Abstract
This paper is a topological study of nanostar dendrimers in the context of topological indices based on reverse degree. Nanostar dendrimers with highly branched structures and potential use in drug delivery and nanotechnology offer a challenge to structural analysis in terms of their complexity. The main aim of the work is to calculate and discuss some topological indices in reverse form, which can be used to obtain the necessary information concerning the molecular structure, the pattern of branches, and the connectivity. The indices are a good source of quantitative information on the hidden topology of nanostar dendrimers and their effects on chemical and physical characteristics. The findings indicate that reverse degree-based indices are very useful in defining the complex topological distributions of dendrimers, hence providing a strong mathematical framework, which contributes to subsequent developments in nanomaterials studies and their potential applications. Entropy-based topological measurements of polypropylenimine octaamine dendrimers and fullerene dendrimers have great advantages to chemists as a quantitative characterization of the molecular complexity, branching behaviors, and structural anomalies. Entropy measures based on degree-based edge partitions give the distribution of atomic connectivity and provide a rigorous method of comparing highly branched dendritic networks and cage-like carbon structures. We compute molecular entropies for polypropylenimine octaamine dendrimers and fullerene dendrimer networks.
Keywords: Nanostar dendrimers, Reverse degree indices, Molecular topology, Nanomaterials science, Nanobiotechnology, Entropies
Subject terms: Chemistry, Materials science, Mathematics and computing, Nanoscience and technology
Introduction
The recent, progressive branch of nanobiotechnology is an amalgamation of the concepts of biology and nanotechnology to design nanoscale instruments and machines that may be used to investigate and engineer biological mechanisms with relevant precision as never before. With advanced nanofabrication technology, scientists can create some of the most innovative devices that can respond molecularly to cells, proteins, and other biomolecules, in addition to drug delivery systems, molecular imaging devices, and nanosensors, to name a few, through nanofabrication technology1. Biology and nanotechnology can totally change the process of therapeutics, healthcare, and diagnostics, as they offer new insights into the complexity of biological processes and diseases2.
Using the concepts of nanofabrication, the creative and quickly developing discipline of nanobiotechnology creates state-of-the-art tools for researching and engaging with biosystems at the molecular level3. Concepts from biology, chemistry, materials science, biophysics, and nanotechnology are all combined in the interdisciplinary field of nanobiotechnology. Its goal is to develop nanoscale functional devices that enable in-depth molecular research of biological systems4. Because nanoscale production techniques offer such fine control, researchers can engineer materials and structures with remarkable precision. A major area of interest for nanobiotechnology is dendrimers. They are artificial macromolecules made by a methodical process called nanoscale fabrication, which results in a highly organised and complex architecture to Combinatorial Chemistry High Throughput Screening.
Dendrimers are similar due to their regular and uniform structure branches of trees, which branch out of a main central core. Because dendrimers are extremely proud of their great qualities and powers diversifiable and demanded in several applications because of their distinct structure5.
One of the features that distinguishes the dendrimers is their monodispersity compared to that of the conventional polymeric systems. The uniform size, shape, and chemical structure of dendrimers distinguish them from standard. polymers. Conversely, regular polymers can exhibit a range of sizes and topologies. Their behavior and characteristics may be more accurately controlled because of this trait, which makes them valuable in a range of scientific and technological domains. There are many applications for dendrimers, including pharmaceutical delivery systems6, gene therapy vectors7, 8, biomaterials9, nanosensors, catalysis, and imaging agents. Analyses of some structural properties on a class of Hierarchical scale-free networks and statistical analyses of a class of random pentagonal chain networks with respect to several topological properties are discussed in Refs.10–15.
With the careful functionalization of dendrimers with desirable molecules or patterns, researchers can develop customized systems that are superior. characteristics, including improved solubility, biocompatibility, or targeted. responses to particular cell types or tissues. The adaptability of dendrimers and the multidisciplinary essence of nanobiotechnology. offer tremendous possibilities for discoveries in a multitude of areas and sectors, such as biotechnology, materials science, medicine, and pharmaceuticals. The earlier research16–19 offers more thorough insights into the various uses and possibilities of dendrimers, clarifying the current findings and research in this fascinating field of scientific inquiry.
The structural characteristics of medicinal drugs are represented mathematically in medicine; the structure is represented as an undirected graph. In this graph, each vertex stands for an atom, and the edges represent the chemical bonds that bind these atoms together. Lets look at a straightforward graph that represents the organisation of medicine and is designated as G. In G, the edges are denoted by E(G) and the set of vertices by V(G). If two atoms have a chemical link, they are regarded as neighbouring. The number of chemical bonds that surround an atom in the undirected medicinal structure is known as its degree. The degree of an atom x is denoted by 
. Over the past 40 years, scientists have created a variety of crucial indices to assess and measure the properties of therapeutic substances. These indexes consist of the eccentric index, Zagreb index, PI index, and Wiener index. Every index is a real valued function that converts a medicine structure into a certain range of real numbers so that its characteristics can be described. Hakeem et al. computed the degree-based indices of Y-graphene and Zigzag graphene nanoribbons20, Huilgol et al. examined the structure activity connections, and Gao et al. investigated the use of topological indices in pharmaceuticals21.calculated the fullerenes topological indices22. Prior research has also examined a variety of dendrimer topological indices23, 24. Degree-based entropies were used by Hui et al. to predict the physicochemical characteristics of antiemetic medications25, and Hayat and Imran talked about the topological indices of specific networks26. This article’s core vocabulary and notation are taken from an earlier study27. Synthesis and properties of novel fullerene derivatives having dendrimer units and taking advantage of the electronic excited states of [60]-fullerenes are discussed in Refs.28, 29.
Kulli presented the idea of the reverse vertex degree in a graph G in30. The reverse degree of any vertex x, is defined as: 
, where 
is the maximum degree of the graph. Some of the reverse degree-based topological indices, including the reverse augmented Zagreb index for metal-organic networks (TM-TCNB), reverse atom bond connectivity index, reverse Balaban index, reverse Zagreb type indices, and reverse general Randic index, were calculated by Zhao et al.31. For TUC4 [m,n], Jung et al.32 calculated the reverse atomic-bond connectivity index, reverse geometric-arithmetic index, first and second reverse Zagreb indices, and first and second reverse hyper Zagreb indices.
For the Remdisivir compound used to treat coronavirus (COVID 19), Wei et al.33 computed a number of reverse topological indices, including the reverse general Randic index, reverse atom bond connectivity index, reverse geometric arithmetic index, reverse forgotten index, reverse Balaban index, and reverse Zagreb type indices. The QSPR of the reverse degree-based topological indices was examined by Ravi et al.34. For metal-organic networks, Haoer and Virk35 calculated the reverse Zagreb indices, reverse hyper indices, and associated polynomials. The reverse ABC index was calculated for several dendrimer generations by Hashmi et al.36.
For Zanamivir and Oseltamivir, Prasanna Poojari et al.37 calculated a few reverse degree-based topological indices. Decitabine, guadecitabine, and azacitidine are three succinct drug structures that are utilised in the treatment of cancer. Monjit Chamua et al.38 calculated the M-polynomial and neighbourhood M-polynomial of these drugs. Additionally, a few degree-based and neighbourhood degree sum-based topological indices were calculated using these polynomials. Studies on the QSPR analysis of these cancer medications, however, were not undertaken. To close this gap, several reverse degree-based topological indices of the molecular structures of azacitidine, decitabine, and guadecitabine are computed in this research in order to analyse the QSPR analysis.
The reverse-degree version of the general Randic index 
for 
 |
1 |
The reverse-degree version of the well-known atom-bond connectivity index:
 |
2 |
The further reverse degree indices are defined in the upcoming section. The flowchart of the proposed methodology is given in Fig. 1.
Fig. 1.
Flowchart of proposed methodology.
Methods and calculations
Our useful graph invariant was the augmented Zagreb index to aid in the understanding of the heat of formation in octanes better than heptanes. This new topological index has been used in chemical graph theory and has a higher predictive power than the atom-bond connectivity index. In-depth studies have revealed that when compared to popular ones, it is effective. Topological indices based on degrees, like the AZI index, have improved greater sensitivity and characteristics of correlation. Its application has been especially useful in the study of the nature of the compounds present, providing very interesting information about their formation and heat. The enhanced Zagreb index contains much in chemical graph theory from the perspective of molecular structure research and fundamental character prediction.
Polypropylenimine octaamine dendrimers
One family of highly branched macromolecular structures with a tree-like structure is called polypropylenimine octaamine dendrimers (
). The possible uses of these dendrimers are numerous. Drugs that are intended to be administered to particular cells or tissues can be attached to the dendrimer using its amine end groups. In order to treat illnesses or make proteins, they can be used to transfer genes to cells. Additionally, certain compounds or ions can be detected using the dendrimer. Table 1 displays 
’s edge partition according to reverse degree. Figure 2 of 
and for further detail for 
, see39.
Table 1.
The edge partition of 
based on reverse degree of vertices.
|
Frequency |
|---|---|
| (3,2) |
|
| (2,2) |
|
| (3,1) |
|
| (2,1) |
|
Fig. 2.
Polypropylenimine octaamine dendrimers (
).
Now, we determined our first result on the reverse-degree atom-bond connectivity index as:
 |
3 |
The reverse degree geometric arithmetic index GA of a graph G is introduced by Vukic et al.40 in 2009 and we determine 
for 
as:
 |
4 |
The first and second Zagreb index is introduced in 1972 by Gutman41, 42 and calculated for 
as:
 |
5 |
 |
6 |
Shirdel et al.43 introduced Hyper-Zagreb index in 2013 and calculated for 
as:
 |
7 |
Furtula and Gutman44, presented Forgotten topological indices in 2015 and calculated for 
as:
 |
8 |
Furtula et al.45 defined Augmented zagreb index in 2010 and calculated for 
as:
 |
9 |
Vukicevic and Gasperov46, 47 define the Inverse sum index and calculated for 
as: :
 |
10 |
The sum-connectivity index was introduced by Bo Zhou and Nenad Trinajstić48 to be described and calculated for 
as:
 |
11 |
Ranjini et al.48, defined, the redefined first, second and third Zagreb indices for a graph G in 2013 and calculated for 
as:
 |
12 |
 |
13 |
 |
14 |
Gutman et al.24, recently, introduced the reduced reciprocal Randić index and reduced second Zagreb index and calculated for 
as:
 |
15 |
Gutman et al.24, recently, introduced reduced reciprocal Randić index and reduced second Zagreb index and calculated for 
as:
 |
16 |
The first Gourava coindex49 determined for 
as:
 |
17 |
The second Gourava coindex50 determined for 
as:
 |
18 |
The first and second hyper-Gourava indices and hyper-Gourava coindices of a molecular graph G is defined in51 and calculated for 
as:
 |
19 |
 |
20 |
SK index, 
index, and 
index of a molecular graph G defined in52 and calculated for 
as:
 |
21 |
 |
22 |
 |
23 |
Figure 3 illustrates the graphical comparison of various computed topological indices for polypropylenimine octaamine dendrimers as a function of generation parameter t. In Fig. 3a, degree–based and connectivity-related indices such as ABC, GA, M1, M2, HM, F, AZI, ISI, SCI, and M3 are plotted against increasing values of t. The graph demonstrates that all indices exhibit a monotonic increasing trend as the generation grows, reflecting the rapid expansion of the dendrimer structure. However, the growth rates differ significantly: certain indices (such as AZI and HM) increase more sharply, indicating higher sensitivity to structural branching and edge proliferation, whereas others show comparatively moderate growth behavior.
Fig. 3.
The graphical comparison among computed topological indices for polypropylenimine octaamine dendrimers (
).
Figure 3b presents another comparative analysis of selected indices including M4, RRR, GO1, GO2, HGO1, HGO2, SK, SK1, and SK2. Similar to Fig. 3a, all indices increase exponentially with respect to the dendrimer generation t, but the magnitude and steepness of growth vary considerably. Some indices demonstrate extremely rapid escalation for higher generations, highlighting their strong dependence on the increasing number of vertices and edges in higher-order dendrimer structures.
Fullerene dendrimer
Fullerene dendrimers, or fullerene-based dendrimers, or fullerene nanoclusters are a particular type where the basis or center is a fullerene molecule. Dendrimers are extremely branched, treelike. structure: macromolecular structures of well-defined and controlled structure. Fullerene, as the central core, is utilized and gives special properties to these dendrimers. The Fullerene dendrimers are interesting because of the following characteristics. conjunction of the special features of fullerenes and the dendritic design. They may be highly soluble in diverse solvents, have superior electron-accepting qualities, and satisfactory thermal stability. These are the promising characteristics. use in different applications such as organic electronics, photovoltaics, catalysis, drug delivery systems, etc., and nanotechnology. Take another type of dendrimer, such as a fullerene dendrimer, as denoted by 
(Fig. 4), where t is the number of steps of growth. The edge partition of 
based on reverse degree is shown in Table 2, and for further details of 
, see39.
Fig. 4.
Fullerene dendrimer (
).
Table 2.
The edge partition of 
based on reverse degree of vertices.
|
Frequency |
|---|---|
| (4,2) |
|
| (3,3) |
|
| (3,2) |
|
| (2,2) | 86 |
| (2,1 ) | 6 |
| (1,1) | 3 |
Now, we determine the reverse degree indices. The reverse degree atom-bond connectivity index for 
as:
 |
24 |
The reverse degree geometric arithmetic index GA for 
as:
 |
25 |
The first and second the reverse degree Zagreb index for 
as41, 42:
 |
26 |
 |
27 |
The reverse degree Hyper-Zagreb index for 
as:
 |
28 |
The reverse degree Forgotten topological index for 
as:
 |
29 |
The reverse degree Inverse sum index for 
as:
 |
30 |
The reverse degree sum-connectivity index for 
as:
 |
31 |
The reverse degree redefined first, second and third Zagreb indices for 
as:
 |
32 |
 |
33 |
 |
34 |
The reverse degree reduced reciprocal Randić index and reduced second Zagreb index for 
as:
 |
35 |
 |
36 |
The reverse degree first Gourava coindex of a graph G49.
 |
37 |
The reverse degree second Gourava coindex for 
as:
 |
38 |
The reverse degree first and second hyper-Gourava indices and hyper-Gourava coindices for 
as:
 |
39 |
 |
40 |
The reverse degree SK index, 
index, and 
index ofor 
as:
 |
41 |
 |
42 |
 |
43 |
Figure 5 presents the graphical comparison of several computed topological indices for the fullerene dendrimer (NSF(t)) as a function of the generation parameter t. The figure is divided into two subfigures, (a) and (b), each illustrating different families of degree-based and connectivity-based indices. In Fig. 5a, classical and reverse degree-based indices including ABC, GA, M1, M2, HM, F, AZI, ISI, REZG1, and REZG2 are plotted against increasing values of t. The horizontal axis represents the dendrimer generation t, while the vertical axis denotes the numerical value of the corresponding index. All indices demonstrate a strictly increasing trend as the generation increases, reflecting the rapid structural growth of the fullerene dendrimer. However, the rate of growth differs significantly among the indices. Certain indices such as AZI, HM, and M2 show a much steeper rise, indicating higher sensitivity to vertex degree distribution and branching complexity. In contrast, indices like ISI, REZG1, and REZG2 grow more moderately, suggesting comparatively lower sensitivity to structural expansion. The divergence between curves becomes more pronounced at higher generations, highlighting the increasing discriminative power of these indices for large dendritic structures.
Fig. 5.
The graphical comparison among computed topological indices for fullerene dendrimer (
).
In Fig. 5b, another set of indices including REZG3, RRR, GO1, GO2, HGO1, HGO2, SK, SK1, and SK2 is illustrated. Similar to subfigure (a), all indices exhibit monotonic and rapidly accelerating growth with respect to t. The magnitudes of these indices are significantly larger compared to those in subfigure (a), reaching values on the order of 
, which indicates a stronger dependence on the exponential increase in vertices and edges within the fullerene dendrimer. Among these, certain indices (such as HGO2 and REZG3) display extremely steep growth curves, reflecting their high responsiveness to structural complexity and higher-order degree interactions. Other indices like SK, SK1, and SK2 increase steadily but at comparatively lower rates.
Overall, Fig. 5 clearly demonstrates that all computed topological indices are highly sensitive to the generation growth of the fullerene dendrimer. The comparative graphical analysis not only confirms the exponential expansion behavior of the nanostructure but also provides insight into which indices are more effective in distinguishing higher generations. These graphical trends validate the theoretical formulations and emphasize the applicability of reverse degree–based and higher-order indices in characterizing large dendritic nanostructures.
Entropies for polypropylenimine octaamine dendrimers and fullerene dendrimer networks
The research on entropy-based topological measurements of Polypropylenimine Octaamine Dendrimers and Fullerene Dendrimer has great advantages to chemists as a quantitative characterisation of the molecular complexity, branching behaviours, and structural anomalies. Entropy measures based on degree-based edge partitions give the distribution of atomic connectivity and provide a rigorous method of comparing highly branched dendritic networks and cage-like carbon structures. To chemists, these entropy values are of great assistance in the determination of physicochemical properties, including stability, reactivity, surface activity, and molecular compactness. In addition, the entropy-based indices have been useful in structure-property relationship (SPR) and quantitative structure-activity relationship (QSAR) investigations to convert the complex molecular structure into numerical values that can be computed. This allows the efficient screening, comparison, and optimization of dendrimeric materials for applications in drug delivery, nanotechnology, catalysis, and materials science. Therefore, the entropy analysis not only adds to the theoretical knowledge about molecular topology, but it also assists in making a valuable decision when designing and producing chemicals. There are well-known modified entropies taken from24, 53 and their mathematical expression is given in Table 3.
Table 3.
Entropies and their mathematical expressions.
| Entropies | Mathematical expression |
|---|---|
|
|
|
|
|
|
Where
 |
Lemma 1
Let
be apolypropylenimine octaamine dendrimer network, then we have
 |
44 |
 |
45 |
 |
46 |
Proposition 1
Let
be apolypropylenimine octaamine dendrimer network, the entropies are
 |
47 |
 |
48 |
 |
49 |
Where;
 |
Lemma 2
Let
be a fullerene dendrimer network, then we have
 |
50 |
 |
51 |
Proposition 2
Let
be a fullerene dendrimer network, the entropies are
 |
52 |
 |
53 |
Where;
 |
Conclusion
The Reverse Degree Topological Index (RDTI) for two distinct classes of nanostar dendrimers has been precisely calculated in this work. The head-to-core, core-to-tail, and overall architectures of these intricate macromolecules are revealed in a number of new ways by this study. The structure-property link for nanostar dendrimers starts to take shape as the RDTI values start to appear. Our understanding of dendrimer designs is improved by these RDTI computations, which also provide a theoretical foundation for additional real-world dendrimer applications in materials science, medicine, and nanobiotechnology. This research offers a significant prospect for expansion. It may be possible to look into even more topological indices in order to comprehend dendrimer structures and their functions, both in terms of their comprehensibility and WWII comprehensibility. Our understanding of the connection between topological characteristics and physicochemical properties may be greatly improved by this investigation, which may also advance predictive modelling skills for dendrimer-based applications. It may be possible to validate and enhance our theoretical models by collaborating with experimental researchers to create and analyse dendrimers that have the characteristics we anticipate. Additionally, it would guarantee the dependability of these models in real-world scenarios. This theoretical study serves as a springboard for upcoming empirical research in the rapidly expanding field of nanobiotechnology. Our goal is to close the gap between applications of nanobiotechnology and topological theory. Therefore, the synthesis and functionalisation of dendrimers nanosized polymers having a tree-like branched structure as vehicles for targeted drug delivery, are the focus of our initial set of studies. Background and theory are discussed in detail, and all our computational results are presented with the required accuracy and precision. The numerical comparison of some RDTI is shown in Tables 4, 5, 6, 7, and their graphical comperson are shown in Figs. 6, 7, 8, and 9, respectively.
Table 4.
The numerical compcomparison of some RDTI of 
.
| t | ABC | GA | M1 | M2 | HM | F | AZI | ISI | SCI | M3 |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 25.1856 | 33.5826 | 130 | 116 | 498 | 266 | 261.5 | 5.2164 | 20.3422 | 42.6654 |
| 2 | 71.3152 | 94.8282 | 362 | 316 | 1362 | 730 | 736.5 | 123.081 | 58.3934 | 122.664 |
| 3 | 163.574 | 217.319 | 826 | 716 | 3090 | 1658 | 1686.5 | 1707.03 | 134.496 | 282.661 |
| 4 | 348.093 | 462.302 | 1754 | 1516 | 6546 | 3514 | 3586.5 | 28136.8 | 286.701 | 602.654 |
| 5 | 717.13 | 952.267 | 3610 | 3116 | 13458 | 7226 | 7386.5 | 534894. | 591.11 | 1242.64 |
| 6 | 1455.2 | 1932.2 | 7322 | 6316 | 27282 | 14650 | 14986.5 |
|
1199.93 | 2522.62 |
| 7 | 2931.35 | 3892.06 | 14746 | 12716 | 54930 | 29498 | 30186.5 |
|
2417.57 | 5082.56 |
| 8 | 5883.64 | 7811.77 | 29594 | 25516 | 110226 | 59194 | 60586.5 |
|
4852.84 | 10202.5 |
| 9 | 11788.2 | 15651.2 | 59290 | 51116 | 220818 | 118586 | 121387. |
|
9723.4 | 20442.3 |
| 10 | 23597.4 | 31330.1 | 118682 | 102316 | 442002 | 237370 | 242987. |
|
19464.5 | 40921.8 |
| 11 | 47215.8 | 62687.8 | 237466 | 204716 | 884370 | 474938 | 486187. |
|
38946.7 | 81881. |
| 12 | 94452.5 | 125403. | 475034 | 409516 | 1769106 | 950074 | 972587. |
|
77911.1 | 163799. |
Table 5.
The numerical comparison of some RDTI of 
.
| t | M4 | RRR | GO1 | GO2 | HGO1 | HGO2 | SK | SK1 | SK2 |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 30.1338 | 18.6568 | 246 | 460 | 1062 | 8008 | 65 | 58 | 124.5 |
| 2 | 83.6006 | 48.3136 | 678 | 1228 | 3374 | 19912 | 181 | 158 | 340.5 |
| 3 | 190.534 | 107.627 | 1542 | 2764 | 7998 | 43720 | 413 | 358 | 772.5 |
| 4 | 404.401 | 226.254 | 3270 | 5836 | 17246 | 91336 | 877 | 758 | 1636.5 |
| 5 | 832.136 | 463.509 | 6726 | 11980 | 35742 | 186568 | 1805 | 1558 | 3364.5 |
| 6 | 1687.6 | 938.018 | 13638 | 24268 | 72734 | 377032 | 3661 | 3158 | 6820.5 |
| 7 | 3398.54 | 1887.04 | 27462 | 48844 | 146718 | 757960 | 7373 | 6358 | 13732.5 |
| 8 | 6820.42 | 3785.07 | 55110 | 97996 | 294686 | 1519816 | 14797 | 12758 | 27556.5 |
| 9 | 13664.2 | 7581.14 | 110406 | 196300 | 590622 | 3043528 | 29645 | 25558 | 55204.5 |
| 10 | 27351.7 | 15173.3 | 220998 | 392908 | 1182494 | 6090952 | 59341 | 51158 | 110501. |
| 11 | 54726.7 | 30357.6 | 442182 | 786124 | 2366238 | 12185800 | 118733 | 102358 | 221093. |
| 12 | 109477. | 60726.1 | 884550 | 1572556 | 4733726 | 24375496 | 237517 | 204758 | 442277. |
Table 6.
The numerical comparison of some RDTI of 
.
| t | ABC | GA | M1 | M2 | HM | F | ISI | REZG1 | REZG2 | SCI |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 88.8512 | 127.944 | 548 | 589 | 2402 | 1224 | 159.356 | 128. | 134.633 | 62.7881 |
| 2 | 116.973 | 167.069 | 756 | 849 | 3490 | 1792 | 237.634 | 160.333 | 184.367 | 64.2024 |
| 3 | 173.218 | 245.32 | 1172 | 1369 | 5666 | 2928 | 394.189 | 225. | 283.833 | 67.0308 |
| 4 | 285.707 | 401.821 | 2004 | 2409 | 10018 | 5200 | 707.299 | 354.333 | 482.767 | 72.6876 |
| 5 | 510.684 | 714.822 | 3668 | 4489 | 18722 | 9744 | 1333.52 | 613. | 880.633 | 84.0013 |
| 6 | 960.639 | 1340.83 | 6996 | 8649 | 36130 | 18832 | 2585.96 | 1130.33 | 1676.37 | 106.629 |
| 7 | 1860.55 | 2592.83 | 13652 | 16969 | 70946 | 37008 | 5090.84 | 2165. | 3267.83 | 151.884 |
| 8 | 3660.37 | 5096.84 | 26964 | 33609 | 140578 | 73360 | 10100.6 | 4234.33 | 6450.77 | 242.393 |
| 9 | 7260.01 | 10104.9 | 53588 | 66889 | 279842 | 146064 | 20120.1 | 8373. | 12816.6 | 423.413 |
| 10 | 14459.3 | 20120.9 | 106836 | 133449 | 558370 | 291472 | 40159.1 | 16650.3 | 25548.4 | 785.451 |
| 11 | 28857.9 | 40153. | 213332 | 266569 | 1115426 | 582288 | 80237.2 | 33205. | 51011.8 | 1509.53 |
| 12 | 57655. | 80217.2 | 426324 | 532809 | 2229538 | 1163920 | 160393. | 66314.3 | 101939. | 2957.68 |
Table 7.
The numerical comparison of some RDTI of 
.
| t | REZG3 | RRR | GO1 | GO2 | HGO1 | HGO2 | SK | SK1 | SK2 |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 2654 | 125.614 | 1137 | 2654 | 10719 | 70556 | 274 | 294.5 | 600.5 |
| 2 | 4022 | 172.543 | 1605 | 4022 | 16275 | 120236 | 378 | 424.5 | 872.5 |
| 3 | 6758 | 266.399 | 2541 | 6758 | 27387 | 219596 | 586 | 684.5 | 1416.5 |
| 4 | 12230 | 454.112 | 4413 | 12230 | 49611 | 418316 | 1002 | 1204.5 | 2504.5 |
| 5 | 23174 | 829.538 | 8157 | 23174 | 94059 | 815756 | 1834 | 2244.5 | 4680.5 |
| 6 | 45062 | 1580.39 | 15645 | 45062 | 182955 | 1610636 | 3498 | 4324.5 | 9032.5 |
| 7 | 88838 | 3082.09 | 30621 | 88838 | 360747 | 3200396 | 6826 | 8484.5 | 17736.5 |
| 8 | 176390 | 6085.5 | 60573 | 176390 | 716331 | 6379916 | 13482 | 16804.5 | 35144.5 |
| 9 | 351494 | 12092.3 | 120477 | 351494 | 1427499 | 12738956 | 26794 | 33444.5 | 69960.5 |
| 10 | 701702 | 24105.9 | 240285 | 701702 | 2849835 | 25457036 | 53418 | 66724.5 | 139593. |
| 11 | 1402118 | 48133.2 | 479901 | 1402118 | 5694507 | 50893196 | 106666 | 133285. | 278857. |
| 12 | 2802950 | 96187.6 | 959133 | 2802950 | 11383851 | 101765516 | 213162 | 266405. | 557385. |
Fig. 6.
The numerical comparison of certain topological indices for 
.
Fig. 7.
The numerical comparison of certain topological indices for 
.
Fig. 8.
The numerical comparison of certain topological indices for 
..
Fig. 9.
The numerical comparison of certain topological indices for 
.
Author contributions
Ashfaq Ahmed Qummer conceptualized the study and supervised the research. Muhammad Saqib carried out the formal analysis and computational work. Shahbaz Ali contributed to the methodology, mathematical modeling, and validation of results. Mamo Abebe Ashebo contributed to data interpretation and assisted in drafting the manuscript. Sardor Jumaniyazov contributed to the investigation, literature review, and manuscript revision. All authors reviewed and approved the final version of the manuscript.
Data availability
All data generated or analyzed during this study are included in this published article.
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Contributor Information
Shahbaz Ali, Email: shahbazali@iub.edu.pk.
Mamo Abebe Ashebo, Email: mamoa@wollegauniversity.edu.et.
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