# Distance-Based Structure Characterization of PAMAM-Related Dendrimers Nanoparticle

Authors: D. Antony Xavier; S. Akhila; Ammar Alsinai; K. Julietraja; Hanan Ahmed; Arul Amirtha Raja; Eddith Sarah Varghese
Journal: Journal of Nanomaterials
 (2022)

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## Abstract

Dendrimers are well-defined nanoparticles, which have far-reaching application in the field of chemistry. Many efforts have been devoted to development of dendrimer due to thier unique structure and various properties and broad application. It helps in varieties of purposes as a catalyst in drug delivery and drug design. The topological descriptor analyze the structure–property relationship of chemical compounds. In this paper, some numerical expressions have been obtained to understand the behavioral pattern of chiral polyamidoamine (PAMAM) dendrimer and PAMAM anthracene moieties dendrimer. The analytical expression has been plotted and compared with varieties of indices to show how it varies between each indices.


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## Body

## 1. Introduction

The topological descriptor helps to analyze the chemical structure and their physical properties. It is essential to understand the operational efficiency, performance, and side effects of new drugs in order to track the process and improve the impact. This helps the medicinal chemistry to overcome the insufficient funds and reach their target at reduced cost. New techniques could help to overcome these challenges.Nanoparticles have been in use for decades in various fields. The word “nano” refers to their size of 1–100 nm. They are capable of carrying drugs and biomolecules and are used in nanotherapy. Vogtle and his coworkers introduced a peculiar nanostructure, dendrimer, in the year 1978. Tomalia et al. [1], Newkome et al. [2], and numerous research group contributed to the development of a new dendrimer. Dendrimers have been extensively investigated and are currently applied in various fields of drug delivery [3, 4], catalysis [5], light harvesting, and porous material [6]. Dendrimers are composite structure, but are well-defined multibranched chemical composition, with high degree of order containing selected chemical units in predetermined sites of their tree-like structure [7]. Dendrimers have a branched architecture and three-dimensional spatial arrangement, and their extension is still predictable. From a topological standpoint, dendrimers generally consist of three structural phases: single central core, surrounded by bridges called generations composed of repeating units and terminated by functional/peripheral groups. Dendrimers are synthesized using a stepwise repetitive reaction sequence, which happens in two approaches: divergent synthesis and convergent synthesis. In divergent synthesis, the dendrimers are prepared from the core as starting point and buildup the generation. By using the same approach, the first dendrimer polyamidoamines (PAMAMs) had been synthesized. Convergent synthesis starts from the surface and ends up at the core. This approach does not allow to form the higher generations because steric problems occur in reaction. With the emergence of a large number of applications in anticancer, anti-inflammatory, antimicrobial activities there is a critical need to study and understand its properties. For application of dendrimer and further details, see [8–12].Topological indices are numerical parameters that determine the physicochemical properties of compounds. Harry Wiener introduced the method called Wiener index in the year 1947 [13]. In evolving QSAR/QSPRs, the structural properties of chemical compounds such as photophysical, pharmacological, and pharmacokinetic are obtained by topological indices. The chemical properties such as molecular refraction, critical temperature, boiling point, critical pressure, partition coefficients, osmotic coefficients, and viscosity can be acquired using topological indices [14, 15]. Therefore, each topological indices help to analyze and quantify different physical compositions of a chemical structure and also the chemical networks [16].In graph theory, the molecular structure of drug can be represented as graph. In chemical drug compounds, how the topological indices are applied and used were explained by García et al. [15]. Dendrimers are demanded in the industry of drug delivery and design due to its properties such as hyperbranching, photosensitizer, targeted delivery, multivalency, structure uniformity, solubility, and low toxicity and also used as anticancer and antitumor therapies.Chiral dendrimers have also been prepared by employing synthesis of monomers. To date, the chiral dendrimer can be categorized into different classes: dendrimer possessing a chiral core and achiral branches; dendrimer possesing chiral peripheral surface group; dendrimer constructed from an achiral core and constitutionally different branches; dendrimer featuring chiral branched units; and dendrimers featuring chiral core, asymmetric branching units, and optically active chiral core. In this study, the dendrimers possesing chiral peripheral surface group: chiral PAMAM dendrimer and PAMAM anthracene moieties dendrimer have been considered.Chiral PAMAM dendrimer is a nanocatalysis tetracarboxylic acid core with four branches, as shown in Figure1 [17]. PAMAM anthrancene moieties are a blue light-emitting ethylenediamine core with four branches, as shown in Figure 2 [18]. Both were peripherally substituted PAMAM dendrimers. The surface of the PAMAM structure is achiral and hydrophobic in nature and it has both light-emitting and electrodeposition properties. The nth generation of chiral PAMAM dendrimer and PAMAM anthracene moieties dendrimer has been denoted as CDn and ADn, respectively. By shrinking the original graph into quotient graph, the distance-based topological indices of CDn and ADn can be determined.Figure 1
Chiral PAMAM dendrimerCD3.Figure 2
PAMAM anthracene moieties dendrimerAD2.


## 2. Topological Descriptor

In this paper, let us consider connected graphs. For basic terminologies, refer to Bondy and Murty [19]. For an edge h=cd∈EG, Nch|G and Mch|G be the set of vertices and edges near to the vertex c than d and their cardinalities are denoted as nch|G and mch|G, respectively. The strength-weighted graph concept Gsw=G,wv,sv,se is introduced by Arockiaraj et al. [20], where wv is the vertex weight, sv is the vertex strength, and se is the edge strength. In the strength-weighted graph, we define Nch|Gsw=Nch|G and Mch|Gsw=Mch|G and the cardinality of nch|Gsw=∑r∈Nch|Gwvr and mch|Gsw=∑r∈Nch|Gsvr+∑s∈Mch|Gses, where h=cd∈EG. Let tch|G=nch|G+mch|G. We represent nch|Gsw as nch|G.Some preliminary concepts are defined, which are the key tool in finding the topological indices. Let us consider a graphG, the Djoković–Winkler’s relation on EG, Θ [21, 22] is as follows. If dx,w+dy,z≠dx,z+dw,y, then e=xy∈EG is Θ related with f=wz∈EG. The relation Θ is always reflexive and symmetric and its transitive closure Θ∗ is an equivalence relation. Using partition, we convert the original graph into the quotient graph as given by Klavžar et al. [23]. A partition X=X1,X2,…,Xc of EG is coarser than Y= Y1,Y2,…,Yd, if Xi is the union of one or more sets in Y. We refer to [13, 21, 24–27] for the basic definitions of distance-based topological indices. For the distance-based topological indices of strength-weighted graph, refer to Table 1.Table 1
Distance-based topological indices ofGsw.
Distance-based indicesMathematical formulaWienerWG=∑c,d⊆VGwvcwvddGc,dSzegedSzvG=∑h=cd∈EGsehnch|Gndh|GEdge SzegedSzeG=∑h=cd∈EGsehmch|Gmdh|GEdge-vertex SzegedSzevG=12∑h=cd∈EGsehnch|Gmdh|G+ndh|Gmch|GTotal SzegedSztG=SzvG+SzeG+2SzevGMostarMoG=∑h=cd∈EGseh|nch|G−ndh|G|Edge MostarMoeG=∑h=cd∈EGseh|mch|G−mdh|G|Total MostarMotG=∑h=cd∈EGseh|tch|G−tdh|G|Padmakar–IvanPIG=∑h=cd∈EGsehmch+mdhTheorem 1 (see [27]).
For a connected strength-weighted graphGsw=G,wv,sv,se, let E=F1,F2,…,Fk be a partition of EG coarser than F. Let X=W,Szv,Sze,PI,S and Gut. Then,(1)XGsw=∑i=1kXG/Fi,wvi,svi,sei,where(i)
wvi:VG/Fi⟶R+ is defined by wviC=∑x∈Cwvx,∀C∈G/Fi(ii)
svi:VG/Fi⟶R+ is defined by sviC=∑x∈Csvx+∑xy∈Csexy,∀C∈G/Fi(iii)
sei:EG/Fi⟶R+ is defined as the number of edges in Fi such that one end in C and the other end in D, for any two connected components C and D of G/Fik-rooted product of graphsGandH, denoted byG∘kH, is the graph obtained by joining the root nodes ofkcopies ofHto each vertex ofG.Theorem 2 (see [28]).
LetG=K1∘mH be a connected weighted graph, and let hi be the root node of Hi, where Hi≅H, i∈m;m≥2. Then,(2)WG=mWH+mm−1∑x∈VH1∑y∈VH2wxwy+∑x∈VH1wxmwede,x+mm−12∑y∈VH2wydh2,y+dx,h1,where e∈VK1.Chiral PAMAM dendrimer, denoted byCDn, has 42×2n+1−31 vertices and 22×2n+2−32 edges. CDn consists of 19×2n+2−32Θ∗ classes. Figure 3 represents the demonstration of chiral PAMAM dendrimer of 2nd generation. We have the sets E1,Ei:2≤i≤n+1,En+2,En+3 coarser than the Θ∗ classes. By shrinking the original graphs to quotient graphs, four types of quotient graphs are obtained. The four types are demonstrated in Figures 4–7.Figure 3
Demonstration ofEi: 1≤i≤5, which is coarser than the set of Θ∗ classes in CD2. Each color represents each Ei.Figure 4
The strength weight of first quotient graph,CDn/E1.Figure 5
CDn/Ei: 2≤i≤n+1, the second quotient graph of chiral PAMAM dendrimer.Figure 6
CDn/En+2, the third quotient graph of chiral PAMAM dendrimer.Figure 7
The strength weight of fourth quotient graph,CDn/En+3.The first quotient graph,CDn/E1, has five vertices and four edges. In that, three vertices have vertex weight 1 and vertex strength 0, where other two vertices have vertex weight 21×2n+1−17 and vertex strength 22×2n+1−18, respectively, as shown in Figure 4.The second quotient graph,CDn/Ei, where 2≤i≤n+1, contains 9×2i+1 vertices and 9×2i edges. It has 2i branches that are connected to ki with wvki,svki=9×2i−31,9×2i−32. The vertex weight and vertex strength of the vertices of all branches are 1 and 0, except vertices xj; j∈2i2≤i≤n+1 with wvxj,svxj=21×2n−i+2−17,22×2n−i+2−18, respectively, as shown in Figure 5.The third quotient graph,CDn/En+2, contains 6×2n+2+1 vertices and 6×2n+2 edges. It has 2n+2 branches which are connected to n with wvn,svn=9×2n+2−31,9×2n+2−32. Except the vertices lj; j∈2n+2 with wvlj,svlj=7,7, the vertex weight and vertex strength of all the vertices of branches are 1 and 0, as shown in Figure 6.The fourth quotient graph,CDn/En+3, contains 6×2n+2+1 vertices and 6×2n+2 edges. It has 2n+2 hexagons that are merged to o with wvo,svo=15×2n+2−31,15×2n+2−32. The vertex weight and vertex strength of all the vertices are 1 and 0, respectively, as shown in Figure 7.Theorem 3.
Ifn≥1, then WCDn=46058×2n−3276×2n×n−27132×22n+49392×22n×n−5372.Proof.
The Wiener index of each quotient graph is calculated as follows:(3)WCDn/E1=231×2n+1+321×2n+1−172−183.
We use the Theorem2 to compute the Wiener index of the following quotient graphs. For i=2,3,…,n+1 and j∈2i,(4)WCDn/Ei=2i76+28wx+2i2i−1×64+16wx+x2+2i32wk+7wkwx+2i2i−12384+144wx+12wx2,where wk=wki=9×2i−31;wx=wxj=21×2n−i+2−17.(5)WCDn/Ei=−22i24696×22n−757×22i+3654×22i×2n−24696×22n×2i+1638×2i×2n.
Therefore,(6)∑i=2n+1WCDn/Ei=59984×2n−3276×2n×n−53928×22n+49392×22n×n−6056.
Also,(7)WCDn/En+2=5882n+22−4792n+2+492n+2wn,where wn=9×2n+2−31=24×2n686×2n−333.(8)WCDn/En+3=902n+22−632n+2+152n+2wo,where wo=15×2n+2−31.(9)WCDn/En+3=48×2n105×2n−44.
Therefore,(10)WCDn=WCDn/E1+∑i=2n+1WCDn/Ei+WCDn/En+2+WCDn/En+3=46058×2n−3276×2n×n−27132×22n+49392×22n×n−5372.
In the next theorem, we compute Szeged-related indices.Theorem 4.
Ifn≥1, then SzvCDn=44834×2n−3276×2n×n−24108×22n+49392×22n×n−5372SzeCDn=52836×2n−3432×2n×n−31856×22n+54208×22n×n−6396SzevCDn=48699×2n−3354×2n×n−27832×22n+51744×22n×n−5868SztCDn=195068×2n−13416×2n×n−111628×22n+207088×22n×n−23504.Proof.
The vertex Szeged index ofCDn is calculated by evaluating vertex Szeged index of each quotient graph.(11)SzvCDn/E1=5292×22n−3822×2n+684.
For2≤i≤n+1,(12)SzvCDn/Ei=2i30wk+wx2i−1+82i−1+7wxwk+wx2i−1+82i−1+26wx+2wk+wx2i+82i−1+76,where wk=wki=9×2i−31;wx=wxj=21×2n−i+2−17.(13)SzvCDn/Ei=−12i49392×22n−1514×22i+7308×22i×2n−49392×22n×2i+3276×2i×2n,(14)∑i=2n+1SzvCDn/Ei=59984×2n−3276×2n×n−53928×22n+49392×22n×n−6056,(15)SzvCDn/En+2=2n+248wn+72n+2−1+52n+2−1+wn+72n+2+52n+2−1+4+98,where wn=9×2n+2−31.(16)SzvCDn/En+2=24×2n686×2n−333,(17)SzvCDn/En+3=2n+224wo+62n+2−1+54,where wo=15×2n+2−31.(18)SzvCDn/En+3=24×2n336×2n−139.
Therefore,(19)SvCDn=SzvCDn/E1+∑i=2n+1SzvCDn/Ei+SzvCDn/En+2+SzvCDn/En+3=44834×2n−3276×2n×n−24108×22n+49392×22n×n−5372.
The edge Szeged index ofCDn is calculated by evaluating edge Szeged index of each quotient graph.(20)SzeCDn/E1=5808×22n−4356×2n+812.
For2≤i≤n+1,(21)SzeCDn/Ei=2i30sk+sx2i−1+92i−1+7sxsk+sx2i−1+92i−1+26sx+62,where sk=ski=9×2i−32.(22)sx=sxj=22×2n−i+2−18=−12i54208×22n−1802×22i+8448×22i×2n−54208×22n×2i+3432×2i×2n,(23)∑i=2n+1SzeCDn/Ei=68104×2n−3432×2n×n−60896×22n+54208×22n×n−7208,(24)SzeCDn/En+2=2n+248sn+72n+2−1+62n+2−1+98,where sn=9×2n+2−32.(25)SzeCDn/En+2=2n+21056×2n+2−2062,(26)SzeCDn/En+3=2n+218so+72n+2−1+36,where so=15×2n+2−32.(27)SzeCDn/En+3=72×2n88×2n−37.
Therefore,(28)SzeCDn=SeCDn/E1+∑i=2n+1SzeCDn/Ei+SzeCDn/En+2+SzeCDn/En+3=52836×2n−3432×2n×n−31856×22n+54208×22n×n−6396.
The edge-vertex Szeged index ofCDn is calculated by evaluating edge-vertex Szeged index of each quotient graph.(29)SzevCDn/E1=126sawa+9wa+11sa+7,where wa=21×2n+1−17;sa=22×2n+1−18.(30)SzevCDn/E1=5544×22n−4081×2n+746.
For2≤i≤n+1,(31)SzevCDn/Ei=122i7wk+wx2i−1+82i−1sx+30wk+wx2i−1+82i−1+26sx+30sk+sx2i−1+92i−1+7sk+sx2i−1+92i−1wx+26wx+2sk+sx2i+92i−1+8+124,where wx=wxj=21×2n−i+2−17;sx=sxj=22×2n−i+2−18;wk=wki=9×2i−31andsk=ski=9×2i−32.(32)SzevCDn/Ei=−12i51744×22n−3307×22i2+7860×22i×2n−51744×22n×2i+3354×2i×2n,(33)∑i=2n+1SzevCDn/Ei=63926×2n−3354×2n×n−57312×22n+51744×22n×n−6614,(34)SzevCDn/En+2=122n+248wn+72n+2−1+52n+2−1+48sn+72n+2−1+62n+2−1+sn+72n+2+62n+2−1+201,where wn=9×2n+2−31;sn=9×2n+2−32.(35)SzevCDn/En+2=2×2n8344×2n−4061,(36)SzevCDn/En+3=122n+218wo+62n+2−1+24so+72n+2−1+90,where wo=15×2n+2−31;so=15×2n+2−32.(37)SzevCDn/En+3=48×2n151×2n−63.
Therefore,(38)SzevCDn=SzevCDn/E1+∑i=2n+1SzevCDn/Ei+SzevCDn/En+2+SzevCDn/En+3=48699×2n−3354×2n×n−27832×22n+51744×22n×n−5868.
The total Szeged index ofCDn is calculated by evaluating total Szeged index of each quotient graph.(39)SztCDn/E1=22188×22n−16340×2n+2988,(40)∑i=2n+1SztCDn/Ei=255940×2n−13416×2n×n−229448×22n+207088×22n×n−26492SztCDn/En+2=4×2n16684×2n−8121SztCDn/En+3=48×2n602×2n−251.
Therefore,(41)SztCDn=SztCDn/E1+∑i=2n+1SztCDn/Ei+SztCDn/En+2+SztCDn/En+3=195068×2n−13416×2n×n−111628×22n+207088×22n×n−23504.
In the next theorem, we compute Mostar-related indices.Theorem 5.
Ifn≥1, then MoCDn=7392×22n−1176×2n×n−5556×2n+394MoeCDn=7744×22n−1232×2n×n−5768×2n+394MotCDn=15136×22n−2408×2n×n−11324×2n+788.Proof.
The Mostar index ofCDn is calculated by evaluating Mostar index of each quotient graph.(42)MoCDn/E1=42×2n+1−26.
For2≤i≤n+1,(43)MoCDn/Ei=2i7wk+wx2i−1+82i−1−7wx+2wk+wx2i+82i−1+7−6,where wk=wki=9×2i−31;wx=wxj=21×2n−i+2−17.(44)MoCDn/Ei=756×2i×2n−1176×2n−105×2i,(45)∑i=2n+1MoCDn/Ei=3024×22n−1176×2n×n−3444×2n+420,(46)MoCDn/En+2=2n+25wn+72n+2−1+52n+2−1+wn+72n+2+52n+2−1−33,where wn=9×2n+2−31.(47)MoCDn/En+2=2n+2126×2n+2−284,(48)MoCDn/En+3=2n+27wo+62n+2−1−6,where wo=15×2n+2−31.(49)MoCDn/En+3=2n+2147×2n+2−265,
Therefore,(50)MoCDn=MoCDn/E1+∑i=2nMoCDn/Ei+MoCDn/En+1+MoCDn/En+2=7392×22n−1176×2n×n−5556×2n+394.
The edge Mostar index ofCDn is calculated by evaluating edge Mostar index of each quotient graph.(51)MoeCDn/E1=22×2n+2−26.
For2≤i≤n+1,(52)MoeCDn/Ei=2i7sk+sx2i−1+92i−1−7sx+2sk+sx2i+92i−1+8−4,where sk=ski=9×2i−32.(53)sx=sxj=22×2n−i+2−18=792×2i×2n−1232×2n−105×2i,(54)∑i=2n+1MoeCDn/Ei=3168×22n−1232×2n×n−3588×2n+420,(55)MoeCDn/En+2=2n+25sn+72n+2−1+62n+2−1+sn+72n+2+62n+2−1−31,where sn=9×2n+2−32.(56)MoeCDn/En+2=2n+2132×2n+2−294,(57)MoeCDn/En+3=2n+27so+72n+2−1,where so=15×2n+2−32.(58)MoeCDn/En+3=2n+2154×2n+2−273,
Therefore,(59)MoeCDn=MoeCDn/E1+∑i=2n+1MoeCDn/Ei+MoeCDn/En+2+MoeCDn/En+3=7744×22n−1232×2n×n−5768×2n+394.
The total Mostar index ofCDn is calculated by evaluating total Mostar index of each quotient graph.(60)MotCDn/E1=86×2n+1−52,(61)∑i=2n+1MotCDn/Ei=6192×22n−2408×2n×n−7032×2n+840MotCDn/En+2=8×2n516×2n−289MotCDn/En+3=8×2n602×2n−269.
Therefore,(62)MotCDn=MotCDn/E1+∑i=2n+1MotCDn/Ei+MotCDn/En+2+MotCDn/En+3=15136×22n−2408×2n×n−11324×2n+788.Theorem 6.
Ifn≥1, then PICDn=7744×22n−5744×2n+1056.Proof.
The Padmakar–Ivan index ofCDn is calculated by evaluating edge Padmakar–Ivan index of each quotient graph.(63)PICDn/E1=88×2n+2−132.
For2≤i≤n+1,(64)PICDn/Ei=2i7sk+sx2i−1+92i−1+7sx+2sk+sx2i+92i−1+8+56,where sk=ski=9×2i−32and;sx=sxj=22×2n−i+2−18.(65)PICDn/Ei=99×2i8×2n−3,(66)∑i=2n+1PICDn/Ei=3168×22n−4356×2n+1188,(67)PICDn/En+2=2n+25sn+72n+2−1+62n+2−1+sn+72n+2+62n+2−1+65,where sn=9×2n+2−32.(68)PICDn/En+2=2n+2132×2n+2−198,(69)PICDn/En+3=2n+27so+72n+2−1+36,where so=15×2n+2−32.(70)PICDn/En+3=2n+2154×2n+2−237.
Therefore,(71)PICDn=PICDn/E1+∑i=2n+1PICDn/Ei+PICDn/En+2+PICDn/En+3(72)PICDn=7744×22n−5744×2n+1056.PAMAM anthracene moieties dendrimer, denoted byADn, has 62×2n−28 vertices and 68×2n−29 edges. ADn consists of 25×2n−29Θ∗ classes. Figure 8 represents the demonstration of PAMAM anthracene moieties dendrimer of 2nd generation. We have three sets E1,Ei:2≤i≤n+1,En+2 coarser than the Θ∗ classes. By shrinking the original graph to quotient graph, three types of quotient graphs are obtained. The three types are indicated in Figures 9–11.Figure 8
Demonstration ofEi:1≤i≤4, which is coarser than the set of Θ∗ classes in AD2. Each color represents each Ei.Figure 9
The strength weight of first quotient graph,ADn/E1.Figure 10
ADn/Ei; 2≤i≤n+1, the second quotient graph of PAMAM anthracene moieties dendrimer.Figure 11
The third quotient graph,ADn/En+2.The first quotient graph,ADn/E1, has four vertices and three edges. In that, two vertices have vertex weight 1 and vertex strength 0, where other two vertices have vertex weight 31×2n−15 and vertex strength 34×2n−16, respectively, as shown in Figure 9.The second quotient graph,ADn/Ei, where 2≤i≤n+1, contains 8×2i+1 vertices and 8×2i edges. It has 2i branches that are connected to pi with wvpi,svpi=32×2i−2−28,32×2i−2−29. The vertex weight and vertex strength of the vertices of all branches are 1 and 0, except vertices yj; j∈2i, 2≤i≤n+1 with wvyj,svyj=31×2n−i+1−15,34×2n−i+1−16, respectively, as shown in Figure 10.The third quotient graph,ADn/En+2, contains 15×2n+1+1 vertices and 18×2n+1 edges. It has 2n+1 hexagon that is merged to q with wvq,svq=32×2n−28,32×2n−29. The vertex weight and vertex strength of all the vertices are 1 and 0, respectively, as shown in Figure 11.Theorem 7.
Ifn≥1, then(73)WADn=184223×22n−107508×2n×n−138762×2n−338423.Proof.
The proof is similar to Theorem3.Theorem 8.
Ifn≥1, then SzvADn=27856×2n−2108×2n×n−16895×22n+26908×22n×n−4454SzeADn=34902×2n−2312×2n×n−22916×22n+32368×22n×n−5295SzevADn=31197×2n−2210×2n×n−19712×22n+29512×22n×n−4860SztADn=125152×2n−8840×2n×n−79235×22n+118300×22n×n−19469.Proof.
The proof is similar to Theorem4.Theorem 9.
Ifn≥1, then MoADn=4216×22n−868×2n×n−3684×2n+260MoeADn=4624×22n−952×2n×n−3964×2n+260MotADn=8840×22n−1820×2n×n−7648×2n+520.Proof.
The proof is similar to Theorem5.Theorem 10.
Ifn≥1, then PIADn=4624×22n−4060×2n+870.Proof.
The proof is similar to Theorem6.


## 3. Discussion

In this section, the numerical results for all the indices of whose analytical expressions are determined. Refer to Tables2–5 for the same. The graphical comparison between Wiener, Padmakar–Ivan, Szeged, and Mostar indices helps to understand the molecular structure of CDn and ADn and its properties. Refer to Figures 12–15 for the same. The graphical behavior of topological indices has been plotted using MATLAB.Figure 12
Graphical comparison of Wiener and Szeged indices ofCDn.Figure 13
Graphical comparison graph of Padmakar–Ivan and Mostar indices ofCDn.Figure 14
Graphical comparison of Wiener and Szeged indices ofADn.Figure 15
Graphical comparison of Padmakar–Ivan and Mostar indices ofADn.Table 2
Numerical values for Wiener and Szeged indices ofCDn.
nWnSzvnSzenSzevnSztn1169232178880181820180470721640212990841342572140245213725925490208380312848215028870307684568283383176044415350844908068479731884641952418572030452260481962291056042460596522374451729500556006110440947611167174441203790852115948360446394755047522309376452724823085698759940548163880421934519856824122586372243204538922633937997225310352404101280538672Table 3
Numerical values for Padmakar–Ivan and Mostar indices ofCDn.
nMonMoenMotnPIn1164981737033868205442870349137017840410198434008104202988211084507204172858618117223540308189161657203850754855414752404774710462947085830877578603484363135286471193460581250359142443819721261435208480611722503511434984123156506041376Table 4
Numerical values for Wiener and Szeged indices ofADn.
nWnSzvnSzenSzevnSztn170170870949769392314369415249953868084278493773124029282593268871442528584966465459641218412147412675514235350022768350525526868102272243555656282121018138142999825131556196527130355623479324259305794670295265764568966825873899397968062362281088937033391560493063719044122774835078393916841013003120282154725256491418439731656844440563Table 5
Numerical values for Padmakar–Ivan and Mostar indices ofADn.
nMonMoenMotnPIn1802089241694411246246036507729680858614321978024163646141626432649650601059652202471211196545406067644560688516744460592661669990818320900350208081868093476782592474399492142225416752408068273579268300074244573653512301999974


## 4. Conclusion

The distance-based topological indices have been analyzed forCDn and ADn, which help to determine its structural characterization and properties. Dendrimers are emerging in diverse areas such as material science and biomedical science. The numerical results of CDn and ADn help to integrate the indices, which ease to examine the synthesis of dendrimer.



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*Source: 2911196-2022-12-22.xml*