{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T00:43:21Z","timestamp":1759970601027,"version":"build-2065373602"},"reference-count":12,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,1,23]],"date-time":"2025-01-23T00:00:00Z","timestamp":1737590400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Chemical graph theory connects the analysis of molecular structures with mathematical graph theory, allowing for the prediction of chemical and physical properties through the use of topological indices. Among these, the recently introduced Harmonic\u2013Arithmetic (HA) index, proposed by Abeer M. Albalahi et al. in 2023, offers a novel method to quantify molecular and graph structures. It is defined as HA(G)=\u2211\u03bc\u03c9\u2208E(G)4dG(\u03bc)dG(\u03c9)(dG(\u03bc)+dG(\u03c9))2, where dG(\u03bc) and dG(\u03c9) are degrees of nodes \u03bc and \u03c9 in G. In this paper, the HA index examines the bounds for a tree T of order n, with a maximum degree \u2206. The application of the HA index extends to QSPR\/QSAR analyses, where topological indices play a crucial role in predicting the relationship between molecular structures and physicochemical properties, such as in Parkinson\u2019s, disease-related antibiotics by calculating their topological indices and analyzing them using QSPR models. Comparative analyses were performed between linear regression models and curvilinear-approach quadratic and cubic regression models to identify the minimal RMSE and enhance predictive accuracy for physicochemical properties. The results demonstrate that the HA index effectively connects mathematical graph theory with molecular characterization, offering reliable predictions, dependable bounds for tree graphs, and meaningful insights into molecular properties. These findings highlight the HA index\u2019s potential as a versatile and innovative tool in advancing chemical graph theory and its applications to real-world problems in chemistry.<\/jats:p>","DOI":"10.3390\/sym17020167","type":"journal-article","created":{"date-parts":[[2025,1,23]],"date-time":"2025-01-23T04:54:31Z","timestamp":1737608071000},"page":"167","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Harmonic\u2013Arithmetic Index: Trees with Maximum Degrees and Comparative Analysis of Antidrugs"],"prefix":"10.3390","volume":"17","author":[{"given":"Kalpana","family":"Ramesh","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur 603203, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6401-6533","authenticated-orcid":false,"given":"Shobana","family":"Loganathan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur 603203, India"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,23]]},"reference":[{"key":"ref_1","unstructured":"Trinajstic, N. (1992). Chemical Graph Theory, CRC Press. [2nd ed.]."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"673","DOI":"10.1038\/s41586-023-05905-z","article-title":"Computational approaches streamlining drug discovery","volume":"616","author":"Sadybekov","year":"2023","journal-title":"Nature"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"351","DOI":"10.5562\/cca2294","article-title":"Degree-based topological indices","volume":"86","author":"Gutman","year":"2013","journal-title":"Croatica Chem. Acta"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1021\/ja01193a005","article-title":"Structural determination of paraffin boiling points","volume":"69","author":"Wiener","year":"1947","journal-title":"J. Am. Chem. Soc."},{"key":"ref_5","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."},{"key":"ref_6","unstructured":"Albalahi, A.M., Ali, A., Alanazi, A.M., Bhatti, A.A., and Hamza, A.E. (2023). Harmonic-arithmetic index of (molecular) trees. arXiv."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1007\/s13370-024-01183-8","article-title":"Inequalities involving the harmonic-arithmetic index","volume":"35","author":"Albalahi","year":"2024","journal-title":"Afr. Mat."},{"key":"ref_8","first-page":"3","article-title":"Lanzhou index of trees with fixed maximum degree","volume":"86","author":"Dehgardi","year":"2021","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"e32397","DOI":"10.1016\/j.heliyon.2024.e32397","article-title":"A QSPR analysis and curvilinear regression models for various degree-based topological indices: Quinolone antibiotics","volume":"10","author":"Kirana","year":"2024","journal-title":"Heliyon"},{"key":"ref_10","first-page":"239","article-title":"QSPR analysis of antiparkinson\u2019s drugs with degree based topological indices","volume":"6","author":"Ramesh","year":"2024","journal-title":"Montes Taurus J. Pure Appl. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"008862","DOI":"10.1101\/cshperspect.a008862","article-title":"The history of Parkinson\u2019s disease: Early clinical descriptions and neurological therapies","volume":"1","author":"Goetz","year":"2011","journal-title":"Cold Spring Harb. Perspect. Med."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2291933","DOI":"10.1080\/27684830.2023.2291933","article-title":"First irregularity Sombor index of trees with fixed maximum degree","volume":"11","author":"Dehgardi","year":"2024","journal-title":"Res. 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