{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,4]],"date-time":"2026-09-04T16:16:22Z","timestamp":1788538582938,"version":"build-2803163510"},"reference-count":98,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/501100009558","name":"University Natural Science Research Project of Anhui Province","doi-asserted-by":"publisher","award":["2025AHGXZK30908"],"award-info":[{"award-number":["2025AHGXZK30908"]}],"id":[{"id":"10.13039\/501100009558","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100009558","name":"University Natural Science Research Project of Anhui Province","doi-asserted-by":"publisher","award":["2025AHGXZK50019"],"award-info":[{"award-number":["2025AHGXZK50019"]}],"id":[{"id":"10.13039\/501100009558","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Journal of Molecular Graphics and Modelling"],"published-print":{"date-parts":[[2026,11]]},"DOI":"10.1016\/j.jmgm.2026.109546","type":"journal-article","created":{"date-parts":[[2026,8,17]],"date-time":"2026-08-17T23:11:16Z","timestamp":1787008276000},"page":"109546","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"C","title":["Closed-form connection-number-based topological indices for porphyrazine and tetrakis-porphyrazine networks"],"prefix":"10.1016","volume":"148","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2780-0820","authenticated-orcid":false,"given":"Muhammad Mudassar","family":"Hassan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Abu","family":"Bakkar","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5301-8739","authenticated-orcid":false,"given":"Xiang-Feng","family":"Pan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"issue":"5","key":"10.1016\/j.jmgm.2026.109546_b1","doi-asserted-by":"crossref","first-page":"262","DOI":"10.1080\/0025570X.1983.11977056","article-title":"Leonhard euler, 1707\u20131783","volume":"56","author":"Burckhardt","year":"1983","journal-title":"Math. Mag."},{"issue":"1","key":"10.1016\/j.jmgm.2026.109546_b2","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1038\/scientificamerican0753-66","article-title":"Leonhard Euler and the K\u00f6nigsberg bridges","volume":"189","author":"Euler","year":"1953","journal-title":"Sci. Am."},{"issue":"1","key":"10.1016\/j.jmgm.2026.109546_b3","first-page":"573","article-title":"The seven bridges of K\u00f6nigsberg","volume":"1","author":"Euler","year":"1956","journal-title":"World Math."},{"issue":"2","key":"10.1016\/j.jmgm.2026.109546_b4","first-page":"242","article-title":"An overview of application of graph theory","volume":"9","author":"Prathik","year":"2016","journal-title":"Int. J. ChemTech Res."},{"issue":"2","key":"10.1016\/j.jmgm.2026.109546_b5","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1039\/rr9710400173","article-title":"Graph theory in chemistry","volume":"4","author":"Rouvray","year":"1971","journal-title":"R. Inst. Chem. Rev."},{"issue":"23","key":"10.1016\/j.jmgm.2026.109546_b6","doi-asserted-by":"crossref","first-page":"6609","DOI":"10.1021\/ja00856a001","article-title":"Characterization of molecular branching","volume":"97","author":"Randic","year":"1975","journal-title":"J. Am. Chem. Soc."},{"issue":"38","key":"10.1016\/j.jmgm.2026.109546_b7","doi-asserted-by":"crossref","first-page":"7482","DOI":"10.1021\/jp0346561","article-title":"Generalized graph matrix, graph geometry, quantum chemistry, and optimal description of physicochemical properties","volume":"107","author":"Estrada","year":"2003","journal-title":"J. Phys. Chem. A"},{"issue":"2","key":"10.1016\/j.jmgm.2026.109546_b8","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1049\/iet-syb.2015.0007","article-title":"Graph theory and stability analysis of protein complex interaction networks","volume":"10","author":"Huang","year":"2016","journal-title":"IET Syst. Biol."},{"issue":"11","key":"10.1016\/j.jmgm.2026.109546_b9","doi-asserted-by":"crossref","first-page":"1525","DOI":"10.1021\/j100474a032","article-title":"Graph theoretical ordering of structures as a basis for systematic searches for regularities in molecular data","volume":"83","author":"Randic","year":"1979","journal-title":"J. Phys. Chem."},{"key":"10.1016\/j.jmgm.2026.109546_b10","doi-asserted-by":"crossref","DOI":"10.1016\/j.chemolab.2024.105168","article-title":"Enhancing Hansen solubility predictions with molecular and graph-based approaches","volume":"251","author":"Cvetkovic","year":"2024","journal-title":"Chemom. Intell. Lab. Syst."},{"issue":"3","key":"10.1016\/j.jmgm.2026.109546_b11","doi-asserted-by":"crossref","first-page":"334","DOI":"10.1021\/ci00047a033","article-title":"Applications of graph theory in chemistry","volume":"25","author":"Balaban","year":"1985","journal-title":"J. Chem. Inf. Comput. Sci."},{"issue":"5","key":"10.1016\/j.jmgm.2026.109546_b12","doi-asserted-by":"crossref","first-page":"1715","DOI":"10.1021\/acs.jcim.9b00002","article-title":"From quantum chemistry to networks in biology: A graph spectral approach to protein structure analyses","volume":"59","author":"Gadiyaram","year":"2019","journal-title":"J. Chem. Inf. Model."},{"issue":"10","key":"10.1016\/j.jmgm.2026.109546_b13","doi-asserted-by":"crossref","first-page":"188","DOI":"10.3390\/computation11100188","article-title":"Graph theoretical analysis of biological networks: A survey","volume":"11","author":"Erciyes","year":"2023","journal-title":"Computation"},{"issue":"10","key":"10.1016\/j.jmgm.2026.109546_b14","doi-asserted-by":"crossref","first-page":"2562","DOI":"10.1007\/s10910-023-01511-6","article-title":"Applications of graph theory in studying protein structure, dynamics, and interactions","volume":"62","author":"Zhou","year":"2023","journal-title":"J. Math. Chem."},{"issue":"9","key":"10.1016\/j.jmgm.2026.109546_b15","doi-asserted-by":"crossref","first-page":"3737","DOI":"10.1039\/C0CP01446J","article-title":"Defining rules of aromaticity: A unified approach to the H\u00fcckel, Clar and Randi\u0107 concepts","volume":"13","author":"Ciesielski","year":"2011","journal-title":"Phys. Chem. Chem. Phys."},{"issue":"3","key":"10.1016\/j.jmgm.2026.109546_b16","doi-asserted-by":"crossref","first-page":"863","DOI":"10.1002\/slct.201602080","article-title":"Aromaticity, the H\u00fcckel 4n+2 rule and magnetic current","volume":"2","author":"Zhao","year":"2017","journal-title":"ChemistrySelect"},{"issue":"6","key":"10.1016\/j.jmgm.2026.109546_b17","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1038\/s41557-022-00961-w","article-title":"Aromaticity rules","volume":"14","author":"Sola","year":"2022","journal-title":"Nat. Chem."},{"issue":"2","key":"10.1016\/j.jmgm.2026.109546_b18","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1002\/qsar.19910100207","article-title":"Studies of performance of graph theoretical indices in QSAR analysis","volume":"10","author":"Osamialowski","year":"1991","journal-title":"Quant. Struct.-Act. Relat."},{"key":"10.1016\/j.jmgm.2026.109546_b19","first-page":"257","article-title":"Recent advances in the open access cheminformatics toolkits, software tools, workflow environments, and databases","author":"Ambure","year":"2016","journal-title":"Comput.-Aided Drug Discov."},{"issue":"1","key":"10.1016\/j.jmgm.2026.109546_b20","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1186\/s13321-023-00786-w","article-title":"PDBe CCDUtils: An RDKit-based toolkit for handling and analysing small molecules in the protein data bank","volume":"15","author":"Kunnakkattu","year":"2023","journal-title":"J. Cheminform."},{"key":"10.1016\/j.jmgm.2026.109546_b21","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1016\/j.comcom.2022.09.029","article-title":"Deep reinforcement learning meets graph neural networks: Exploring a routing optimization use case","volume":"196","author":"Almasan","year":"2022","journal-title":"Comput. Commun."},{"issue":"9","key":"10.1016\/j.jmgm.2026.109546_b22","doi-asserted-by":"crossref","first-page":"1129","DOI":"10.1142\/S1088424604000490","article-title":"The chemistry of porphyrazines: An overview","volume":"8","author":"Rodr\u00edguez-Morgade","year":"2004","journal-title":"J. Porphyr. Phthalocyanines"},{"key":"10.1016\/j.jmgm.2026.109546_b23","doi-asserted-by":"crossref","DOI":"10.1016\/j.rineng.2025.107448","article-title":"Topological indices, entropy measures, and graph energy prediction of porphyrazine and tetrakis porphyrazine","volume":"28","author":"Kallesh","year":"2025","journal-title":"Results Eng."},{"key":"10.1016\/j.jmgm.2026.109546_b24","doi-asserted-by":"crossref","DOI":"10.1016\/j.bioorg.2020.103634","article-title":"S-seco-porphyrazine as a new member of the seco-porphyrazine family\u2013synthesis, characterization and photocytotoxicity against cancer cells","volume":"96","author":"Mlynarczyk","year":"2020","journal-title":"Bioorg. Chem."},{"issue":"4","key":"10.1016\/j.jmgm.2026.109546_b25","doi-asserted-by":"crossref","first-page":"304","DOI":"10.1021\/ci00008a008","article-title":"Molecular topological index: A relation with the Wiener index","volume":"32","author":"Klein","year":"1992","journal-title":"J. Chem. Inf. Comput. Sci."},{"issue":"2","key":"10.1016\/j.jmgm.2026.109546_b26","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1351\/pac198855020199","article-title":"Topological indices based on topological distances in molecular graphs","volume":"55","author":"Balaban","year":"1983","journal-title":"Pure Appl. Chem."},{"issue":"3","key":"10.1016\/j.jmgm.2026.109546_b27","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1016\/j.akcej.2017.09.006","article-title":"Randi\u0107 index and information","volume":"15","author":"Gutman","year":"2018","journal-title":"AKCE Int. J. Graphs Comb."},{"issue":"2","key":"10.1016\/j.jmgm.2026.109546_b28","first-page":"113","article-title":"The Zagreb indices 30 years after","volume":"76","author":"Nikolic","year":"2003","journal-title":"Croat. Chem. Acta"},{"issue":"1","key":"10.1016\/j.jmgm.2026.109546_b29","first-page":"550","article-title":"On the edge version of topological indices for certain networks","volume":"47","author":"Husin","year":"2022","journal-title":"Ital. J. Pure Appl. Math."},{"issue":"1","key":"10.1016\/j.jmgm.2026.109546_b30","first-page":"5","article-title":"On Zagreb indices and coindices","volume":"74","author":"Gutman","year":"2015","journal-title":"MATCH Commun. Math. Comput. Chem."},{"issue":"1","key":"10.1016\/j.jmgm.2026.109546_b31","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1080\/09720529.2022.2038931","article-title":"Enumeration of spanning trees in a chain of diphenylene graphs","volume":"25","author":"Modabish","year":"2022","journal-title":"J. Discrete Math. Sci. Cryptogr."},{"issue":"1","key":"10.1016\/j.jmgm.2026.109546_b32","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1080\/09720510.2022.2043621","article-title":"On sombor indices of line graph of silicate carbide Si2C3-I [p, q]","volume":"25","author":"Asif","year":"2022","journal-title":"J. Discrete Math. Sci. Cryptogr."},{"issue":"6","key":"10.1016\/j.jmgm.2026.109546_b33","doi-asserted-by":"crossref","first-page":"1253","DOI":"10.1080\/09720529.2020.1816695","article-title":"Zagreb indices and Zagreb coindices of the line graphs of the subdivision graphs","volume":"23","author":"Saidi","year":"2020","journal-title":"J. Discrete Math. Sci. Cryptogr."},{"issue":"10","key":"10.1016\/j.jmgm.2026.109546_b34","doi-asserted-by":"crossref","first-page":"2889","DOI":"10.1007\/s10910-024-01584-x","article-title":"Bonding alkane attributes with topological indices: A statistical intervention","volume":"62","author":"Raja","year":"2024","journal-title":"J. Math. Chem."},{"issue":"7","key":"10.1016\/j.jmgm.2026.109546_b35","doi-asserted-by":"crossref","first-page":"2059","DOI":"10.1080\/09720529.2021.1984561","article-title":"M-polynomials and degree-based topological indices of tadpole graph","volume":"24","author":"Chaudhry","year":"2021","journal-title":"J. Discrete Math. Sci. Cryptogr."},{"issue":"11","key":"10.1016\/j.jmgm.2026.109546_b36","doi-asserted-by":"crossref","first-page":"2443","DOI":"10.3390\/math11112443","article-title":"Topological properties and entropy calculations of aluminophosphates","volume":"11","author":"Vijay","year":"2023","journal-title":"Mathematics"},{"issue":"12","key":"10.1016\/j.jmgm.2026.109546_b37","doi-asserted-by":"crossref","first-page":"4726","DOI":"10.3390\/molecules28124726","article-title":"Computation of entropy measures for metal\u2013organic frameworks","volume":"28","author":"Imran","year":"2023","journal-title":"Molecules"},{"issue":"7","key":"10.1016\/j.jmgm.2026.109546_b38","doi-asserted-by":"crossref","first-page":"1305","DOI":"10.1080\/09720529.2019.1691329","article-title":"Some reversed degree-based topological indices for graphene","volume":"22","author":"Kwun","year":"2019","journal-title":"J. Discrete Math. Sci. Cryptogr."},{"issue":"7","key":"10.1016\/j.jmgm.2026.109546_b39","doi-asserted-by":"crossref","first-page":"1289","DOI":"10.1080\/09720529.2019.1700921","article-title":"Reverse degree based indices of some nanotubes","volume":"22","author":"Jung","year":"2019","journal-title":"J. Discrete Math. Sci. Cryptogr."},{"issue":"1","key":"10.1016\/j.jmgm.2026.109546_b40","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2021\/4540276","article-title":"On the study of reverse degree-based topological properties for the third type of pth chain hex-derived network","volume":"2021","author":"Koam","year":"2021","journal-title":"J. Math."},{"issue":"4","key":"10.1016\/j.jmgm.2026.109546_b41","first-page":"923","article-title":"Reverse Wiener indices","volume":"73","author":"Balaban","year":"2000","journal-title":"Croat. Chem. Acta"},{"issue":"13","key":"10.1016\/j.jmgm.2026.109546_b42","doi-asserted-by":"crossref","first-page":"1433","DOI":"10.1016\/j.dam.2010.04.003","article-title":"Hyper- and reverse-Wiener indices of F-sums of graphs","volume":"158","author":"Metsidik","year":"2010","journal-title":"Discrete Appl. Math."},{"issue":"2","key":"10.1016\/j.jmgm.2026.109546_b43","doi-asserted-by":"crossref","first-page":"256","DOI":"10.1080\/03081087.2021.2025199","article-title":"Reverse Wiener spectral radius of trees","volume":"71","author":"Oboudi","year":"2023","journal-title":"Linear Multilinear Algebra"},{"issue":"8","key":"10.1016\/j.jmgm.2026.109546_b44","doi-asserted-by":"crossref","first-page":"5249","DOI":"10.1080\/10406638.2021.1934045","article-title":"Quantitative structure analysis of some molecules in drugs used in the treatment of COVID-19 with topological indices","volume":"42","author":"Havare","year":"2022","journal-title":"Polycycl. Aromat. Compd."},{"issue":"6","key":"10.1016\/j.jmgm.2026.109546_b45","doi-asserted-by":"crossref","first-page":"3792","DOI":"10.1080\/10406638.2021.1873807","article-title":"Topological study of hydroxychloroquine conjugated molecular structure used for novel coronavirus (COVID-19) treatment","volume":"42","author":"Rauf","year":"2022","journal-title":"Polycycl. Aromat. Compd."},{"issue":"7","key":"10.1016\/j.jmgm.2026.109546_b46","first-page":"647","article-title":"On the certain topological indices of titania nanotube TiO2 [m, n]","volume":"72","author":"Javaid","year":"2017","journal-title":"J. Nat. Sci. A"},{"issue":"2","key":"10.1016\/j.jmgm.2026.109546_b47","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1139\/cjc-2016-0486","article-title":"Topological indices of rhombus type silicate and oxide networks","volume":"95","author":"Javaid","year":"2017","journal-title":"Can. J. Chem."},{"issue":"1","key":"10.1016\/j.jmgm.2026.109546_b48","first-page":"9","article-title":"On the degree based topological indices of benzene ring embedded in P-type-surface in 2D network","volume":"47","author":"Ahmad","year":"2018","journal-title":"Hacet. J. Math. Stat."},{"issue":"11","key":"10.1016\/j.jmgm.2026.109546_b49","doi-asserted-by":"crossref","first-page":"6","DOI":"10.3390\/sym10110619","article-title":"On degree-based topological indices of symmetric chemical structures","volume":"10","author":"Liu","year":"2018","journal-title":"Symmetry"},{"issue":"1","key":"10.1016\/j.jmgm.2026.109546_b50","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1007\/s40840-017-0463-2","article-title":"Zagreb indices and multiplicative Zagreb indices of Eulerian graphs","volume":"42","author":"Liu","year":"2019","journal-title":"Bull. Malays. Math. Sci. Soc."},{"issue":"3","key":"10.1016\/j.jmgm.2026.109546_b51","first-page":"3605","article-title":"Topological properties of cellular neural networks","volume":"37","author":"Imran","year":"2019","journal-title":"J. Intell. Fuzzy Syst."},{"issue":"6","key":"10.1016\/j.jmgm.2026.109546_b52","doi-asserted-by":"crossref","first-page":"680","DOI":"10.33945\/SAMI\/ECC.2020.6.5","article-title":"On molecular topological descriptors of certain families of nanostar dendrimers","volume":"2","author":"Imran","year":"2020","journal-title":"Eurasian Chem. Commun."},{"issue":"23","key":"10.1016\/j.jmgm.2026.109546_b53","doi-asserted-by":"crossref","DOI":"10.1002\/qua.26016","article-title":"Correlation between the Estrada index electronic energies for benzenoid hydrocarbons with applications to boron nanotubes","volume":"119","author":"Hayat","year":"2019","journal-title":"Int. J. Quantum Chem."},{"issue":"5","key":"10.1016\/j.jmgm.2026.109546_b54","doi-asserted-by":"crossref","first-page":"530","DOI":"10.1007\/s11244-024-01918-8","article-title":"Graph theory in chemical kinetics practice problems","volume":"67","author":"Szukiewicz","year":"2024","journal-title":"Top. Catal."},{"issue":"10","key":"10.1016\/j.jmgm.2026.109546_b55","doi-asserted-by":"crossref","first-page":"9479","DOI":"10.1080\/10406638.2022.2164315","article-title":"Topological indices and QSPR analysis of NSAID drugs","volume":"43","author":"Gnanaraj","year":"2023","journal-title":"Polycycl. Aromat. Compd."},{"issue":"5","key":"10.1016\/j.jmgm.2026.109546_b56","doi-asserted-by":"crossref","DOI":"10.1002\/qua.26520","article-title":"Mostar indices of SiO2 nanostructures andporphyrazine nanostructures","volume":"121","author":"Akhter","year":"2021","journal-title":"Int. J. Quantum Chem."},{"issue":"1","key":"10.1016\/j.jmgm.2026.109546_b57","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1515\/mgmc-2021-0006","article-title":"Some topological properties of uniform subdivision of Sierpi\u0144ski graphs","volume":"44","author":"Liu","year":"2021","journal-title":"Main Group Met. Chem."},{"issue":"7","key":"10.1016\/j.jmgm.2026.109546_b58","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1016\/j.dam.2021.10.004","article-title":"The Kirchhoff index and spanning trees of M\u00f6bius\/cylinder octagonal chain","volume":"30","author":"Liu","year":"2022","journal-title":"Discrete Appl. Math."},{"issue":"3","key":"10.1016\/j.jmgm.2026.109546_b59","doi-asserted-by":"crossref","first-page":"528","DOI":"10.2174\/1386207324666210217143114","article-title":"Graph indices for cartesian product of-sum of connected graphs","volume":"25","author":"Liu","year":"2022","journal-title":"Comb. Chem. High Throughput Screen."},{"issue":"1","key":"10.1016\/j.jmgm.2026.109546_b60","doi-asserted-by":"crossref","first-page":"755","DOI":"10.1080\/10406638.2021.2022726","article-title":"On topological properties of planar octahedron networks","volume":"43","author":"Liu","year":"2023","journal-title":"Polycycl. Aromat. Compd."},{"issue":"3","key":"10.1016\/j.jmgm.2026.109546_b61","doi-asserted-by":"crossref","first-page":"439","DOI":"10.2174\/1386207325666220104223136","article-title":"Novel applications of graph theory in chemistry and drug designing","volume":"25","author":"Liu","year":"2022","journal-title":"Comb. Chem. High Throughput Screen."},{"issue":"2","key":"10.1016\/j.jmgm.2026.109546_b62","doi-asserted-by":"crossref","DOI":"10.1016\/j.progsurf.2023.100704","article-title":"Epitaxial growth of borophene on substrates","volume":"98","author":"Li","year":"2023","journal-title":"Prog. Surf. Sci."},{"issue":"1","key":"10.1016\/j.jmgm.2026.109546_b63","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1038\/s41557-021-00813-z","article-title":"Synthesis of bilayer borophene","volume":"14","author":"Chen","year":"2022","journal-title":"Nat. Chem."},{"issue":"21","key":"10.1016\/j.jmgm.2026.109546_b64","doi-asserted-by":"crossref","first-page":"1873","DOI":"10.2174\/0113852728271631231121061349","article-title":"Connection number-based multiplicative Zagreb indices of chemical structures","volume":"27","author":"Hassan","year":"2023","journal-title":"Curr. Org. Chem."},{"issue":"6","key":"10.1016\/j.jmgm.2026.109546_b65","doi-asserted-by":"crossref","first-page":"563","DOI":"10.1038\/nchem.2491","article-title":"Experimental realization of two-dimensional boron sheets","volume":"8","author":"Feng","year":"2016","journal-title":"Nat. Chem."},{"key":"10.1016\/j.jmgm.2026.109546_b66","doi-asserted-by":"crossref","DOI":"10.1080\/00268976.2023.2226242","article-title":"Connection-based modified Zagreb indices of boron triangular sheet BTS (m, n)","author":"Hassan","year":"2023","journal-title":"Mol. Phys."},{"key":"10.1016\/j.jmgm.2026.109546_b67","first-page":"1","article-title":"Computation of connection-based Zagreb indices in chain graphs and triangular sheets","author":"Hassan","year":"2024","journal-title":"J. Coord. Chem."},{"issue":"4","key":"10.1016\/j.jmgm.2026.109546_b68","first-page":"1","article-title":"Improved QSAR methods for predicting drug properties utilizing topological indices and machine learning models","volume":"48","author":"Sardar","year":"2025","journal-title":"Eur. Phys. J. E"},{"issue":"1","key":"10.1016\/j.jmgm.2026.109546_b69","first-page":"50","article-title":"Study of middle graph for certain classes of graph by applying degree-based topological indices","volume":"5","author":"Sardar","year":"2023","journal-title":"Eurasian Chem. Commun."},{"key":"10.1016\/j.jmgm.2026.109546_b70","first-page":"378","article-title":"The coherence and robustness analysis for a family of unbalanced networks","volume":"11","author":"Liu","year":"2025","journal-title":"IEEE Trans. Signal Inf. Process. Netw."},{"issue":"1","key":"10.1016\/j.jmgm.2026.109546_b71","first-page":"1","article-title":"On irregularity indices and main eigenvalues of graphs and their applicability","volume":"1","author":"Arif","year":"2023","journal-title":"J. Appl. Math. Comput."},{"issue":"3","key":"10.1016\/j.jmgm.2026.109546_b72","doi-asserted-by":"crossref","DOI":"10.1016\/j.arabjc.2021.102994","article-title":"Quality testing of spectrum-based distance descriptors for polycyclic aromatic hydrocarbons with applications to carbon nanotubes and nanocones","volume":"14","author":"Hayat","year":"2021","journal-title":"Arab. J. Chem."},{"issue":"27","key":"10.1016\/j.jmgm.2026.109546_b73","first-page":"1","article-title":"Statistical significance of valency-based topological descriptors for correlating thermodynamic properties of benzenoid hydrocarbons with applications","volume":"12","author":"Hayat","year":"2023","journal-title":"Comput. Theor. Chem."},{"issue":"1","key":"10.1016\/j.jmgm.2026.109546_b74","doi-asserted-by":"crossref","first-page":"7080","DOI":"10.1038\/s41598-024-54962-5","article-title":"Valency based novel quantitative structure property relationship (QSPR) approach for predicting physical properties of polycyclic chemical compounds","volume":"14","author":"Raza","year":"2024","journal-title":"Sci. Rep."},{"issue":"7","key":"10.1016\/j.jmgm.2026.109546_b75","doi-asserted-by":"crossref","first-page":"4718","DOI":"10.1109\/TSMC.2025.3559326","article-title":"Coherence analysis for a class of polygon networks with the noise disturbance","volume":"55","author":"Liu","year":"2025","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"10.1016\/j.jmgm.2026.109546_b76","doi-asserted-by":"crossref","DOI":"10.3389\/fphy.2024.1348407","article-title":"Exploring spectrum-based descriptors in pharmacological traits through quantitative structure property (QSPR) analysis","volume":"12","author":"Raza","year":"2024","journal-title":"Front. Phys."},{"issue":"7","key":"10.1016\/j.jmgm.2026.109546_b77","doi-asserted-by":"crossref","first-page":"4386","DOI":"10.1080\/10406638.2021.1891105","article-title":"On reverse degree based topological indices of polycyclic metal organic network","volume":"42","author":"Zhao","year":"2022","journal-title":"Polycycl. Aromat. Compd."},{"issue":"7","key":"10.1016\/j.jmgm.2026.109546_b78","doi-asserted-by":"crossref","first-page":"1289","DOI":"10.1080\/09720529.2019.1700921","article-title":"Reverse degree based indices of some nanotubes","volume":"22","author":"Jung","year":"2019","journal-title":"J. Discrete Math. Sci. Cryptogr."},{"issue":"2","key":"10.1016\/j.jmgm.2026.109546_b79","first-page":"267","article-title":"Reverse Sombor index","volume":"12","author":"Swamy","year":"2022","journal-title":"Bull Int Math Virtual Inst"},{"key":"10.1016\/j.jmgm.2026.109546_b80","doi-asserted-by":"crossref","DOI":"10.1016\/j.compbiomed.2025.110101","article-title":"Role of eccentricity based topological descriptors to predict anti-HIV drugs attributes with supervised machine learning algorithms","volume":"190","author":"Zaman","year":"2025","journal-title":"Comput. Biol. Med."},{"issue":"1","key":"10.1016\/j.jmgm.2026.109546_b81","doi-asserted-by":"crossref","first-page":"15159","DOI":"10.1038\/s41598-023-42340-6","article-title":"Mathematical modeling and topological graph description of dominating david-derived networks based on edge partitions","volume":"13","author":"Zaman","year":"2023","journal-title":"Sci. Rep."},{"issue":"8","key":"10.1016\/j.jmgm.2026.109546_b82","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1140\/epje\/s10189-024-00446-3","article-title":"Mathematical modeling for prediction of physicochemical characteristics of cardiovascular drugs via modified reverse degree topological indices","volume":"47","author":"Arockiaraj","year":"2024","journal-title":"Eur. Phys. J. E"},{"issue":"2","key":"10.1016\/j.jmgm.2026.109546_b83","doi-asserted-by":"crossref","first-page":"1172","DOI":"10.1080\/10406638.2022.2026418","article-title":"Computing reverse degree based topological indices of vanadium carbide","volume":"43","author":"Wei","year":"2023","journal-title":"Polycycl. Aromat. Compd."},{"issue":"7","key":"10.1016\/j.jmgm.2026.109546_b84","doi-asserted-by":"crossref","first-page":"4300","DOI":"10.1080\/10406638.2021.1887299","article-title":"On topological indices of remdesivir compound used in treatment of coronavirus (COVID-19)","volume":"42","author":"Wei","year":"2022","journal-title":"Polycycl. Aromat. Compd."},{"issue":"1","key":"10.1016\/j.jmgm.2026.109546_b85","doi-asserted-by":"crossref","first-page":"47","DOI":"10.22457\/apam.v16n1a6","article-title":"Reverse Zagreb and reverse hyper-Zagreb indices and their polynomials of rhombus silicate networks","volume":"16","author":"Kulli","year":"2018","journal-title":"Ann. Pure Appl. Math."},{"issue":"10","key":"10.1016\/j.jmgm.2026.109546_b86","doi-asserted-by":"crossref","first-page":"6932","DOI":"10.1080\/10406638.2021.1993941","article-title":"On topological descriptors and curvilinear regression analysis of antiviral drugs used in COVID-19 treatment","volume":"42","author":"Ravi","year":"2022","journal-title":"Polycycl. Aromat. Compd."},{"issue":"1","key":"10.1016\/j.jmgm.2026.109546_b87","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1515\/mgmc-2022-0009","article-title":"On computation of the reduced reverse degree and neighbourhood degree sum-based topological indices for metal\u2013organic frameworks","volume":"45","author":"Ravi","year":"2022","journal-title":"Main Group Met. Chem."},{"issue":"1","key":"10.1016\/j.jmgm.2026.109546_b88","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1038\/s41598-023-28416-3","article-title":"Curvilinear regression analysis of benzenoid hydrocarbons and computation of some reduced reverse degree based topological indices for hyaluronic acid\u2013paclitaxel conjugates","volume":"13","author":"Ravi","year":"2023","journal-title":"Sci. Rep."},{"issue":"7","key":"10.1016\/j.jmgm.2026.109546_b89","doi-asserted-by":"crossref","DOI":"10.1088\/1402-4896\/ad5648","article-title":"Investigation of reduced reverse degree based polynomials and indices of gold crystals","volume":"99","author":"Khan","year":"2024","journal-title":"Phys. Scr."},{"issue":"9","key":"10.1016\/j.jmgm.2026.109546_b90","doi-asserted-by":"crossref","first-page":"6165","DOI":"10.1080\/10406638.2023.2274473","article-title":"Topological descriptors of molecular networks via reverse degree","volume":"44","author":"Hassan","year":"2024","journal-title":"Polycycl. Aromat. Compd."},{"key":"10.1016\/j.jmgm.2026.109546_b91","doi-asserted-by":"crossref","DOI":"10.1016\/j.jmgm.2024.108917","article-title":"Molecular networks via reduced reverse degree approach","volume":"135","author":"Hassan","year":"2025","journal-title":"J. Mol. Graph. Model."},{"issue":"7","key":"10.1016\/j.jmgm.2026.109546_b92","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1140\/epje\/s10189-024-00442-7","article-title":"Molecular structure of DNA via Zagreb connection descriptors","volume":"47","author":"Hassan","year":"2024","journal-title":"Eur. Phys. J. E"},{"issue":"4","key":"10.1016\/j.jmgm.2026.109546_b93","doi-asserted-by":"crossref","first-page":"256","DOI":"10.1007\/s11082-025-08170-8","article-title":"Exploring triangular prism networks TP(s) through the connection number approach","volume":"57","author":"Hassan","year":"2025","journal-title":"Opt. Quantum Electron."},{"issue":"2","key":"10.1016\/j.jmgm.2026.109546_b94","doi-asserted-by":"crossref","DOI":"10.1088\/1402-4896\/acb327","article-title":"Zagreb connection topological descriptors and structural property of the triangular chain structures","volume":"98","author":"Ullah","year":"2023","journal-title":"Phys. Scr."},{"key":"10.1016\/j.jmgm.2026.109546_b95","doi-asserted-by":"crossref","first-page":"1521","DOI":"10.1007\/s11696-025-04468-9","article-title":"Comparative analysis of planar octahedron molecular structure via Zagreb connection indices","volume":"80","author":"Hassan","year":"2026","journal-title":"Chem. Pap."},{"key":"10.1016\/j.jmgm.2026.109546_b96","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1140\/epje\/s10189-024-00418-7","article-title":"Fundamental aspects of the molecular topology of fuchsine acid dye with connection numbers","volume":"47","author":"Koam","year":"2024","journal-title":"Eur. Phys. J. E"},{"key":"10.1016\/j.jmgm.2026.109546_b97","doi-asserted-by":"crossref","DOI":"10.1038\/s41598-026-50541-y","article-title":"Spectral energies of redefined Zagreb indices and comparative QSPR applications to anticancer and alcohol datasets","author":"Zeren","year":"2026","journal-title":"Sci. Rep."},{"key":"10.1016\/j.jmgm.2026.109546_b98","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1007\/s11227-026-08347-2","article-title":"Mathematical quantification of nanostructures using degree-based reduced reverse indices","volume":"82","author":"Hassan","year":"2026","journal-title":"J. Supercomput."}],"container-title":["Journal of Molecular Graphics and Modelling"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S109332632600272X?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S109332632600272X?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,9,4]],"date-time":"2026-09-04T15:35:45Z","timestamp":1788536145000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S109332632600272X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,11]]},"references-count":98,"alternative-id":["S109332632600272X"],"URL":"https:\/\/doi.org\/10.1016\/j.jmgm.2026.109546","relation":{},"ISSN":["1093-3263"],"issn-type":[{"value":"1093-3263","type":"print"}],"subject":[],"published":{"date-parts":[[2026,11]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Closed-form connection-number-based topological indices for porphyrazine and tetrakis-porphyrazine networks","name":"articletitle","label":"Article Title"},{"value":"Journal of Molecular Graphics and Modelling","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.jmgm.2026.109546","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"109546"}}