
==== Front
Sci Prog
Sci Prog
SCI
spsci
Science Progress
0036-8504
2047-7163
SAGE Publications Sage UK: London, England

39212153
10.1177/00368504241271719
10.1177_00368504241271719
Physics
Degree-based topological indices and entropies of diamond crystals
https://orcid.org/0000-0002-4709-3860
Khan Abdul Rauf 1
Ullah Zafar 2
Imran Muhammad 3
Salman Muhammad 4
Zia Arooj 1
Tchier Fairouz 5
Hussain Shahid 6
1 Department of Mathematics, Faculty of Sciences, Ghazi University, Dera Ghazi Khan, Pakistan
2 Division of Science and Technology, Department of Mathematics, University of Education, Lahore, Pakistan
3 Department of Mathematical Sciences, 11239 United Arab Emirates University , Al Ain, United Arab Emirates
4 Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur, Pakistan
5 Mathematics Department, College of Science, 37850 King Saud University , Riyadh, Saudi Arabia
6 Department of Engineering Science and Mathematics, Energy Engineering Division, Lulea University of Technology, Lulea, Sweden
Abdul Rauf Khan, Department of Mathematics, Faculty of Sciences, Ghazi University, 32200, Dera Ghazi Khan, Pakistan. Emails: khankts@gmail.com; imrandhab@gmail.com
30 8 2024
Jul-Sep 2024
107 3 00368504241271719© The Author(s) 2024
2024
SAGE Publications
https://creativecommons.org/licenses/by-nc/4.0/ This article is distributed under the terms of the Creative Commons Attribution-NonCommercial 4.0 License (https://creativecommons.org/licenses/by-nc/4.0/) which permits non-commercial use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access page (https://us.sagepub.com/en-us/nam/open-access-at-sage).
High hardness, low friction coefficient and chemical resistance are only a few of the exceptional mechanical qualities of diamond. Diamonds can be artificially created to have different levels of conductivity, or they can be single, micro or nanocrystalline and highly electrically insulating. It also has high biocompatibility and is famous for being mechanically robust. Due to its high hardness, lack of ductility and difficulty in welding, diamond is a challenging material to construct devices with. Diamonds have experienced a rise in attention as a biological material in recent decades due to new synthesis and fabrication techniques that have eliminated some of these disadvantages. In general, entropic measurements are used for investigating the chemical or biological properties of molecular structures. This study calculates several important K -Banhatti entropies, redefined Zagreb entropies and atom-bond sum connectivity entropy for diamond crystals. We also present a numeric and graphical explanations of obtain indices.

Degree
topological index
entropy
diamond
molecular structure
mathematical modelling
typesetterts19
cover-dateJuly-September 2024
==== Body
pmcIntroduction

Diamond is regarded as a suitable strengthening stage of metal matrix composites because it possesses the maximum heat conductivity of any natural substance, a low rate of thermal expansion, extreme durability and exceptional chemical solidity.1,2 The artificial diamond market has grown recently, and this has culminated in a large drop in price, which encourages the use of artificial diamond in composite materials. 2

Due to its outstanding properties, diamond is a valued material for a wide range of uses. For instance, this carbon allotrope's tetrahedral geometry with covalent carbon bonds allows for the greatest thermal conductivity, the lowest heat expansion coefficient (0.8 × 10−6 K-1), and exceptionally high hardness (98 GPa).1,3–6

Additionally, diamond is biocompatible and has a strong chemical resistance to bases and acids. 7 Carbon is one of the lightest elements and makes up diamonds as well. An isotope effect's size is directly related to the square root of the mass ratio among the different isotopes. Where this ratio is biggest, the lighter elements are where isotope effects are most prominent. The stable isotopes of carbon are 12C (98.9 percentage) and 13C (1.1 percentage), which occur naturally.

Simple cryogenic distillation can be used to separate carbon isotopes due to the comparatively large percentage variation in mass. This has led to the commercial availability of a wide range of carbon compounds with high isotopic purity. 8 The lattice of tightly coupled, extremely symmetric crystals formed by covalent bonds is formed by the lowest mass element, which is diamond. The material is currently attracting enormous interest for a variety of optical applications due to its exceptional qualities that arise from recent advancements in its synthesis.

There is a tantalizing possibility of much increased capability because the optical properties are distinct from those of other materials in many ways. The assistance of optical design greatly depends on having a thorough understanding of how electromagnetic radiation interacts with diamond's bulk and surface. 9 The results of the simulation showed that too much oxygen increases the oxidation of diamond, which raises the substance's mechanical rigidity and roughness rate. The lower rigidity of the oxidized diamond allowed for a larger surface deformation.10–12

Due to its advantageous characteristics, such as its malleability and mechanical workability, the alloy of copper is widely used in diamond instruments. 13 Copper is a typical bond material used for numerous diamond-cutting instruments due to its inexpensive cost and low sintering temperature. 14 The diamond swords can be properly squeezed and self-sharpen when a suitable ratio of diamonds is established and enough cutting edges are scattered across the working surface of the diamond tool. In addition, the cutting life and efficiency of diamond particles is directly affected by their size.15–17

Several methods exist for converting a chemical network into a graph. Examine the types of compounds that are present in the network. 18 These may involve other chemicals, ions or radicals. 19 Find out which species in the network are involved in which chemical processes. This can entail looking over experimental results, chemical equations or other knowledge sources. 20

Al-Raee 21 initiated the formula for calculating the specific bond volume of the Morse potential is universal and can be used for many materials. Shi et al. studied the stability of the interface is determined by the type of interfacial border and bond, as well as the populations and numbers of interfacial bonds. 22 The layer-projected density of states analysis indicates that all the examined contacts display metallic properties. 22 Zhang et al. presents the prediction of a new carbon phase, called pentapeptide diamond, which exhibits orthorhombic crystal structure and sp3 bonding. 23 The prediction was made by integrating the particle swarm optimization method with first-principle calculations. The phonon spectra, total energy and elastic constants calculations of the pentapeptide diamond provide evidence that it is dynamically, thermally and mechanically stable under zero pressure. It has a large bulk modulus of 385 GPa and a Vickers hardness of 72.6 GPa, which is like that of diamond. The electronic band structure simulations indicate that the pentapeptide diamond possesses a direct band gap of 4.18 electron volts (eV). 23

Make an edge connecting the points that represent the reactants and outcomes of each chemical reaction in the graph. 24 Complete the graph by adding any other details, such as rates of reactions or thermodynamic characteristics, that might be essential to the chemical network.25,26 Once the chemical network is represented as a graph, its composition can be examined using a range of graph-theoretical methods and procedures.27–29 For instance, strongly connected node groups can be detected using community detection techniques, and important nodes in the network can be located using centrality metrics.30,31

This method can offer guidance for the development of novel chemical processes and reactions as well as knowledge about the chemistry of intricate chemical structures. 32 Topological indices play an essential role in providing guidance for the treatment of tumours or malignancies.33,34 Experimental or numerical methods can be used to find these indices. Computer analysis offers a time- and money-efficient solution because experimental data, despite their high cost, are important. The eccentricity-based topological indices are a key component of the theory of chemical graphs. 35 Chemist Wiener created the first topological index in 1947. 36

Methodology

Consider a graph with the form G=(V,E) with the vertex position V=V(G) and the edge group E=E(G) . The number of edges that join a vertex Ci defines its degree, and we represent it by the notation αCi . For any edge e=C1∼C2 of G, the degree of e is defined as αe=αC1+αC2−2 . A molecular graph, often mention to as a chemical network in graph theory, is a straightforward network with vertices and edges that represent individual atoms and chemical bonds. 37 We use eight different degree-based topological indices to find the entropy in this paper. The formula of each of index is presented in Table 1 and they are introduced as follows: four Banhatti indices were first computed by Kulli in 2016;38,39 concepts of atom-bond and sum connectivity indices were collected by Ali et al., which led to the create another molecular descriptor known as the atom-bond sum-connectivity index in. 40 We use the method of edge partition to compute our results. By observing the degrees of vertices and edges of the underline structure (graph) G, we partition the edge group E(G) into parts of the form: E(x,y)={e=C1∼C2∈E(G)|x=αC1,y=αC2}.

Table 1. Degree-based topological indices.

Index	Symbol	Formula	
1st redefined Zagreb	ReZG1	∑C1∼C2αC1+αC2αC1×αC2	
2nd redefined Zagreb	ReZG2	∑C1∼C2αC1×αC2αC1+αC2	
3rd redefined Zagreb	ReZG3	∑C1∼C2(αC1×αC2)(αC1+αC2)	
Atom-bond sum connectivity	ABS	∑C1∼C2(αC1+αC2−2)(αC1+αC2)	
1st K -Banhatti	B1	∑e=C1∼C2(αC1+αe)	
2nd K -Banhatti	B2	∑e=C1∼C2(αC1×αe)	
1st K -hyper Banhatti	HB1	∑e=C1∼C2(αC1+αe)2	
2nd K -hyper Banhatti	HB2	∑e=C1∼C2(αC1×αe)2	

Shannon introduced the concept of entropy in 1948,41–43 which measures the molecular disorganization of the system as well.19,44 The measurement of edge-weighted graph entropy was initially presented in 2009 18 for an edge-weighted graph G=((V,E),ψ(Ps˙1Ps˙2)) , where ψ(Ps˙1Ps˙2) denotes the edge-weight of an edge (Ps˙1Ps˙2) . As demonstrated in a series of papers,45–49 we perform the entropy calculations based on the scalar multiplicative indices instead of standard multiplicative indices to show the discriminating power of different indices considered in this study. The entropy of a G is defined by the formula: (1) ENTψ(G)=−∑s˙1∼s˙2ψ(Ps˙1Ps˙2)∑s˙1∼s˙2ψ(Ps˙1Ps˙2)log{ψ(Ps˙1Ps˙2)∑s˙1∼s˙2ψ(Ps˙1Ps˙2)}

Results

In this section, we determine exact values of all the degree-based topological indices and entropies tabulated in Tables 1 and 2. For this, first we provide the edge partition of diamond crystal CBRℓ as shown in Figures 1 and 2 on the base of degree of vertices and edges in Table 3.

Figure 1. Step-by-step structure of a diamond.

Figure 2. Tetrahedral structure of a diamond.

Table 2. Formulae to compute the entropy on the base of indices tabulated in Table 1.

Entropy	Symbol	Formula	
1st redefined Zagreb	ENTReZG1	log(ENTReZG1)−1ENTReZG1log{∏C1∼C2[αC1+αC2αC1αC2][αC1+αC2αC1αC2]}	
2nd redefined Zagreb	ENTReZG2	log(ENTReZG2)−1ENTReZG2log{∏C1∼C2[αC1αC2αC1+αC2][αC1αC2αC1+αC2]}	
3rd redefined Zagreb	ENTReZG3	log(ENTReZG2)−1ENTReZG2log{∏C1∼C2[(αC1αC2)(αC1+αC2)][(αC1αC2)(αC1+αC2)]}	
ABS Connectivity	ENTABS	log(ENTABS)−1ENTABSlog{∏C1∼C2[αC1+αC2−2αC1+αC2][αC1+αC2−2αC1+αC2]}	
1st K-Banhatti	ENTB1	log(ENTB1)−1ENTB1log{∏e=C1∼C2[αC1+αe][αC1+αe]}	
2nd K-Banhatti	ENTB2	log(ENTB2)−1ENTB2log{∏e=C1∼C2[αC1αe][αC1αe]}	
1st K-hyper Banhatti	ENTHB1	log(ENTB1)−1ENTB1log{∏e=C1∼C2[αC1+αe]2[αC1+αe]2}	
2nd K-hyper Banhatti	ENTHB2	log(ENTB2)−1ENTB2log{∏e=C1∼C2[αC1αe]2[αC1αe]2}	

Table 3. Edge partition with edge degree of CBRℓ .

Edge part	E(1,4)	E(2,4)	E(3,4)	E(4,4)	
Edge degree	3	4	5	6	
Frequency	4	12(l−1)	6l(l−3)+12	13(2l3−12l2+22l−12)	

By using the edge partition of Table 3 in each formula of Table 1, we have the following equations: (2) ReZG1(CBRℓ)=(4)φ54+(12(l−1))φ68+(6l(l−3)+12)φ712+(λ)φ816

(3) ReZG2(CBRℓ)=(4)φ45+(12(l−1))φ86+(6l(l−3)+12)φ127+(λ)φ168

(4) ReZG3(CBRℓ)=(4)φ20+(12(l−1))φ48+(6l(l−3)+12)φ84+(λ)φ128

(5) ABS(CBRℓ)=(4)φ35+(12(l−1))φ46+(6l(l−3)+12)φ57+(λ)φ68

(6) B1(CBRℓ)=(4)φ11+(12(l−1))φ14+(6l(l−3)+12)φ17+(λ)φ20

(7) B2(CBRℓ)=(4)φ15+(12(l−1))φ24+(6l(l−3)+12)φ35+(λ)φ40

(8) HB1(CBRℓ)=(4)φ65+(12(l−1))φ100+(6l(l−3)+12)φ145+(λ)φ200

(9) HB2(CBRℓ)=(4)φ153+(12(l−1))φ320+(6l(l−3)+12)φ625+(λ)φ1152

After evaluating the first derivative of the right hand side of each equation form (2) to (9) at φ=1 , we get the following indices: (10) ReZG1(CBRℓ)=13l3+32l2+136l+1

(11) ReZG2(CBRℓ)=43l3+167l2−421l−835

(12) ReZG3(CBRℓ)=2563l3−8l2+83l

(13) ABS(CBRℓ)=(4)35+(12(l−1))46+(6l(l−3)+12)57+(2l3−12l2+22l−123)68

(14) B1(CBRℓ)=403l3+22l2+263l

(15) B2(CBRℓ)=32l3+18l2−10l

(16) HB1(CBRℓ)=4003l3+70l2+1703l

(17) HB2(CBRℓ)=768l3−858l2+1038l−336

Now, by using indices from equations (10) to (17) in formulae of entropy, given in Table 2, we obtain the following entropies: ENTReZG1(CBRℓ)=log(13l3+32l2+136l+1)−1(13l3+32l2+136l+1)log{(4)(54)54×(12(l−1))(68)68×(6l(l−3)+12)(712)712×(λ)(816)816}

ENTReZG2(CBRℓ)=log(43l3+167l2−421l−835)−143l3+167l2−421l−835log{(4)(45)45×(12(l−1))(86)86×(6l(l−3)+12)(127)127×(λ)(168)168}

ENTReZG3(CBRℓ)=log(2563l3−8l2+83l)−12563l3−8l2+83llog{(4)(20)20×(12(l−1))4848×(6l(l−3)+12)8484×(λ)128128}

ENTABS(CBRℓ)=log(ABS)−1ABSlog{(4)(35)35×(12(l−1))(46)46×(6l(l−3)+12)(57)57×(λ)(68)68}

ENTB1(CBRℓ=log(403l3+22l2+263l)−1403l3+22l2+263llog{(4)(11)11×(12(l−1))(14)14×(6l(l−3)+12)(17)17×(λ)(20)20}

ENTB2(CBRℓ=log(32l3+18l2−10l)−132l3+18l2−10llog{(4)1515×(12(l−1))2424×(6l(l−3)+12)3535×(λ)4040}

ENTHB1(CBRℓ)=log(4003l3+70l2+1703l)−14003l3+70l2+1703llog{(4)(6565)(12(l−1))(100100)×(6l(l−3)+12)(145145)×(λ)(200200)}

ENTHB2(CBRℓ)=log(768l3−858l2+1038l−336)−1768l3−858l2+1038l−336log{(4)(153)153×(12(l−1))320320×(6l(l−3)+12)625625×(λ)11521152}

Comparison

This part presents numeric and graphical comparisons of the computed results as presented in Table 4 and their graphical analysis as shown in Figure 3. For the sake of comparison of results, we take values l=2,3,…,12 . We observed that as the value of l increased the index and entropy values are also increased gradually. We utilize Corel Draw and Origin for figure creation, and Maple for numeric computations.

Figure 3. Graphical expression of TIs of CBRℓ .

Table 4. Numeric expressions of TIs of CBRℓ .

l	ReZG1	ReZG2	ReZG3	ABS	B1	B2	HB1	HB2	
2	14	19.2	656	12.896	212	348	1460	4452	
3	30	55.77	2240	32.84	584	1056	4400	15792	
4	55	120.91	5344	66.382	1240	2376	9880	39240	
5	91	222.63	10480	116.998	2260	4500	18700	79404	
6	140	368.91	18160	188.15	3724	7620	31660	140892	
7	204	567.77	28896	283.296	5712	11928	49560	228312	
8	285	827.2	43200	405.91	8304	17616	73200	346272	
9	385	1155.2	61584	559.44	11580	24876	103380	499380	
10	506	1559.77	84560	747.37	15620	33900	140900	692244	
11	650	2048.91	112640	973.15	20504	44880	186560	929472	
12	819	2630.63	146336	1240.25	26312	58008	241160	1215672	

Concluding remarks

This manuscript presents key findings regarding the entropy of eight degree-based topological indices for a molecular structure of diamonds. By the reason of rich conception, we presented numeric and graphical analysis of obtained results of toplogical indices for few initial values of the used parameter as shown in Table 4 and Figure 3. From obtained results and their provided comparisons, each of the calculated topological indices can be regarded as a finest/moral or nastiest/immoral descriptor for a topological characterization for the molecular structure of diamond according to the following values-based hierarchy (this pyramid is just based upon our mathematical computations, numeric and graphical analysis. However, the novelty of each topological index can be followed after investigating that which index is best correlating with which of the chemical property): ReZG1<ABS<ReZG2<B1<B2<ReZG3<HB1<HB2.

Acknowledgements

This research was supported by the researchers Supporting Project Number (RSP2024R401), King Saud University, Riyadh, Saudi Arabia.

Authors’ note: Future work: We intend to characterize the molecular structure of a diamond crystal via distance-based topological indices and entropies.

Authors contribution: This work was equally contributed by all authors.

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding: The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD: Abdul Rauf Khan https://orcid.org/0000-0002-4709-3860
==== Refs
References

1 Ren X She D Peng Z . Fabrication of diamond enhanced WC-Ni composites by spark plasma sintering. Int J Refract Metals Hard Mater 2022; 102 : 105732.
2 Liu Q Wang T Jiao J , et al. The microstructures and properties of diamond reinforced Cu-Ni-Si-Ti alloys. Mater Sci Eng A 2023; 862 : 144478.
3 Williams OA . Nanocrystalline diamond. Diam Relat Mater 2011; 20 : 621–640.
4 Williams OA Douhéret O Daenen M , et al. Enhanced diamond nucleation on monodispersed nanocrystalline diamond. Chem Phys Lett 2007; 445 : 255–258.
5 Pobedinskas P Degutis G Dexters W , et al. Nanodiamond seeding on plasma-treated tantalum thin films and the role of surface contamination. Appl Surf Sci 2021; 538 : 148016.
6 Chen YC Tzeng Y Cheng AJ , et al. Inkjet printing of nanodiamond suspensions in ethylene glycol for CVD growth of patterned diamond structures and practical applications. Diam Relat Mater 2009; 18 : 146–150.
7 Avvaru B Patil MN Gogate PR , et al. Ultrasonic atomization: effect of liquid phase properties. Ultrasonics 2006; 44 : 146–158.16321416
8 Anthony TR Banholzer WF . Properties of diamond with varying isotopic composition. Diam Relat Mater 1992; 1 : 717–726.
9 Mildren RP . Intrinsic optical properties of diamond. Optic Eng Diamond 2013; 1 : 1–34.
10 Rysaeva LK Lisovenko DS Gorodtsov VA , et al. Stability, elastic properties and deformation behavior of graphene-based diamond-like phases. Comput Mater Sci 2020; 172 : 109355.
11 Baimova J Rysaeva LK Rudskoy AI . Deformation behavior of diamond-like phases: molecular dynamics simulation. Diam Relat Mater 2018; 81 : 154–160.
12 Gou R Luo X Chen J , et al. Study on the tribological properties of diamond and SiC interactions using atomic scale numerical simulations. Tribol Int 2023; 178 : 108093.
13 Fateh A Aliofkhazraei M Rezvanian AR . Review of corrosive environments for copper and its corrosion inhibitors. Arab J Chem 2020; 13 : 481–544.
14 Ma B Pang Q Lou J . Rod-like brazed diamond tool fabricated by supersonic-frequency induction brazing with Cu-based brazing alloy. Int J Refract Metals Hard Mater 2014; 43 : 25–29.
15 Liu DM Tseng WJ . Influence of debinding rate, solid loading and binder formulation on the green microstructure and sintering behaviour of ceramic injection mouldings. Ceram Int 1998; 24 : 471–481.
16 Wang Y Zhang L Li X , et al. On hot isostatic pressing sintering of fused filament fabricated 316L stainless steel – evaluation of microstructure, porosity, and tensile properties. Mater Lett 2021; 296 : 129854.
17 He T Zhang S Kong X , et al. Influence of diamond parameters on microstructure and properties of copper-based diamond composites manufactured by fused deposition modeling and sintering (FDMS). J Alloys Compd 2023; 931 : 167492.
18 Hu M Ali H Binyamin MA , et al. On distance-based topological descriptors of chemical interconnection networks. J Math 2021; 1 : 5520619.
19 Hayat S . Computing distance-based topological descriptors of complex chemical networks: new theoretical techniques. Chem Phys Lett 2017; 688 : 51–58.
20 Hayat S Khan S Imran M . Quality testing of spectrum-based distance descriptors for polycyclic aromatic hydrocarbons with applications to carbon nanotubes and nanocones. Arab J Chem 2021; 14 : 102994.
21 Al-Raeei M . Morse potential specific bond volume: a simple formula with applications to dimers and soft-hard slab slider. J Phys Condens Matter 2022; 34 : 284001.
22 Shi XR Huang S Huang Y , et al. Atomic structures and electronic properties of Ni or N modified Cu/diamond interface. J Phys Condens Matter 2020; 32 : 225001.31910398
23 Zhang C Yang X Lv R , et al. Pentaheptite diamond: a new carbon allotrope. J Phys Condens Matter 2022; 34 : 184003.
24 Khan AR Mutlib A Campeña FJH , et al. Investigation of reduced reverse degree based polynomials & indices of gold crystals. Phys Scr 2024; 99 : 075259.
25 Malik MYH Hayat S Khan S , et al. Predictive potential of spectrum-based topological descriptors for measuring the π-electronic energy of benzenoid hydrocarbons with applications to boron triangular and boron α-nanotubes. Math Methods Appl Sci 2021.
26 Khan AR Ullah Z Imran M , et al. Molecular temperature descriptors as a novel approach for QSPR analysis of Borophene nanosheets. PLoS One 2024; 19 : e0302157.
27 Ghani MU Sultan F Tag El Din ESM , et al. A paradigmatic approach to find the valency-based K-Banhatti and redefined Zagreb entropy for niobium oxide and a metal–organic framework. Molecules 2022; 27 : 6975.36296567
28 Khan AR Awan N ul H Ghani MU , et al. Fundamental aspects of skin cancer drugs via degree-based chemical bonding topological descriptors. Molecules 2023; 28 : 3684.37175093
29 Khan AR Ghani MU Ghaffar A , et al. Characterization of temperature indices of silicates. Silicon 2023; 15 : 6533–6539.
30 Imran M Khan AR Husin MN , et al. Computation of entropy measures for metal-organic frameworks. Molecules 2023; 28 : 4726.37375281
31 Khan AR Awan NUH Tchier F , et al. An estimation of physiochemical properties of bladder cancer drugs via degree-based chemical bonding topological descriptors. J Biomol Struct Dyn 2023: 1–9.
32 Alsaadi FE Salman M Rehman MU , et al. On the geodesic identification of vertices in convex plane graphs. Math Probl Eng 2020; 2020 : 1–13.
33 Khan AR Zia A Campeña FJH , et al. Investigations of entropy double & strong double graph of silicon carbide. Silicon 2024; 16 : 4187–4197.
34 Husin MN Khan AR Awan NUH , et al. Multicriteria decision making attributes and estimation of physicochemical properties of kidney cancer drugs via topological descriptors. PLoS One 2024; 19 : e0302276.
35 Chu YM Khan AR Ghani MU , et al. Computation of Zagreb polynomials and Zagreb indices for benzenoid triangular & hourglass system. Polycycl Aromat Compd 2023; 43 : 4386–4395.
36 Wiener H . Structural determination of paraffin boiling points. J Am Chem Soc 1947; 69 : 17–20.20291038
37 Gowtham KJ Husin MN . Multiplicative reverse product connectivity and multiplicative reverse sum connectivity of silicate network. EDUCATUM J Sci Math Technol 2023; 10 : 90–100.
38 V R K . Hyper Zagreb-K-Banhatti indices of graphs. Int J Math Trends Technol 2020; 66 : 123–130.
39 Kulli VR . Graph indices. In: Handbook of research on advanced applications of graph theory in modern society. IGI Global, 2020, pp.66–91.
40 Saeed N Long K Mufti ZS , et al. Degree-based topological indices of boron B12. J Chem 2021; 2021 : 1–6.
41 Ali A Furtula B Redžepović I , et al. Atom-bond sum-connectivity index. J Math Chem 2022; 60 : 2081–2093.
42 Shannon CE . A mathematical theory of communication. Bell Syst Tech J 1948; 27 : 379–423.
43 Zaman S Hakami KH Rasheed S , et al. Reduced reverse degree-based topological indices of graphyne and graphdiyne nanoribbons with applications in chemical analysis. Sci Rep 2024; 14 : 547.
44 Yu G Siddiqui MK Hussain M , et al. On topological indices and entropy measures of beryllonitrene network via logarithmic regression model. Sci Rep 2024; 14 . doi:10.1038/s41598-024-57601-1
45 Arockiaraj M Jency J Maaran A , et al. Refined degree bond partitions, topological indices, graph entropies and machine-generated boron NMR spectral patterns of borophene nanoribbons. J Mol Struct 2024; 1295 : 136524.
46 Jacob K Clement J Arockiaraj M , et al. Topological characterization and entropy measures of tetragonal zeolite merlinoites. J Mol Struct 2023; 1277 : 134786.
47 Paul D Arockiaraj M Jacob K Clement J . Multiplicative versus scalar multiplicative degree based descriptors in QSAR/QSPR studies and their comparative analysis in entropy measures. Eur Phys J Plus 2023; 138 . doi:10.1140/epjp/s13360-023-03920-7
48 Kavitha SRJ Abraham J Arockiaraj M , et al. Topological characterization and graph entropies of tessellations of Kekulene structures: existence of isentropic structures and applications to thermochemistry, nuclear magnetic resonance, and electron spin resonance. J Phys Chem A 2021; 125 : 8140–8158.34469167
49 Abraham J Arockiaraj M Jency J , et al. Graph entropies, enumeration of circuits, walks and topological properties of three classes of isoreticular metal organic frameworks. J Math Chem 2022; 60 : 695–732.
