advantageous in biological systems. e structure of dendrimers is made up of three components, the multivalent surface, the outer shell, and a core which is protected by the dendritic branches in higher generations of dendrimers. Dendrimers are synthesized by the use of two approaches, divergent, and convergent. Nowadays, dendrimers are considered to be the notably manufactured macromolecules with applicability in the domain of biomedical science including gene transfection, tissue engineering, drug delivery, contrast intensification for magnetic resonance imaging, and immunology, for details see [1][2][3].
ey are extensively employed in the formation of chemical sensors, colored glass, nanolatex, nanotubes, and micro-/ macrocapsules. Due to their wide ranging applications in distinct areas, dendrimers are attaining valuable contemplation from the researchers. ey are trying to specify these molecular structures by the use of numerical graphs descriptors. e numerical graph descriptors or topological indices are the trending topological approach in computational and mathematical chemistry to characterize or signalize the topology of molecular structures. ese graph descriptions or invariants have countless utilizations in quantitative structure-activity relationship (QSAR) and quantitative structure property relationship (QSPR) studies appropriate for hazard analysis of chemicals, the discovery of drugs, and novel molecular designs [4]. Topological index (TI) is a numeric measure which helps to correlate the distinct psychochemical properties of molecular structures like freezing point, melting point, volatility, density, stability, flammability, and strain energy of molecular compounds. Topological indices (TIs) are classified on the basis of distance, degree, and polynomial. Wiener [5] put forward the innovational conception of distance-based TI which is known by Wiener index. Aslam et al. [6] compute the TIs of some interconnection networks. After the invention of Wiener index, a large number of other distance-based TIs have been investigated and considered by many analysts in the chemical and mathematico chemical literature, for details see [7][8][9].
Gutman and Trinajstic [10] initiated the innovational notion of first ZI (1 st ZI) in 1972. In 1975, Gutman et al. [11] proposed the conception of second ZI (2 nd ZI). ese classical ZIs have been utilized broadly in the study of chemical graph theory. Furthermore, the conception of third ZI (3 rd ZI), also called forgotten index, was explored by Furtula and Gutman [12]. ese degree-based TIs have great significance in the field of cheminformatics, as one can see [13][14][15]. In 2003, Nikolic et al. [16] explored the new index, namely, modified ZI. Hao [17] compared these introduced ZIs and considered the outcomes concerning these indices in well-mannered way. Das et al. [18] investigated some MZIs of graph operations.
Recently, Ali and Trinnajstic [19] explored a new way to study the psychochemical properties of compounds by introducing the connection number (CN) of the vertex and initiated Zagreb connection indices (ZCIs). e number of those vertices which are distance two from a certain vertex is said to be a CN of that vertex. ey reported that the newly proposed connection-based ZIs have better applicability to forecast the psychochemical properties of various molecular structures instead of the classical ZIs. After the invention of CN, many researchers started work to explore new connection-based indices. Multiplicative leap ZIs were investigated by Haoer et al. [20]. Du et al. [21] utilized connection-based modified FZI to find the extremal alkanes. Recently, Sattar et al. [22] computed the general expressions to compute MZCI of dendrimer nanostars. Furthermore, in 2020, Ali et al. [23] worked out to calculate the modified ZCIs for T-sum graphs. Javaid et al. [24] calculated multiplicative ZIs for some wheel graphs. In 2019, Nisar et al. [25] computed ZCIs of two types of dendrimer nanostars. Ye et al. [26] calculated ZCIs of nanotubes and regular hexagonal lattice. Bokhary et al. [27] studied the topological properties of some nanostars. Bashir et al. [28] computed the 3 rd ZI of a dendrimer nanostar. Gharibi et al. [29] developed the conception of Zagreb polynomials of nanotubes and nanocones. For the other information, we recommend the readers to study [30,31]. e motivation for this article is as follows: (1) Topological indices (TIs), the numerical descriptors, are efficient enough to characterize the topology of molecular structures and also assist to correlate their distinct psychochemical properties.
(2) Dendrimers are symmetric, versatile, and well-defined chemical polymers forming a tree-like structure. ese nanoparticles are signalized by a numerous attributes which make them advantageous for wide ranging utilizations in various fields of science.
(3) e connection-based ZIs have better applicability to predict the various psychochemical properties of distinct molecular structures in chemistry rather than the other classical ZIs present in literature.
In this paper, we present the general expressions to compute the ZCIs and MZCIs of nanostar. is manuscript is organized as follows: in Section 2, the elementary definitions are discussed which helps the readers to fully understand the main idea of this article. In Section 3, we present the general expressions to compute ZCIs, namely, 1 st ZCI, 2 nd ZCI, modified 1 st ZCI, modified 2 nd ZCI, and modified 3 rd ZCI. Section 3 involves general expressions to compute the MZCIs, namely, 1 st MZCI, 2 nd MZCI, 3 rd MZCI, 3 rd MZCI, modified 1 st MZCI, modified 2 nd MZCI, and modified 3 rd MZCI. Section 4 covers the concluding remarks.

Preliminaries
is section involves some useful primary definitions from the literature to understand the main result of this manuscript.

Definition 1.
Let Ω � (R(Ω), S(Ω)) be a graph, where R(Ω) and S(Ω) be the set of vertices and set of edges, respectively. en, the degree-based Zagreb indices are defined as follows: Here, d Ω (t) and d Ω (x) denote the degree of the vertex t and x, respectively. ese degree-based indices, discovered by Gutman and Trinajstic [10], are known as first ZI (1 st ZI) and second ZI (2 nd ZI), respectively.
Definition 2. For a graph Ω, connection-based Zagreb indices are given as Here, φ Ω (t) and φ Ω (x) indicate the connection number (CN) of the vertex t and x, respectively. ese connectionbased indices were discovered by Ali and Trinajstic [19] and are known as the first Zagreb connection index (1 st ZCI) and second Zagreb connection index (2 nd ZCI), respectively. Definition 3. For a graph Ω, the modified ZCIs can be given as follows: ese modified ZIs, proposed by Ali [19] and Ali et al. [23], are known as the modified 1 st ZCI, modified 2 nd ZCI, and modified 3 rd ZCI, respectively.

ZCIs of Nanostar Dendrimer D[k]
is section involves the expressions to obtain connectionbased ZIs, namely, 1 st ZCI, 2 nd ZCI, modified 1 st ZCI, modified 2 nd ZCI, and modified 3 rd ZCI of the nanostar dendrimer. e molecular structure of D[k] for k � 1, 2, 3 together with connection number of each vertex is presented in Figure 1, 2 and 3. e molecular structure of D[k] for k � 1, 2, 3 together with degree of each vertex is presented in Figures 4, 5, and 6. First, in order to compute all ZCIs, we rewrite the abovementioned ZIs as follows.
Definition 6. For a graph Ω, the 1st ZCI can be rewritten as where |F α (Ω)| is the total number of vertices in Ω with CN α. Furthermore, we can rewrite the 2 nd ZCI as Similarly, the modified 1 st ZCI can be rewritten as where |F α,τ (Ω)| is the total number of edges in Ω with CNs (α, τ). e modified 2 nd ZCI can be rewritten as e modified 3 rd ZCI can be rewritten as (5) where |F (μ,])(α,τ) (Ω)| is the total number of edges in Ω with degrees (μ, ]) and CNs (α, τ).
Before computing the general results of this article, we first categorize the D[k] into term-hexagon, pivot-hexagon, primary vertices, and (e, f)− type edges.
(1) Term-hexagon: a hexagon is named as term-hexagon if its five vertices have degree 2.
(2) Pivot-hexagon: a hexagon which is not term-hexagon is said to be pivot-hexagon.
Definition 7. For a graph Ω, the 1st MZCI can be rewritten as where |F α (Ω)| are the total vertices in Ω with CN α. e 2nd MZCI is rewritten as where |F α,τ (Ω)| are the total edges with CNs α and τ. e 3 rd MZCI can be rewritten as where |F ((c,α)) (Ω)| are the total vertices with degree c and CN α. Similarly, the 4 th MZCI can be rewritten as where |F (α,τ) (Ω)| are the total edges in Ω with CNs (α, τ). e modified 1 st MZCI can be rewritten as 8 Mathematical Problems in Engineering e modified 2 nd MZCI can be rewritten as e modified 3 rd MZCI can be rewritten as MZC * 3 (Ω) � equal to two times of number of term-hexagons of Ω. So, F (2,2) ′ (Ω) � 9 2 k− 1 .
Furthermore, we find |F (2,3) ′ (Ω)|. It can be easily seen that the every term-hexagon of Ω contains two vertices having degree 2 and CN 3 while pivothexagon contains 4 vertices. e total amount of such vertices in term-hexagon and pivot-hexagon of Ω are 6(2 k− 1 ) and 24(2 k− 1 ) − 1, respectively. us,  Future directions: in future, we are interested to compute the connection-based Zagreb indices for the other type of dendrimers.

Data Availability
e data used to support the findings of this study are included within this article. However, the reader may contact the corresponding author for more details on the data.