# Connection-Based Multiplicative Zagreb Indices of Dendrimer Nanostars

Authors: Aqsa Sattar; Muhammad Javaid; Ebenezer Bonyah
Journal: Journal of Mathematics
 (2021)

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## Abstract

The field of graph theory is broadly growing and playing a remarkable role in cheminformatics, mainly in chemistry and mathematics in developing different chemical structures and their physicochemical properties. Mathematical chemistry provides a platform to study these physicochemical properties with the help of topological indices (TIs). A topological index (TI) is a function that connects a numeric number to each molecular graph. Zagreb indices (ZIs) are the most studied TIs. In this paper, we establish general expressions to calculate the connection-based multiplicative ZIs, namely, first multiplicative ZIs, second multiplicative ZIs, third multiplicative ZIs, and fourth multiplicative ZIs, of two renowned dendrimer nanostars. The defined expressions just depend on the step of growth of these dendrimers. Moreover, we have compared our calculated for both type of dendrimers with each other.


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## Body

## 1. Introduction

TIs are the numerical numbers which are linked with different chemical structures of molecular graphs and predict the structural, toxicological, biological, and physicochemical properties of the existing chemical compounds. A graph in which the vertices represent atoms while the edges correspond to the covalent bonds between atoms is known as a molecular graph. TIs are extensively used in the study of quantitative structure-activity relationships and quantitative structure-property relationships [38]. Many researchers have worked on TIs [1, 18, 25]. TIs are classified into three distinct TIs, namely, degree-based TI, distance-based TI, and polynomial-based TI. A distance-based TI is a TI which is based on the distance between the vertices. In 1947, Wiener [39] developed the innovative conception of degree-based TI. Furthermore, Dankelmann et al. calculated the sharp upper bounds of graphs by utilizing these distance-based TIs in a very comprehensive way. Moreover, for the diameter δ≥2, Mazorodze et al. [33] computed the sharp upper bounds of graphs by using the Gutman index which is also a distance-based TI. Furthermore, Fang et al. [15] discussed the topological properties of Sierpinski network along with its applications.A degree-based TI is concerned with the degree of a vertex. Degree-based TI is further categorized into two subclasses named as degree and connection-based TIs. Gutman and Trinajstić [22] put forward the innovative idea of the well-known TI named as first Zagreb index (FZI). They used FZI to calculate the π−electron energy of the alternant hydrocarbon. Furthermore, second ZI was proposed by Gutman et al. [21] in 1975. The innovative idea of third ZI was proposed by Furtula and Gutman [17]. These first and second ZIs have been studied widely in distinct areas (see [2, 5, 6]). Chu et al. [10] calculated the sharp bounds of ZIs on connected graphs. Gharibi et al. [19] also worked on ZIs and investigated Zagreb polynomials of nanocone and nanotubes. Nikolic et al. [34] initiated modified ZIs in 2003. Hao [23] compared the ZIs and modified ZIs and discussed important results related to these indices in 2011. Furthermore, Dhanalakshmi et al. [12] introduced some modified and multiplicative ZIs (MZIs) on graph operators. Das et al. [11] in 2013 used MZIs to compute the upper bounds for some particular graphs. Fang et al. [14] computed ZIs of the hierarchical hypercube networks. For more details, we refer the readers to [9, 26, 36].Recently, the idea of connection number (CN) has been instilled into the minds of researchers. CN is a number of those of vertices which are at distance 2 from a certain vertex. Ali and Trinajstić [4] investigated the modification of first ZI. In 2019, Tang et al. [37] developed some Zagreb connection indices (ZCIs). Furthermore, Ali et al. [3] computed modified ZCIs of T-sum graphs. Recently, In 2020, Liu et al. [32] gave Zagreb connection numbers of molecular graphs based on operations. Cao et al. [8] made use of ZCIs to calculate both exact and upper bounds of some product related graphs. Furthermore, Javaid et al. [28] initiated novel connection-based ZIs of different wheel related graphs. Haoer et al. [24] investigated the multiplicative leap ZIs.A dendrimer is an artificially synthesized molecular structure made up of monomers (branched units). Dendrimer nanostars are highly branched nanostructures and are considered as the basic element in nanotechnology. The major three architectural parts of dendrimer nanostars are end groups, branches, and cores. Nowadays, dendrimer nanostars are rapidly gaining considerable attention from researchers due to their special chemical and physical characteristics and a broad range of applicability in distinct fields of bioscience, including drug delivery, immunology, and the advancement of antimicrobials, antivirals, and vaccines [29, 31]. Siddiqui et al. [35] introduced Zagreb polynomial of some nanostars in 2016. Furthermore, Bokhary et al. [7] discussed some molecular topological properties of dendrimers. In 2019, Fatima et al. [16] proposed ZCIs of two dendrimer nanostars in a very logical way. For more details about dendrimers, the readers are referred to [13, 30].In this paper, we rewrite some already introduced connection-based MZIs. Further, we establish the general expressions to calculate the MZCIs of two well-known dendrimer nanostars in a very logical and comprehensive way. The proposed expressions only depend upon the step of growth of these dendrimers.This paper is organized as follows. Section1 presents some important definitions which are obligatory to understand the concept of our paper. In Section 2, we establish the general expression to find the connection-based MZIs of first type of dendrimer nanostar. Section 3 holds the general expression to calculate connection-based MZIs of the second type of dendrimer nanostar. Section 4 draws the conclusions.


## 2. Preliminaries

In this section, we define some basic definitions which are useful for the further evaluations.Definition 1.
(see [22]). Let ζ=Mζ,ℕζ be a graph, where Mζ and ℕζ represent the set of vertices and edges, respectively. Then, the first Zagreb index (FZI) can be given as(1)ℤ^1ζ=∑x∈Mζdζ˜x2.
This equation can be rewritten as(2)ℤ^1ζ=∑xy∈ℕζdζ˜x+dζ˜y,where dζ˜x and dζ˜y denote the degree of the vertices x and y, respectively.Definition 2.
(see [21]). For graph ζ, the second Zagreb index (SZI) can be given as(3)ℤ^2ζ=∑xy∈ℕζdζ˜x×dζ˜y,where dζ˜x and dζ˜y denote the degree of the vertices x and y, respectively.Definition 3.
(see [4]). For a graph ζ, the first Zagreb connection index (FZCI) and second Zagreb connection index (SZCI) can be given as(1)
ℤ^ℭ1ζ=∑x∈Mζηζ˜x2(2)
ℤ^ℭ2ζ=∑xy∈ℕζηζ˜x×ηζ˜y,whereηζ˜x and ηζ˜y denote the connection number of the vertices x and y, respectively.Definition 4.
(see [4]). For a graph ζ, the modified first Zagreb connection index can be given as(4)ℤ^ℭ1∗ζ=∑x∈Mζηζ˜x+ηζx.Definition 5.
(see [20]). For a graph ζ, the first multiplicative Zagreb index (FMZI) and second Zagreb index (SMZI) can be given as(1)
ℤ^1Mζ=∏xy∈ℕζdζ˜x+dζ˜y(2)
ℤ^2Mζ=∏xy∈ℕζdζ˜x×dζ˜yDefinition 6.
(see [27]). For a graph ζ, connection-based MZIs can be defined as(1)
ℤ^1Mℭζ=∏x∈Mζηζ˜x2 (first MZCI)(2)
ℤ^2Mℭζ=∏xy∈ℕζηζ˜x×ηζ˜y (second MZCI)(3)
ℤ^3Mℭζ=∏x∈Mζd˜ζx×ηζ˜x (third MZCI)(4)
ℤ^4Mℭζ=∏xy∈ℕζηζ˜x+ηζ˜y (fourth MZCI)Now, before moving towards our main results of this article, first we rewrite the connection-based MZIs given in Definition6.Definition 7.
For a graphζ, the first MZCI can be rewritten as(5)ℤ^1Mℭζ=∏0≤α≤n^−2α2Cαζ,where Cαζ is the total number of vertices in ζ with connection number α.
The second MZCI can be rewritten as(6)ℤ^2Mℭζ=∏0≤α≤β≤n^−2α×βCα,βζ,where Cα,βζ is the total number of edges in ζ with connection numbers α,β.
Similarly, the third MZCI can be given as(7)ℤ^3Mℭζ=∏0≤γ≤α≤n^−2γ×αCγ,α′ζ,where Cγ,α′ζ is the total number of vertices with degree γ and CN α.
The fourth MZCI can be written as(8)ℤ^4Mℭζ=∏0≤α≤β≤n^−2α+βCα,βζ.


## 3. MZCIs of First Type of Dendrimer Nanostar

In this section, we establish the general expressions to calculate the MZCIs of first type of dendrimer nanostar. First, we provide the construction of the dendrimer nanostar of the first type, i.e.,D1m^, by labeling the vertices with degrees and CNs. The skeletal formulas of D1m^ along with connection number for m^=1,2,3 are shown in Figures 1–3.Figure 1
D11 along with the connection of each vertex.Figure 2
D12 along with the connection of each vertex.Figure 3
D11 along with the degree of each vertex.The skeletal formulas of dendrimer nanostarD1m^ along with degrees are shown in Figures 4–6.Figure 4
D12 along with the degree of each vertex.Figure 5
D12 along with the degree of each vertex.Figure 6
D13 along with the degree of each vertex.Before presenting the main results our paper, we first classify the hexagons ofD1m^ with the help the degrees of the vertices into terminal hexagon, initial hexagon, and λ−hexagon.(i)
Terminal Hexagon. A hexagon in which the degree of exactly five vertices is two is said to be terminal hexagon.(ii)
Initial Hexagon. A hexagon which is in the center of D1m^ is said to be initial hexagon.(iii)
λ-Hexagon. A hexagon which is neither initial nor terminal is said to be λ-hexagon.All the remaining vertices which do not lie in any of the above mentioned hexagons are said to beμ-type vertices. By r,s−type edges, we mean the edge joining the vertices with CNs r and s.Table 1
Total number of vertices with CN 3 inζ.
Types of hexagonsNumber of vertices with CN 3 in hexagon (x)Total number of hexagons (y)Total number of vertices with CN 3 in hexagonsx×yTerminal22m^+12×2m^+1Initial616λ52m^+1−452m^+1−4μ-type000Table 2
Total number of vertices with CN 4 inζ.
Types of hexagonsNumber of vertices with CN 4 in hexagon (x)Total number of hexagons (y)Total number of vertices with CN 4 in hexagonsx×yTerminal12m^+12m^+1Initial010λ12m^+1−42m^+1−4Table 3
Total number of vertices with degree 2 and CN 2 inζ.
Types of hexagonsNumber of vertices with degree 2 and CN 2 in hexagon (x)Total number of hexagons (y)Total number of vertices with degree 2 and CN 2 in hexagonsx×yTerminal32m^+13×2m^+1Initial010λ02m^+1−40Table 4
Total number of vertices with degree 2 and CN 3 inζ.
Types of hexagonsNumber of vertices with degree 2 and CN 3 in hexagon (x)Total number of hexagons (y)Total number of vertices with degree 2 and CN 3 in hexagonsx×yTerminal22m^+12×2m^+1Initial414λ42m^+1−442m^+1−4Theorem 1.
Letζ=D1m^ be a molecular graph for m^=1,2,3. Then, first MZCI and third MZCI of ζ are given below:(1)
ℤ^1Mζ=64κ489776025600κ−2(2)
ℤ^3Mζ=64κ7255941120κ−2whereκ=2m^+1.Proof.
(1)
First, we find the total number of terminal hexagons, initial hexagons, andλ-hexagons. By simple observation, we have(9)total number of terminal hexagons=2m^+1,total number of initial hexagons=1.The total number ofλ−hexagons in ζ for m^=1,2,3,⋯ is 0,4,12,28,…, respectively. The nth term of the sequence is 2m^+1−4. Thus,(10)total number ofλ−hexagons=2m^+1−4.The total number ofμ− type vertices in ζ for m^=1,2,3,⋯ is 4,12,28,…, respectively. The nth term of the sequence is 2×2m^+1−4. Thus,(11)total number ofμ−vertices=2×2m^+1−4.Now, we find the number of vertices having CN 2 inζ, i.e., C2ζ. One can observe easily that all those vertices which have CN 2 are present only in terminal hexagons and no vertices exist having CN 2 in central and λ−hexagons. Every terminal hexagon has exactly 3 vertices with CN 2, and the total number of terminal hexagons in ζ is 2m^+1. Thus, C2ζ must be equal to 3 times the number of terminal hexagons in ζ. Mathematically, we have(12)C2ζ=3×2m^+1.Now, we find the number of vertices having CN 3 inζ, i.e., C3ζ. In Table 1, we have calculated the total number of vertices with CN 3 in ζ.The total number of vertices with CN 3 inζ is the sum of the number of vertices with CN 3 in terminal hexagon, initial hexagon, and λ−hexagon of ζ. So,(13)C3ζ=2×2m^+1+52m^+1−4+6=2×2m^+1+52m^+1−20+6=7×2m^+1−14.Next, we calculate the total number of vertices with CN 4 inζ, i.e., C4ζ. Table 2 shows the total number of vertices with CN 4 in ζ.On can easily observe that half of theμ-type vertices have CN 4 while the other half has CN 5. We know that the total number of μ-type vertices is 2×2m^+1−4. Thus, the number of vertices having CN 4 must be 2m^+1−2. The total number of vertices with CN 4 in ζ is the sum of the number of vertices with CN 4 in terminal hexagon, initial hexagon, λ−hexagon, and μ−type vertices of ζ. So,(14)C4ζ=2m^+1+0+2m^+1−4+2m^+1−2=3×2m^+1−6.Finally, we calculateC5ζ. So,(15)C5ζ=2m^+1−2.By using equation (5), we have(16)ℤ^1Mℭζ=∏0≤α≤n^−2α2Cαζ=22C2ζ×32C3ζ×42C4ζ×52C5ζ=223×2m^+1×327×2m^+1−14×423×2m^+1−6×522m^+1−2=64κ×489776025600κ−2,whereκ=2m^+1.(2)
Firstly, we calculate number of vertices with degree 2 and CN 2, i.e.,C2,2′ζ. The number of vertices with degree 2 and CN 2 in terminal hexagon, initial hexagon, and λ-type hexagon is given in Table 3.Thus, we have(17)C2,2′ζ=3×2m^+1.Now, we calculate the number of vertices with degree 2 and CN 3 inζ. The number of vertices with degree 2 and CN 3 in terminal hexagon, initial hexagon, and λ-type hexagon is given in Table 4.
Thus,C2,3′ζ will be(18)C2,3′ζ=2×2m^+1+42m^+1−4+4=6×2m^+1−12.
We notice that the number of vertices with degree 2 and CN 4 is present only in half of theμ−type vertices of ζ. Thus, C2,4′ζ will be(19)C2,4′ζ=2m^+1.
Now, we calculate the number of vertices with degree 3 and CN 3 inζ. The number of vertices with degree 3 and CN 3 in terminal hexagon, initial hexagon, and λ-type hexagon is given in Table 5.
Thus,C3,3′ζ will be(20)C3,3′ζ=2+2m^+1−4=2m^+1−2.
Similarly,(21)C3,5′ζ=2m^+1−2.
The number of vertices with degree 3 and CN 4 in terminal hexagon, initial hexagon, andλ-type hexagon is given in Table 6.
Thus,C3,4′ζ will be(22)C3,4′ζ=2m^+1+2m^+1−4+4=2×2m^+1−4.
By substituting all the values ofCα,β′ζ in equation (7), we have(23)ℤ^3Mℭζ=∏0≤γ≤α≤m^−2α×βCα,β′ζ=4C2,2′ζ×6C2,3′ζ×8C2,4′ζ×9C3,3′ζ×12C3,4′ζ×20C3,5′ζ=43×2m^+1×66×2m^+1−12×82m^+1−2×92m^+1−2×122×2m^+1−4×152m^+1−2=43κ×66κ−2×8κ−2×9κ−2×122×κ−2×15κ−2=64κ7255941120κ−2,where κ=2m^+1.Table 5
Total number of vertices with degree 3 and CN 3 inζ.
Types of hexagonsNumber of vertices with degree 3 and CN 3 in hexagon (x)Total number of hexagons (y)Total number of vertices with degree 3 and CN 3 in hexagonsx×yTerminal02m^+10Initial212λ12m^+1−42m^+1−4Table 6
Total number of vertices with degree 3 and CN 4 inζ.
Types of hexagonsNumber of vertices with degree 3 and CN 4 in hexagon (x)Total number of hexagons (y)Total number of vertices with degree 3 and CN 4 in hexagonsx×yTerminal12m^+12m^+1Initial010λ12m^+1−42m^+1−4Table 7
The total number of (3,3)-type edges in hexagons ofζ.
Types of hexagonsNumber of (3,3)-type edges in hexagon (x)Total number of hexagons (y)Total number of (3,3)-type edges in hexagonsx×yTerminal02m^+10Initial616λ42m^+1−442m^+1−4Table 8
The total number of (3,4)-type edges in hexagons ofζ.
Types of hexagonsNumber of (3,4)-type edges in hexagon (x)Total number of hexagons (y)Total number of (3,4)-type edges in hexagonsx×yTerminal22m^+12×2m^+1Initial010λ22m^+1−422m^+1−4Table 9
Types of pentagons and hexagons.
Types of pentagons and hexagonsTotal number of pentagons inζTerminal pentagon1Non-terminal pentagon2m^−1η−hexagon2m^ϕ−hexagon22m^−1−1ϑ−hexagon22m^−1ψ−hexagon2m^−1Theorem 2.
Letζ=D1m^ be a molecular graph for m^=1,2,3. Then, second MZCI and fourth MZCI of ζ are given below(1)
Z^2Mℭζ=576κ1990656000κ−2812κ−5,(2)
Z^4Mℭζ=400κ12252303κ−2362κ−5,whereκ=2m^+1.Proof.
(1)
First, we calculateC2,2ζ, i.e., (2,2)-type edges in ζ. One can easily observe that graph ζ has 2,2−type edges only in terminal hexagon. There are exactly two (2,2)-type and (2,3)-type edges in every terminal hexagon of ζ. Hence, C2,2ζ must be equal to(24)C2,2ζ=2×2m^+1,C2,3ζ=2×2m^+1.Next, the total number of (3,3)-type edges is displayed in Table7.The total number of (3,3)-type edges inζ is the sum of the (3,3)-type edges in terminal hexagon, initial hexagon, λ−hexagon of ζ. So,(25)C3,3ζ=4×2m^+1−10.Now, we calculateC3,4ζ. The total number of (3,4)-type edges in every terminal hexagon, initial hexagon, and λ−hexagons of ζ is displayed in Table 8.The total number of3,4−type edges which do not exist in any of the hexagon of ζ is equal to the number of μ−type vertices with CN 4. Thus, C3,4ζ will be(26)C3,4ζ=2×2m^+1+22m^+1−4+2m^+1−2=5×2m^+1−10.Finally, we calculateC4,5ζ. We observe that 4,5−type edges in ζ are equal to 3 times the number of μ−type vertices of ζ with CN 5. Thus, we have(27)C4,5ζ=32m^+1−2.By using equation (6), we have(28)ℤ^2Mℭζ=∏0≤α≤β≤n^−2α×βCα,βζ=4C2,2ζ×6C2,3ζ×9C3,3ζ×12C3,4ζ×20C4,5ζ=42×2m^+1×62×2m^+1×94×2m^+1−10×125×2m^+1−10×2032m^+1−2=42κ×62κ×94κ−10×125κ−10×203κ−6=576κ1990656000κ−2812κ−5,whereκ=2m^+1.(2)
Now, we drive the second formula. By substituting all the values ofCα,βζ in equation (8), we have(29)ℤ^4Mℭζ=∏0≤α≤β≤m^−2α+βCα,βζ=4C2,2ζ×5C2,3ζ×6C3,3ζ×7C3,4ζ×9C4,5ζ=42×2m^+1×52×2m^+1×64×2m^+1−10×75×2m^+1−10×932m^+1−2=400κ12252303κ−2362κ−5.


## 4. MZCIs of Second Type of Dendrimer Nanostar

In this section, we calculate the multiplicative ZCIs of second type of dendrimer nanostars. First, we provide the construction of the dendrimer nanostar of the second type, i.e.,D2m^, by labeling the vertices with degrees and CNs. The skeletal formulas of D2m^ along with connection number for m^=1,2,3 are shown in Figures 7–9.Figure 7
D21 along with the CN of each vertex.Figure 8
D22 along with the CN of each vertex.Figure 9
D23 along with the CN of each vertex.The skeletal formulas of dendrimer nanostarD2m^ along with degrees of each vertex are shown in Figures 10–12.Figure 10
D21 along with the degree of each vertex.Figure 11
D22 along with the degree of each vertex.Figure 12
D23 along with the degree of each vertex.Before stating our main results about the second type of dendrimers, we classify hexagons and pentagons ofD2m, for m=1,2,3, with the help of degrees of the vertices, into the following:(1)
Terminal Pentagon. A pentagon which has three adjacent vertices of degree 1.(2)
Non-Terminal Pentagon. A pentagon which is not terminal is said to be non-terminal pentagon.(3)
η−Hexagon. A hexagon which has exactly two vertices of degree 3.(4)
ϕ−Hexagon. A hexagon which has exactly three vertices of degree 3.(5)
ϑ−Hexagon. A hexagon which has exactly four vertices of degree 3.(6)
ψ−Hexagon. A hexagon which has exactly six vertices of degree 3.All those vertices which do not lie in any type of pentagon or hexagon are said to beμ−type vertices.Theorem 3.
Letζ=ℕ2m^ be a molecular graph of order n^. Then, the first and third MZCIs of a molecular graph ζ are given as(1)
ℤ^1Mζ=44κ−1166κ−716815125390625κ−1365κ−6(2)
ℤ^3Mζ=16κ22κ−15015306502144κ−1159κ−101852κ−3,whereκ=2m^.Proof.
First, we find the total number of hexagons and pentagons. The total number of pentagons and hexagons of a graphζ is depicted in Table 9.
The vertices which do not exist in any of the above-defined hexagon is said to beμ−type vertices. The total number of μ−type vertices is 2m^+1−1.(1)
After simple calculation, the total number of vertices with CNs 2, 3, 4, 5, and 6, from Figure9, is(30)C2ζ=4×2m^−1,C3ζ=82m^−1,C4ζ=6×2m^−7,C5ζ=42m^−1,C6ζ=5×2m^−6.By putting the above values in equation (5), we have(31)ℤ^1Mζ=44κ−1166κ−716815125390625κ−1365κ−6.(2)
Now, we findCγ,α′ζ. From Figures 9 and 12, we have(32)C1,2′ζ=2m^+1−1,C2,2′ζ=2×2m^,C2,3′ζ=82m^−1,C3,4′ζ=62m^−1,C3,5′ζ=9×2m^−10,C3,6′ζ=52×2m^−3.By putting above values in equation (7), we have(33)ℤ^3Mζ=16κ22κ−15015306502144κ−1159κ−101852κ−3.Theorem 4.
Letζ=ℕ2m^ be a molecular graph of order n^. Then, second and fourth MZCIs of a molecular graph ζ are given as(1)
ℤ^2Mζ=148κ82κ−193κ−4245κ−81804κ−1115202κ−1306κ−1(2)
ℤ^4Mζ=100κ62κ−163κ−4105κ−8344534954841κ−137748736κ−1,whereκ=2m^.Proof.
(1)
Again from Figure9, we have(34)C2,2ζ=2m^,C2,3ζ=2m^+1,C2,4ζ=2m^+1−1,C3,3ζ=3×2m^−4,C3,4ζ=42m^−1,C3,5ζ=42m^−1,C4,4ζ=22m^−1,C4,5ζ=22m^−1,C6,6ζ=22m^−1,C4,6ζ=5×2m^−8,C5,6ζ=62m^−1.By putting above values in equation (6), we have(35)ℤ^2Mζ=148κ82κ−193κ−4245κ−81804κ−1115202κ−1306κ−1.(2)
By putting all the values ofCα,β in equation (8), we have the following expression:(36)ℤ^4Mζ=100κ62κ−163κ−4105κ−8344534954841κ−137748736κ−1.


## 5. Comparison Analysis

In this section, to check the validity and superiority, we compare our calculated values for both the dendrimers with each other. Table10 shows the comparison between the proposed results of dendrimers.Table 10
Comparison between the values ofD1m and D2m.
ZMCIsD1mD2mℤ^1Mζ64κ489776025600κ−244κ−1166κ−716815125390625κ−1365κ−6ℤ^2Mζ576κ1990656000κ−2812κ−5148κ82κ−193κ−4245κ−81804κ−1115202κ−1306κ−1ℤ^3Mζ64κ7255941120κ−216κ22κ−15015306502144κ−1159κ−101852κ−3ℤ^4Mζ400κ12252303κ−2362κ−5100κ62κ−163κ−4105κ−8344534954841κ−137748736κ−1From Table10, it can be seen that D1m has the largest value for ℤ^2Mζ while D2m has the greatest value of ℤ^2Mζ.


## 6. Conclusions

Dendrimers are highly branched organic macromolecules with successive layers or generations of branch units surrounding a central core and are considered as key molecules in nanotechnology. In this research article, we have established the general expressions to find the connection-based MZIs, namely, first multiplicative Zagreb connection index, second multiplicative Zagreb connection index, third multiplicative Zagreb connection index, and fourth multiplicative Zagreb connection index of two well-known dendrimer nanostars. These calculated general expressions just depend upon the step of growth of these dendrimers. Moreover, to check the authenticity, we have also compared the calculated result for both types of dendrimers. In future, we are interested in extending our work for other types of dendrimers which are poly(propyl) ether imine dendrimer and polypropylenimine octamin dendrimer.



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