{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,2]],"date-time":"2026-09-02T11:05:37Z","timestamp":1788347137161,"version":"build-2803163510"},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2026,1,31]],"date-time":"2026-01-31T00:00:00Z","timestamp":1769817600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"},{"start":{"date-parts":[[2026,2,18]],"date-time":"2026-02-18T00:00:00Z","timestamp":1771372800000},"content-version":"vor","delay-in-days":18,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Sci Rep"],"DOI":"10.1038\/s41598-026-37731-4","type":"journal-article","created":{"date-parts":[[2026,1,31]],"date-time":"2026-01-31T05:44:04Z","timestamp":1769838244000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Comparative entropy analysis of 2D transition metal tetrahydroxyquinones via machine learning approaches"],"prefix":"10.1038","volume":"16","author":[{"given":"Muhammad","family":"Irfan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nabeela","family":"Bashir","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"AbdulGuddoos S. A.","family":"Gaid","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Turke","family":"Althobaiti","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hamood Ur","family":"Rehman","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Andrea","family":"Semani\u010dov\u00e1-Fe\u0148ov\u010d\u00edkov\u00e1","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,1,31]]},"reference":[{"key":"37731_CR1","doi-asserted-by":"publisher","unstructured":"Bonchev, D.  Chemical Graph Theory: Introduction and Fundamentals (1st ed.) (Routledge, 1991). https:\/\/doi.org\/10.1201\/9781315139104..","DOI":"10.1201\/9781315139104."},{"key":"37731_CR2","doi-asserted-by":"publisher","unstructured":"Wagner, S., & Wang, H.  Introduction to Chemical Graph Theory (1st ed.) (Chapman and Hall\/CRC, 2018). https:\/\/doi.org\/10.1201\/9780429450532..","DOI":"10.1201\/9780429450532."},{"issue":"5","key":"37731_CR3","doi-asserted-by":"publisher","first-page":"e1729","DOI":"10.1002\/wcms.1729","volume":"14","author":"LS Leite","year":"2024","unstructured":"Leite, L. S., Banerjee, S., Wei, Y., Elowitt, J. & Clark, A. E. Modern chemical graph theory. Wiley Interdiscipl. Rev. Comput. Mol. Sci. 14(5), e1729 (2024).","journal-title":"Wiley Interdiscipl. Rev. Comput. Mol. Sci."},{"key":"37731_CR4","unstructured":"Chen, W. K.  Applied Graph Theory: Graphs and Electrical Networks (2nd ed.)  (Elsevier, 2014)."},{"key":"37731_CR5","doi-asserted-by":"publisher","unstructured":"Gross, J. L., Yellen, J., & Anderson, M.  Graph Theory and its Applications (3rd ed.)  (Chapman and Hall\/CRC, 2018). https:\/\/doi.org\/10.1201\/9780429425134.","DOI":"10.1201\/9780429425134"},{"issue":"32","key":"37731_CR6","doi-asserted-by":"publisher","first-page":"2774","DOI":"10.2174\/1568026619666190208164005","volume":"18","author":"K Balasubramanian","year":"2018","unstructured":"Balasubramanian, K. Mathematical and computational techniques for drug discovery: promises and developments. Curr. Top. Med. Chem. 18(32), 2774\u20132799 (2018).","journal-title":"Curr. Top. Med. Chem."},{"key":"37731_CR7","doi-asserted-by":"publisher","unstructured":"Balasubramanian, K.  Computational and artificial intelligence techniques for drug discovery and administration. In Comprehensive Pharmacology  (ed. Kenakin, T.)  553\u2013616 (Elsevier, 2022). https:\/\/doi.org\/10.1016\/B978-0-12-820472-6.00015-3..","DOI":"10.1016\/B978-0-12-820472-6.00015-3."},{"key":"37731_CR8","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1155\/2021\/5551825","volume":"2021","author":"M Irfan","year":"2021","unstructured":"Irfan, M., Rehman, H., Almusawa, H., Rasheed, S. & Baloch, I. A. M-polynomials and topological indices for line graphs of chain silicate network and H-naphtalenic nanotubes. J. Math. 2021, 1\u201311 (2021).","journal-title":"J. Math."},{"issue":"1","key":"37731_CR9","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1021\/ja01193a005","volume":"69","author":"H Wiener","year":"1947","unstructured":"Wiener, H. Structural determination of paraffin boiling points. J. Am. Chem. Soc. 69(1), 17\u201320 (1947).","journal-title":"J. Am. Chem. Soc."},{"issue":"6","key":"37731_CR10","doi-asserted-by":"publisher","first-page":"102721","DOI":"10.1016\/j.asej.2024.102721","volume":"15","author":"I Masmali","year":"2024","unstructured":"Masmali, I., Naeem, M., Ishaq, M. & Koam, A. N. A. Mathematical modeling of irregularity indices for adriamycin and their statistical analysis for emetic drugs. Ain Shams Eng. J. 15(6), 102721 (2024).","journal-title":"Ain Shams Eng. J."},{"key":"37731_CR11","doi-asserted-by":"publisher","first-page":"136","DOI":"10.3389\/fphy.2024.1369939","volume":"12","author":"MW Rasheed","year":"2024","unstructured":"Rasheed, M. W., Mahboob, A. & Hanif, I. Investigating the properties of octane isomers by novel neighborhood product degree-based topological indices. Front. Phys. 12, 136\u2013156 (2024).","journal-title":"Front. Phys."},{"issue":"1","key":"37731_CR12","doi-asserted-by":"publisher","first-page":"69","DOI":"10.22199\/issn.0717-6279-5843","volume":"43","author":"S Rai","year":"2024","unstructured":"Rai, S. & Das, S. Comparative analysis of M-polynomial based topological indices between poly Hex-derived networks and their subdivisions. Proyecciones J. Math. 43(1), 69\u2013102 (2024).","journal-title":"Proyecciones J. Math."},{"issue":"1","key":"37731_CR13","doi-asserted-by":"publisher","first-page":"556","DOI":"10.1002\/qua.27310","volume":"124","author":"SJ Abirami","year":"2024","unstructured":"Abirami, S. J., Raj, S. A. K., Siddiqui, M. K. & Zia, T. J. Computation of degree-based topological indices for the complex structure of Ruthenium bipyridine. Int. J. Quant. Chem. 124(1), 556\u2013562 (2024).","journal-title":"Int. J. Quant. Chem."},{"key":"37731_CR14","doi-asserted-by":"publisher","first-page":"22478","DOI":"10.1038\/s41598-024-73163-8","volume":"14","author":"R Huang","year":"2024","unstructured":"Huang, R., Hanif, M. F., Siddiqui, M. K., Hanif, M. F. & Gegbe, B. On analysis of silicon dioxide based on topological indices and entropy measure via regression model. Sci. Rep. 14, 22478 (2024).","journal-title":"Sci. Rep."},{"issue":"4","key":"37731_CR15","doi-asserted-by":"publisher","first-page":"101672","DOI":"10.1016\/j.asej.2021.101672","volume":"13","author":"U Asad","year":"2022","unstructured":"Asad, U., Qasim, M., Zaman, S. & Khan, A. Computational and comparative aspects of two carbon nanosheets with respect to some novel topological indices. Ain Shams Eng. J. 13(4), 101672 (2022).","journal-title":"Ain Shams Eng. J."},{"key":"37731_CR16","doi-asserted-by":"publisher","first-page":"535","DOI":"10.1016\/0009-2614(72)85099-1","volume":"17","author":"I Gutman","year":"1972","unstructured":"Gutman, I. & Trinajsti\u0107, N. Graph theory and molecular orbitals, total $$\\varphi$$-electron energy of alternant hydrocarbons. J. Chem. Phys. Lett. 17, 535\u2013538 (1972).","journal-title":"J. Chem. Phys. Lett."},{"issue":"4","key":"37731_CR17","first-page":"973","volume":"41","author":"A Amer","year":"2020","unstructured":"Amer, A. et al. Edge version of topological degree-based indices of boron triangular nanotubes. J. Inf. Optim. Sci. 41(4), 973\u2013990 (2020).","journal-title":"J. Inf. Optim. Sci."},{"key":"37731_CR18","first-page":"1","volume":"2021","author":"A Amer","year":"2021","unstructured":"Amer, A., Irfan, M. & Rehman, H. Computational aspects of line graph of boron triangular nanotubes. Math. Methods Appl. Sci. 2021, 1\u20137 (2021).","journal-title":"Math. Methods Appl. Sci."},{"key":"37731_CR19","doi-asserted-by":"publisher","first-page":"6609","DOI":"10.1021\/ja00856a001","volume":"97","author":"M Randi\u0107","year":"1975","unstructured":"Randi\u0107, M. Characterization of molecular branching. J. Am. Chem. Soc. 97, 6609\u20136615 (1975).","journal-title":"J. Am. Chem. Soc."},{"issue":"4","key":"37731_CR20","first-page":"1454","volume":"80","author":"M Naeem","year":"2018","unstructured":"Naeem, M., Siddiqui, M. K., Qaisar, S., Imran, M. & Farahani, M. R. Computing topological indices of 2-dimensional silicon-carbons. UPB Sci. Bull. Ser. B: Chem. Mater. Sci. 80(4), 1454\u20132331 (2018).","journal-title":"UPB Sci. Bull. Ser. B: Chem. Mater. Sci."},{"issue":"17","key":"37731_CR21","doi-asserted-by":"publisher","first-page":"e27474","DOI":"10.1002\/qua.27474","volume":"124","author":"M Irfan","year":"2024","unstructured":"Irfan, M., Yasmeen, F., Aziz, S. & Mukhtar, M. Neighborhood degree based topological indices of nanotube via direct and NM-polynomial. Int. J. Quant. Chem. 124(17), e27474 (2024).","journal-title":"Int. J. Quant. Chem."},{"key":"37731_CR22","first-page":"849","volume":"37A","author":"E Estrada","year":"1998","unstructured":"Estrada, E., Torres, L., Rodr\u00edguez, L. & Gutman, I. An atom-bond connectivity index: modelling the enthalpy of formation of alkanes. Indian J. Chem. Sect. A: Inorg. Phys. Theor. Anal. Chem. 37A, 849\u2013855 (1998).","journal-title":"Indian J. Chem. Sect. A: Inorg. Phys. Theor. Anal. Chem."},{"key":"37731_CR23","doi-asserted-by":"publisher","first-page":"1369","DOI":"10.1007\/s10910-009-9520-x","volume":"46","author":"D Vuki\u010devi\u0107","year":"2009","unstructured":"Vuki\u010devi\u0107, D. & Furtula, B. Topological index based on the ratios of geometric and arithmetic means of end-vertex degrees of edges. J. Math. Chem. 46, 1369\u20131376 (2009).","journal-title":"J. Math. Chem."},{"key":"37731_CR24","first-page":"125531","volume":"386","author":"KC Das","year":"2020","unstructured":"Das, K. C., Rodr\u00edguez, J. M. & Sigarreta, J. M. On the maximal general ABC index of graphs with given maximum degree. Appl. Math. Comput. 386, 125531 (2020).","journal-title":"Appl. Math. Comput."},{"issue":"6","key":"37731_CR25","doi-asserted-by":"publisher","first-page":"1252","DOI":"10.1007\/s10910-008-9515-z","volume":"46","author":"B Zhou","year":"2009","unstructured":"Zhou, B. & Trinajsti\u0107, N. On a novel connectivity index. J. Math. Chem. 46(6), 1252\u20131270 (2009).","journal-title":"J. Math. Chem."},{"key":"37731_CR26","doi-asserted-by":"crossref","unstructured":"Haynes, K. E., Phillips, F., & Yu, M. C. Y.  Recent applications of entropy in social and environmental sciences. In Handbook on Entropy, Complexity, and Spatial Dynamics   55\u201368 (Springer, 2021).","DOI":"10.4337\/9781839100598.00010"},{"issue":"1","key":"37731_CR27","doi-asserted-by":"publisher","first-page":"512","DOI":"10.1371\/journal.pone.0294580","volume":"19","author":"H Zhou","year":"2024","unstructured":"Zhou, H. et al. On analysis of iron (II) chloride via graph entropy measures and statistical models. PLoS ONE 19(1), 512\u2013536 (2024).","journal-title":"PLoS ONE"},{"key":"37731_CR28","first-page":"168","volume":"50","author":"A Ovais","year":"2024","unstructured":"Ovais, A., Yasmeen, F., Irfan, M., Rasheed, M. W. & Kousar, S. Entropy measures of dendrimers using degree based indices. S. Afr. J. Chem. Eng. 50, 168\u2013181 (2024).","journal-title":"S. Afr. J. Chem. Eng."},{"issue":"1","key":"37731_CR29","doi-asserted-by":"crossref","first-page":"7184","DOI":"10.1038\/s41598-024-57914-1","volume":"14","author":"G Yu","year":"2024","unstructured":"Yu, G. et al. On topological indices and entropy measures of Beryllonitrene network via logarithmic regression model. Sci. Rep. 14(1), 7184 (2024).","journal-title":"Sci. Rep."},{"key":"37731_CR30","doi-asserted-by":"publisher","first-page":"34578","DOI":"10.1038\/s41598-025-18050-6","volume":"15","author":"M Irfan","year":"2025","unstructured":"Irfan, M. et al. Exploring entropy measures in polymer graphs using logarithmic regression model. Sci. Rep. 15, 34578 (2025).","journal-title":"Sci. Rep."},{"issue":"7","key":"37731_CR31","doi-asserted-by":"publisher","first-page":"1092","DOI":"10.3390\/e25071092","volume":"25","author":"S Mondal","year":"2023","unstructured":"Mondal, S. & Das, K. C. Degree-based graph entropy in structure-property modeling. Entropy 25(7), 1092 (2023).","journal-title":"Entropy"},{"key":"37731_CR32","doi-asserted-by":"crossref","first-page":"256","DOI":"10.1038\/s41598-024-77292-y","volume":"15","author":"Y Kara","year":"2025","unstructured":"Kara, Y., Sa\u011flam \u00d6zkan, Y., Bekta\u015f, A. B. & Arockiaraj, M. Applications of Sombor topological indices and entropy measures for QSPR modeling of anticancer drugs: a python-based methodology. Sci. Rep. 15, 256 (2025).","journal-title":"Sci. Rep."},{"issue":"1","key":"37731_CR33","doi-asserted-by":"publisher","first-page":"40903","DOI":"10.1038\/s41598-025-24755-5","volume":"15","author":"J-H Tang","year":"2025","unstructured":"Tang, J.-H. et al. Information-theoretic entropy and topological descriptor analysis of tin oxide $$(SnO_{2})$$ for structural and property prediction. Sci. Rep. 15(1), 40903 (2025).","journal-title":"Sci. Rep."},{"key":"37731_CR34","doi-asserted-by":"publisher","first-page":"78","DOI":"10.1007\/s10791-025-09578-2","volume":"28","author":"W Ahmed","year":"2025","unstructured":"Ahmed, W., Ashraf, T., Zaman, S., Hussain, A. & Belay, M. B. Molecular graphs and entropy based QSPR analysis of drugs by using machine learning. Discover Comput. 28, 78 (2025).","journal-title":"Discover Comput."},{"issue":"3","key":"37731_CR35","doi-asserted-by":"publisher","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","volume":"27","author":"CE Shannon","year":"1948","unstructured":"Shannon, C. E. A mathematical theory of communication. Bell Syst. Tech. J. 27(3), 379\u2013423 (1948).","journal-title":"Bell Syst. Tech. J."},{"issue":"9","key":"37731_CR36","doi-asserted-by":"publisher","first-page":"9252","DOI":"10.1126\/sciadv.aap9252","volume":"3","author":"HB Wu","year":"2017","unstructured":"Wu, H. B. & Lou, X. W. Metal-organic frameworks and their derived materials for electrochemical energy storage and conversion, promises and challenges. Sci. Adv. 3(9), 9252 (2017).","journal-title":"Sci. Adv."},{"issue":"15","key":"37731_CR37","doi-asserted-by":"publisher","first-page":"6267","DOI":"10.1039\/C8CS00268A","volume":"47","author":"M Zhao","year":"2018","unstructured":"Zhao, M. et al. Two-dimensional metal-organic framework nanosheets, synthesis and applications. Chem. Soc. Rev. 47(15), 6267\u20136295 (2018).","journal-title":"Chem. Soc. Rev."},{"issue":"22","key":"37731_CR38","doi-asserted-by":"publisher","first-page":"4049","DOI":"10.3390\/nano12224049","volume":"12","author":"X Zeng","year":"2022","unstructured":"Zeng, X. et al. Two-dimensional (2D) TM-tetrahydroxyquinone metal-organic framework for selective CO$$_2$$ electrocatalysis: a DFT investigation. Nanomaterials 12(22), 4049 (2022).","journal-title":"Nanomaterials"},{"key":"37731_CR39","doi-asserted-by":"publisher","first-page":"17045","DOI":"10.1038\/natrevmats.2017.45","volume":"2","author":"CA Trickett","year":"2017","unstructured":"Trickett, C. A. et al. The chemistry of metal-organic frameworks for CO$$_2$$ capture. Regener. Convers. Nat. Rev. Mater. 2, 17045 (2017).","journal-title":"Regener. Convers. Nat. Rev. Mater."},{"issue":"2","key":"37731_CR40","doi-asserted-by":"publisher","first-page":"673","DOI":"10.1021\/cr300014x","volume":"112","author":"HC Zhou","year":"2012","unstructured":"Zhou, H. C., Long, J. R. & Yaghi, O. M. Introduction to metal-organic frameworks. Chem. Rev. 112(2), 673\u2013674 (2012).","journal-title":"Chem. Rev."}],"container-title":["Scientific Reports"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.nature.com\/articles\/s41598-026-37731-4","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s41598-026-37731-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s41598-026-37731-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,18]],"date-time":"2026-02-18T11:24:49Z","timestamp":1771413889000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.nature.com\/articles\/s41598-026-37731-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,31]]},"references-count":40,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,12]]}},"alternative-id":["37731"],"URL":"https:\/\/doi.org\/10.1038\/s41598-026-37731-4","relation":{},"ISSN":["2045-2322"],"issn-type":[{"value":"2045-2322","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,31]]},"assertion":[{"value":"13 November 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 January 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 January 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare that they have no conflicts of interest.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflicts of Interest"}}],"article-number":"6819"}}