Molecular temperature descriptors as a novel approach for QSPR analysis of Borophene nanosheets

Abstract

Borophene nanosheets appear in various sizes and shapes, ranging from simple planar structures to complicated polyhedral formations. Due to their unique chemical, optical, and electrical properties, Borophene nanosheets are theoretically and practically attractive and because of their high thermal conductivity, boron nanosheets are suitable for efficient heat transmission applications. In this paper, temperature indices of borophene nanosheets are computed and these indices are employed in QSPR analysis of attributes like Young’s modulus, Shear modulus, and Poisson’s ratio of borophene nanosheets and borophene β₁₂ sheets. The regression model for the F-Temperature index is discovered to be the best fit for shear modulus, the reciprocal product connectivity temperature index is discovered to be fit for Poisson’s ratio and the second hyper temperature index is discovered to be fit for Young’s modulus based on the correlation coefficient.

1 Introduction

Boron was simultaneously discovered in 1808 by DAVY in England and France by GAY-LUSSAC, and THENARD used separate methods [1]. A unique element called boron shares a periodic table element with its neighbour, carbon, which can form bonds formed by atoms from the same element. This characteristic enables the generation of various boron hydrides and the heteroatom variants of hydrides, collectively called Borophene nanosheets [2, 3]. As the name indicates, Borophene nanosheets have a polyhedral cage-like structure compared to organic compounds, which are made up of chains and rings. Atoms of boron or heteroatoms that comprise this structure are engaged on multiple edges, ranging from the highly brittle quadrilateral Borophene nanosheet to the compounds built out of the octahedral Borophene nanosheet [4, 5]. This stability is due to the delocalization of bonds of the 3-dimensional aromatic structure of Borophene nanosheet compounds, which differs from the 2D aromaticity of organic compounds [6]. The astonishing structural variety of Borophene nanosheets has recently been uncovered thanks to experimental methods combined with theoretical analysis [7]. Despite the well-established two to three-dimensional transition thresholds, the parameters that determine the geometry that these systems acquire are unknown. Doping frequently causes structural changes that result in the lowest energy structure having an utterly different boron skeleton than the cluster without doping [8]. Borophene nanosheets have remarkable properties in antimicrobial therapy, drug design [9], cancer therapy [10], pharmacophores [11], used as broadband membrane carriers [12], for the synthesis of liquid crystals [13] and luminescent materials [14]. The effectiveness of Borophene nanosheet derivatives against common and multi-drug resistant disease strains is identical. They also exhibit anti-biofilm activity and are less likely to result in medication resistance. Dendrimers, coordinated polymers, metal-organic systems, polymer compounds, and non-linear optical materials have all been made using Borophene nanosheets. These clusters have been employed in nanoscience as the molecules’ fundamental components [15].

In 2015, researchers from a variety of domains were drawn to Borophene, a 2D boron sheet that was successfully synthesized on Ag substrates. It is the lightest 2D material and has similar properties to graphene. Borophene is superconductive and has the highest pressure and external strain among 2D materials. To assess its suitability for use in a variety of disciplines, a great deal of research has been done on its stability, electrical qualities, and chemical and structural complexity [16]. Borophene’s mechanical properties are significant, including low mass density, high strength, and high in-plane stiffness, making it suitable for composite design and flexible nano device fabrication. Its anisotropic structure allows for effective control of magnetic and electronic properties making it a suitable candidate in various applications. Borophene’s polymorphic nature and strong electron-phonon coupling make it a valuable material for future applications due to its resources, low cost, and excellent electrical performance [17, 18]. Despite the wide range of possible uses for Borophene, one of the most difficult tasks that limit its application involves the synthetic methodology and identification of its neoatomic forms with carefully thought-out structure-property connections. Furthermore, in the case of synthetic 2D materials, a variety of parameters, including component elements, processing conditions, and growth substrates, impact the final atomic structure. The assured synthesis of quality specimens and the separation of Borophene from substrates remain difficult and need ongoing theoretical and experimental work to achieve practical applications [19, 20]. With its good metal properties, Borophene has potential in energy-related applications such as lithium batteries and biosensors. Its electrical sensitivities make it suitable for sequencing DNA and gas sensors. However, its NIR absorption needs improvement. Despite extensive research, applications are limited due to low yield and limited nanosheet size [21].

Researchers have characterized the properties of Borophene, a 2-dimensional material, to find its advantages over graphene and other materials. Borophene is a superconductor that is more robust, flexible, and an excellent heat and electrical conductor. Depending on how the voids are arranged and oriented, their properties can be adjusted. Its amazing mechanical capabilities and orientationally adjustable qualities make a lighter, electron-rich Borophene an equivalent contender to graphene. Borophene is also a superior substitute for graphene in composite progress since it has a greater Young’s modulus. Compared to other 2D materials, it is a superior conductor of heat and electricity due to its remarkable anisotropic behaviour [22–24].

Kulli initiated in 2020 a graph’s temperature indices [25]. In chemical graph theory, these indices are used to quantify the chemical characteristics of chemical compounds. These indies are computed for a molecular graph silicate network and a silicate chain network and present valuable results in [26, 27]. Various graph-theoretic techniques and algorithms can be used to analyze the structure and behaviour of the chemical network once it has been converted into a graph [28]. This approach can be used to discover novel chemical reactions or processes and to understand the operation of complex chemical systems [29]. Chemical networks, physical, chemical, and thermal properties, as well as biological and chemical activities, can all be evaluated using chemical graph theory [30]. It is possible to determine a molecule’s chemical, physical, and biological properties using its topological index [31]. Graph theory is extensively employed in modern chemistry, mainly organic chemistry. Mathematical chemistry has demonstrated that polynomials and functions can uncover instructions concealed in the symmetry of chemical graphs. Topological indices can be used to evaluate QSPR, QSAR, and other chemical applications based on theory [32]. Diverse fields of study have investigated topological indices to fathom various types of graphs. In quantitative structure-activity and structure-property interactions, topological indices are employed as numerical descriptions to compare molecules’ biological, physical, and chemical aspects. Numerous researchers have examined various chemical compounds in recent years and computed topological descriptors of different molecular graphs.

Graph indices can be used for various purposes, including chemical record keeping, establishing relationships between structure and activity, property and relationship, and property and relationship [33]. Mathematicians have developed novel concepts with the use of graph indices [34].

We exclusively consider limited, simple, and connected graphs in this work. Assume that K is a graph, where E_K represents the set of edges, and V_K represents the set of vertices. The degree is determined by the number of vertices surrounding a point u. We advise the individual to search for simple terms and symbols. Fajtlowicz presented the temperature of a graph [35]:(1)The first temperature index is defined as below [36]:(2)The second temperature index was introduced by Kulli in 2020 in [37],(3)Kulli defined the first and second hyper temperature indices in [37],(4) (5)The same study provides more related topological indices [37]. The reciprocal product connectivity index, the product connectivity temperature index, and the sum connectivity temperature index were given, respectively as(6) (7) (8)Kulli also examined the F-temperature index of a graph K in [37], as given below(9)

In this article, the above-defined eight temperature indices are constructed by the atom-bonds partition of Borophene nanosheet .

2 Borophene nanosheet

Borophene nanosheets are three-dimensional formations comprised of boron atoms that are small in size. They appear in various sizes and shapes, ranging from simple planar structures to complicated polyhedral formations. Due to their unique chemical, optical, and electrical properties, Borophene nanosheets are theoretically and practically attractive. Two-dimensional Borophene nanosheet, composed of Borophene nanosheets, has recently received greater interest [38]. Due to their high thermal conductivity, boron nanosheets are suitable for efficient heat transmission applications. Due to their triangular arrangement of the boron atoms, strong covalent bonding occurs, providing the sheet tremendous strength and stiffness. Borophene nanosheets have unique electronic properties because of their 2D structure and boron atoms’ electrical configuration, making them useful in electronic and optoelectronic devices. Because of the sheet’s closely bound boron atoms, it is relatively secure and impenetrable to chemical reactions [39]. Various industries, like electronics, optoelectronics, and energy storage, may utilize boron for their products [40–42]. The Borophene nanosheet, borophene’s physical and chemical characteristics, and its application are given in [43–50]. Borophene nanosheet γ-sheet for the l = 6, m = 3 is sketched in Fig 1 and for l = 2, m = 3 is drawn in Fig 1. Here, in Borophene nanosheet , there are nine types of atom-bonds based on the valency of every atom of , which are (3 ∼ 3), (3 ∼ 4), (3 ∼ 5), (3 ∼ 6), (4 ∼ 4), (4 ∼ 5), (4 ∼ 6), (5 ∼ 5) and (5 ∼ 6) in . Tables 1 and 2 represent atom bond partitions based on valency. The atom-bonds partition of given as: The sum of all the atoms and the bonds of :Using edge partition given in Tables 1 and 2 along with Eq (1), we get Tables 3–5 for .

[Figure omitted. See PDF.]

[Figure omitted. See PDF.]

[Figure omitted. See PDF.]

[Figure omitted. See PDF.]

[Figure omitted. See PDF.]

[Figure omitted. See PDF.]

2.1 Temperature indices for Borophene nanosheet

In this section, we present some computed results.

1. 1. Let be a Borophene nanosheet. Then the first temperature index of is .

Tables 3–5 along with Eq (2), implies

After simplification, we get

1. 2. Let be a Borophene nanosheet. Then the second temperature index is .

Tables 3–5 along with Eq (3), gives

After simplification, we get(10)

1. 3. Let be Borophene nanosheet. Then the first hyper temperature index is .

From Tables 3–5 along with Eq (4), we obtain

After simplification, we get

1. 4. Let be Borophene nanosheet. Then the second hyper temperature index is

From Tables 3–5 along with Eq (5), we attain

After simplification, we get

1. 5. Let be Borophene nanosheet. Then the sum connectivity temperature index is

From Tables 3–5 along with Eq (8), we get

After simplification, we get

1. 6. Let be Borophene nanosheet. Then the product connectivity temperature index is

From Tables 3–5 along with Eq (7), we obtain

After simplification, we get

1. 7. Let be Borophene nanosheet. Then the reciprocal product temperature index is

From Tables 3–5 along with Eq (6), we get

After simplification, we get

1. 8. Let be Borophene nanosheet. Then the F-temperature index is

From Tables 3–5 along with Eq (9), we have

After simplification, we get

2.2 Comparative analysis of temperature indices for

Numerical and graphical comparison of computed temperature indices of for l = m is presented in this section. As the value of l or m, increases gradually, the value of temperatures indices HT₂, ST and PT increases gradually while the numerical value of temperature indices T₁, T₂, HT₁, RPT and FT decreases gradually. These changes are also represented in Table 6 and Fig 2.

[Figure omitted. See PDF.]

[Figure omitted. See PDF.]

3 Borophene β₁₂-sheets

Fig 3 is a graph borophene β₁₂-sheet of for the particular values of l = 2 and m = 3. Here, in Borophene β₁₂-sheet , there are eleven types of atom-bonds based on the valency of every atom of , which are (2 ∼ 2), (2 ∼ 4), (3 ∼ 3), (3 ∼ 4), (3 ∼ 5), (3 ∼ 6), (4 ∼ 4), (4 ∼ 5), (4 ∼ 6), (5 ∼ 5) and (5 ∼ 6) in . Based on valency, Tables 7 and 8 provide the partition of the atom-bonds of shown as:

[Figure omitted. See PDF.]

[Figure omitted. See PDF.]

[Figure omitted. See PDF.]

The total number of atoms and atom-bonds :By using Eq (1) and above partition of , as described in Tables 7 and 8.

3.1 Temperature indices for Borophene β₁₂-sheet

1. 9. Let be a Borophene β₁₂-sheet. Then the first temperature index of is .

Tables 9–12 along with Eq (2), gives

[Figure omitted. See PDF.]

[Figure omitted. See PDF.]

[Figure omitted. See PDF.]

[Figure omitted. See PDF.]

After simplification, we get

1. 10. Let be a Borophene β₁₂-sheet. Then the second temperature index is .

Tables 9–12 along with Eq (3), implies

After simplification, we get

1. 11. Let be Borophene β₁₂-sheet. Then the first hyper temperature index is .

From Tables 9–12 along with Eq (4), we obtain

After simplification, we get

1. 12. Let be Borophene β₁₂-sheet. Then the second hyper temperature index is

From Tables 9–12 along with Eq (5), we get

After simplification, we get

1. 13. Let be Borophene β₁₂-sheet. Then the sum connectivity temperature index is .

From Tables 9–12 along with Eq (8), we have

After simplification, we get

1. 14. Let be Borophene β₁₂-sheet. Then the product connectivity temperature index is

From Tables 9–12 along with Eq (7), we obtain

After simplification, we get

1. 15. Let be Borophene β₁₂-sheet. Then the reciprocal product temperature index is

From Tables 9–12 along with Eq (6), we attain

After simplification, we get

1. 16. Let be Borophene β₁₂-sheet. Then the F-temperature index is .

From Tables 9–12 along with Eq (9), we obtain

After simplification, we get

3.2 Comparative analysis of temperature indices for

In this section, we perform a numerical and graphical comparison of temperature indices for n = 2, 3, 4, …, 12 of a Borophene β₁₂-sheet . As we increase the value of l or m, the value of temperatures indices T₁, T₁, T₂, HT₁, RPT and FT decreased gradually while the numerical values of the temperature indices HT₂, ST and PT increased gradually. These changes are also represented in Table 13 and Fig 4.

[Figure omitted. See PDF.]

[Figure omitted. See PDF.]

4 Quantitative structure-activity property relationship model

Topological descriptors are numerical depictions of chemical compounds with information about the structure’s topology. Various descriptors have been learned and are often employed in QSPR to predict chemical characteristics and biological activities. These descriptors include physicochemical, constitutional, electrostatic, geometrical, and topological properties. The process begins with an appropriate molecular topological descriptor and concludes with an inference, hypothesis, or prediction about the molecule’s behaviour, characteristics, and attributes. High-quality experimental data must be available and accurate for this investigation to be successful. Determining a suitable topological descriptor for modeling in the analysis is the essential step in the QSPR/QSAR process. Since there is no consensus regarding the ideal molecular description, all conceivable descriptors are chosen. The capacity to predict a chemical compound’s properties, activities, and behaviour based on the molecular structure of related compounds whose properties, activities, and characteristics have already been assessed is a vital use of QSPR/QSAR models. Numerical values of the properties of borophene nanosheets are given in Table 14.

[Figure omitted. See PDF.]

4.1 Linear regression model

Regression analysis aims to ascertain the significance and strength of the correlations via a numerical value provided by the topological index. Regression models come in a variety of forms, and which one you pick will depend on the kind of data you have and the issue you’re attempting to solve. The following are some popular varieties of regression models: exponential and logarithmic regression models, linear and polynomial regression models such as quadratic and cubic regression models, etc. We employ the linear regression models for a few properties of the borophene nanosheet obtained in this section. For the analysis, the model below is taken into account.Where Y and K represent the property and structural descriptors. The analysis considers characteristics like Young’s modulus, Poisson’s ratio, and shear modulus. Regression analysis was done using the analytical features of Matlab. The regression model for the F-Temperature index is discovered to be the best fit for shear modulus based on the correlation coefficient (R), and the regression equation can be written as:The regression formula for shear modulus for the first hyper temperature index can be written as:The regression model for the reciprocal product connectivity temperature index is discovered to be fit for Poisson’s ratio based on the correlation coefficient (R), and the regression equation can be written as:The regression model for the second hyper temperature index is discovered to be fit for Young’s modulus based on the correlation coefficient (R), and the regression equation can be written as:Figs 5–11 depict regression models that show the relationship between attributes and their best-fitted descriptions. However, it can be concluded that the above-mentioned attributes of various boron variations can be anticipated using regression equations and calculated topological descriptors.

[Figure omitted. See PDF.]

[Figure omitted. See PDF.]

[Figure omitted. See PDF.]

[Figure omitted. See PDF.]

[Figure omitted. See PDF.]

[Figure omitted. See PDF.]

[Figure omitted. See PDF.]

5 Conclusion

In this study, we computed degree-based temperature indices. We employed these indices to perform linear regression analysis to predict three physiochemical properties of borophene nanosheets: shear modulus, Young’s modulus, and Poisson’s ratio. We concluded that the linear regression analysis provides the following best-fit models with R² value for shear modulus: Young’s modulus: and Poisson’s ratio:

Supporting information

S1 File.

https://doi.org/10.1371/journal.pone.0302157.s001

(STY)

Citation: Khan AR, Ullah Z, Imran M, Malik SA, Alamoudi LM, Cancan M (2024) Molecular temperature descriptors as a novel approach for QSPR analysis of Borophene nanosheets. PLoS ONE 19(6): e0302157. https://doi.org/10.1371/journal.pone.0302157

About the Authors:

Abdul Rauf Khan

Contributed equally to this work with: Abdul Rauf Khan, Zafar Ullah, Muhammad Imran, Sidra Aziz Malik, Lamis M. Alamoudi, Murat Cancan

Roles: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Software, Supervision, Validation, Visualization, Writing – review & editing

Affiliation: Department of Mathematics, Faculty of Sciences, Ghazi University, Dera Ghazi Khan, Pakistan

ORICD: https://orcid.org/0000-0002-4709-3860

Zafar Ullah

Contributed equally to this work with: Abdul Rauf Khan, Zafar Ullah, Muhammad Imran, Sidra Aziz Malik, Lamis M. Alamoudi, Murat Cancan

Roles: Conceptualization, Formal analysis, Writing – review & editing

Affiliation: Division of Science, Department of Mathematics, University of Education, Lahore, Pakistan

Muhammad Imran

Contributed equally to this work with: Abdul Rauf Khan, Zafar Ullah, Muhammad Imran, Sidra Aziz Malik, Lamis M. Alamoudi, Murat Cancan

Roles: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Visualization, Writing – review & editing

E-mail: [email protected]

Affiliation: Department of Mathematical Sciences, United Arab Emirates University, Al Ain, United Arab Emirates

ORICD: https://orcid.org/0000-0002-2827-0462

Sidra Aziz Malik

Contributed equally to this work with: Abdul Rauf Khan, Zafar Ullah, Muhammad Imran, Sidra Aziz Malik, Lamis M. Alamoudi, Murat Cancan

Roles: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Software, Writing – original draft

Affiliation: Department of Mathematics, Faculty of Sciences, Ghazi University, Dera Ghazi Khan, Pakistan

Lamis M. Alamoudi

Contributed equally to this work with: Abdul Rauf Khan, Zafar Ullah, Muhammad Imran, Sidra Aziz Malik, Lamis M. Alamoudi, Murat Cancan

Roles: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Software, Writing – review & editing

Affiliation: Department of Statistics, King Abdulaziz University, Jeddah, Saudi Arabia

Murat Cancan

Contributed equally to this work with: Abdul Rauf Khan, Zafar Ullah, Muhammad Imran, Sidra Aziz Malik, Lamis M. Alamoudi, Murat Cancan

Roles: Conceptualization, Formal analysis, Methodology, Software, Writing – review & editing

Affiliation: Faculty of Education, Yuzuncu Yil University, Van, Turkey

ORICD: https://orcid.org/0000-0002-5643-4532

[/RAW_REF_TEXT]

[/RAW_REF_TEXT]

References

1. Cueilleron J, Thevenot F. Chemical properties of boron. In Boron and Refractory Borides 1977 (pp. 203–213). Berlin, Heidelberg: Springer Berlin Heidelberg.

2. Goszczyński TM, Fink K, Boratyński J. Icosahedral boron clusters as modifying entities for biomolecules. Expert opinion on biological therapy. 2018 May 31;18(sup1):205–13. pmid:30063861

3. Fink K, Uchman M. Boron cluster compounds as new chemical leads for antimicrobial therapy. Coordination Chemistry Reviews. 2021 Mar 15;431:213684.

4. King RB. Three-dimensional aromaticity in polyhedral boranes and related molecules. Chemical reviews. 2001 May 9;101(5):1119–52. pmid:11710215

5. Sun X, Liu X, Yin J, Yu J, Li Y, Hang Y, et al. Two‐dimensional boron crystals: structural stability, tunable properties, fabrications and applications. Advanced Functional Materials. 2017 May;27(19):1603300.

6. Poater i Teixidor J, Solà i Puig M, Viñas C, Teixidor Bombardó F. Π[pi] Aromaticity and Three-Dimensional Aromaticity: Two sides of the Same Coin?. Angewandte Chemie International Edition, 2014, vol. 53, p. 12191–12195. 2014. pmid:25223923

7. Li C, Tareen AK, Long J, Iqbal M, Ahmad W, Khan MF, et al. Two dimensional borophene nanomaterials: Recent developments for novel renewable energy storage applications. Progress in Solid State Chemistry. 2023 Aug 15:100416.

8. Barroso J, Pan S, Merino G. Structural transformations in boron clusters induced by metal doping. Chemical Society Reviews. 2022;51(3):1098–123. pmid:35029622

9. Baig N. Two-dimensional nanomaterials: A critical review of recent progress, properties, applications, and future directions. Composites Part A: Applied Science and Manufacturing. 2023 Feb 1;165:107362.

10. Tjarks W. The use of boron clusters in the rational design of boronated nucleosides for neutron capture therapy of cancer. Journal of Organometallic Chemistry. 2000 Dec 8;614:37–47.

11. Lesnikowski ZJ. Boron units as pharmacophores-new applications and opportunities of boron cluster chemistry. Collection of Czechoslovak Chemical Communications. 2007;72(12):1646–58.

12. Barba-Bon A, Salluce G, Lostalé-Seijo I, Assaf KI, Hennig A, Montenegro J, et al. Boron clusters as broadband membrane carriers. Nature. 2022 Mar 24;603(7902):637–42. pmid:35322251

13. Tai G, Xu M, Hou C, Liu R, Liang X, Wu Z. Borophene nanosheets as high-efficiency catalysts for the hydrogen evolution reaction. ACS Applied Materials & Interfaces. 2021 Dec 17;13(51):60987–94. pmid:34918510

14. Mukherjee S, Thilagar P. Recent advances in purely organic phosphorescent materials. Chemical communications. 2015;51(55):10988–1003. pmid:26036713

15. Dash BP, Satapathy R, Maguire JA, Hosmane NS. Polyhedral boron clusters in materials science. New Journal of Chemistry. 2011;35(10):1955–72.

16. Mir SH, Chakraborty S, Jha PC, Wärnå J, Soni H, Jha PK, et al. Two-dimensional boron: Lightest catalyst for hydrogen and oxygen evolution reaction. Applied Physics Letters. 2016 Aug 1;109(5).

17. Li D, He J, Ding G, Tang Q, Ying Y, He J, et al. Stretch‐driven increase in ultrahigh thermal conductance of hydrogenated borophene and dimensionality crossover in phonon transmission. Advanced Functional Materials. 2018 Aug;28(31):1801685.

18. Liu X, Wang L, Li S, Rahn MS, Yakobson BI, Hersam MC. Geometric imaging of borophene polymorphs with functionalized probes. Nature communications. 2019 Apr 9;10(1):1642. pmid:30967559

19. Mannix AJ, Zhang Z, Guisinger NP, Yakobson BI, Hersam MC. Borophene as a prototype for synthetic 2D materials development. Nature nanotechnology. 2018 Jun;13(6):444–50. pmid:29875501

20. Zhang Z, Penev ES, Yakobson BI. Two-dimensional boron: structures, properties and applications. Chemical Society Reviews. 2017;46(22):6746–63. pmid:29085946

21. Xie Z, Meng X, Li X, Liang W, Huang W, Chen K, et al. Two-dimensional borophene: properties, fabrication, and promising applications. Research. 2020 Jun 15. pmid:32607497

22. Bhavyashree M, Rondiya SR, Hareesh K. Exploring the emerging applications of the advanced 2-dimensional material borophene with its unique properties. RSC advances. 2022;12(19):12166–92. pmid:35481099

23. Gao Z, Horiguchi Y, Nakai K, Matsumura A, Suzuki M, Ono K, et al. Use of boron cluster-containing redox nanoparticles with ROS scavenging ability in boron neutron capture therapy to achieve high therapeutic efficiency and low adverse effects. Biomaterials. 2016 Oct 1;104:201–12. pmid:27467416

24. Wojtczak BA, Andrysiak A, Grüner B, Lesnikowski ZJ. “Chemical ligation”: a versatile method for nucleoside modification with boron clusters. Chemistry–A European Journal. 2008 Nov 26;14(34):10675–82. pmid:18942698

25. Kulli VR. Some new temperature indices of oxide and honeycomb networks. Annals of Pure and Applied Mathematics. 2020;21(2):129–33.

26. Khan AR, Ghani MU, Ghaffar A, Asif HM, Inc M. Characterization of temperature indices of silicates. Silicon. 2023 May 31:1–7.

27. Ghani MU, Sultan F, Tag El Din ES, Khan AR, Liu JB, Cancan M. A Paradigmatic Approach to Find the Valency-Based K-Banhatti and Redefined Zagreb Entropy for Niobium Oxide and a Metal–Organic Framework. Molecules. 2022 Oct 17;27(20):6975. pmid:36296567

28. Imran M, Khan AR, Husin MN, Tchier F, Ghani MU, Hussain S. Computation of Entropy Measures for Metal-Organic Frameworks. Molecules. 2023 Jun 13;28(12):4726. pmid:37375281

29. Alsaadi FE, Salman M, Rehman MU, Khan AR, Cao J, Alassafi MO. On the Geodesic Identification of Vertices in Convex Plane Graphs. Mathematical Problems in Engineering. 2020 Dec 27;2020:1–3.

30. Chu YM, Khan AR, Ghani MU, Ghaffar A, Inc M. Computation of zagreb polynomials and zagreb indices for benzenoid triangular & hourglass system. Polycyclic Aromatic Compounds. 2023 May 28;43(5):4386–95.

31. Gutman I. Some basic properties of Sombor indices. Open Journal of Discrete Applied Mathematics. 2021 Dec 31;4(1):1–3.

32. Khan AR, Awan NU, Ghani MU, Eldin SM, Karamti H, Jawhari AH, et al. Fundamental Aspects of Skin Cancer Drugs via Degree-Based Chemical Bonding Topological Descriptors. Molecules. 2023 Apr 24;28(9):3684. pmid:37175093

33. Khan AR, Awan NU, Tchier F, Alahmari SD, Khalel AF, Hussain S. An estimation of physiochemical properties of bladder cancer drugs via degree-based chemical bonding topological descriptors. Journal of Biomolecular Structure and Dynamics. 2023 Dec 9:1–9. pmid:38095491

34. Liu JB, Imran M, Baby S, Siddiqui H, Shafiq MK. Graph Indices for Cartesian Product of-sum of Connected Graphs. Combinatorial Chemistry & High Throughput Screening. 2022 Mar 1;25(3):528–35.

35. Fajtlowicz S. On conjectures of Graffiti. In Annals of Discrete Mathematics 1988 Jan 1 (Vol. 38, pp. 113–118). Elsevier.

36. Narayankar KP, Kahsay AT, Selvan D. Harmonic temperature index of certain nanostructures. Int J Math Trends Technol. 2018;56:159–64.

37. Kulli VR. The (a, b)-ka temperature indices of tetrameric 1, 3-adamantane. Int. J. Recent Sci. Res. 2021;12:40929–33.

38. Koam AN, Ahmad A, Azeem M. Boron clusters sheets with algebraic properties. Molecular Physics. 2023 Jun 24:e2228424.

39. Hassan MM, Jabeen S, Ali H, Ali P. Connection-based modified Zagreb indices of Boron triangular sheet BTS (m, n). Molecular Physics. 2023 Jun 17:e2226242.

40. Wu X, Dai J, Zhao Y, Zhuo Z, Yang J, Zeng XC. Two-dimensional boron monolayer sheets. ACS nano. 2012 Aug 28;6(8):7443–53. pmid:22816319

41. Liu Y, Tai G, Hou C, Wu Z, Liang X. Chemical Vapor Deposition Growth of Few-Layer β12-Borophane on Copper Foils toward Broadband Photodetection. ACS Applied Materials & Interfaces. 2023 Mar 10;15(11):14566–74. pmid:36897961

42. Cheng Z, Zhang X, Zhang H, Liu H, Yu X, Dai X, et al. Porous-Induced Performance Enhancement of Flat Boron Sheets for Lithium-Ion Batteries. The Journal of Physical Chemistry C. 2022 Dec 16;126(51):21542–9.

43. Xiong Z, Xiao Y, Shen C. Screening of the Transition Metal Single Atom Anchored on α-Borophene Catalysts as a Feasible Strategy for Electrosynthesis of Urea. Chemistry of Materials. 2022 Oct 17;34(21):9402–13.

44. Kang Y, Ma X, Fu J, Yang K, Wang Z, Li H, et al. Substrate-mediated Borophane Polymorphs through Hydrogenation of Two-dimensional Boron Sheets. The Journal of Physical Chemistry Letters. 2022 Oct 27;13(43):10222–9. pmid:36300795

45. Abdi Y, Mazaheri A, Hajibaba S, Darbari S, Rezvani SJ, Cicco AD, et al. A two-dimensional borophene supercapacitor. ACS Materials Letters. 2022 Sep 6;4(10):1929–36.

46. Chen K, Guo H, Zhang J, Wang L, Wu M. 2D/2D Boron/g-C3N4 Nanosheet Heterojunction Boosts Photocatalytic Hydrogen Evolution Performance. ACS Applied Energy Materials. 2022 Aug 17;5(9):10657–66.

47. Radatovic B, Jadrisko V, Kamal S, Kralj M, Novko D, Vujicic N, et al. Macroscopic single-phase monolayer borophene on arbitrary substrates. ACS Applied Materials Interfaces. 2022 May 2;14(18):21727–37. pmid:35500044

48. Xu Y, Xuan X, Yang T, Zhang Z, Li SD, Guo W. Quasi-freestanding bilayer borophene on ag (111). Nano letters. 2022 Mar 26;22(8):3488–94. pmid:35341246

49. Li J, An Q, Liu L. A Strong Two-Dimensional Semiconductor l-B4C with High Carrier Mobility. The Journal of Physical Chemistry C. 2022 Mar 28;126(13):6036–46.

50. Xu Y, Zhang P, Xuan X, Xue M, Zhang Z, Guo W, et al. Borophane polymorphs. The Journal of Physical Chemistry Letters. 2022 Jan 26;13(4):1107–13. pmid:35080410