{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T10:28:36Z","timestamp":1769164116010,"version":"3.49.0"},"reference-count":35,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2021,5,12]],"date-time":"2021-05-12T00:00:00Z","timestamp":1620777600000},"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>Recently, the arithmetic\u2013geometric index (AG) was introduced, inspired by the well-known and studied geometric\u2013arithmetic index (GA). In this work, we obtain new bounds on the arithmetic\u2013geometric index, improving upon some already known bounds. In particular, we show families of graphs where such bounds are attained.<\/jats:p>","DOI":"10.3390\/sym13050857","type":"journal-article","created":{"date-parts":[[2021,5,12]],"date-time":"2021-05-12T22:46:14Z","timestamp":1620859574000},"page":"857","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Some Properties of the Arithmetic\u2013Geometric Index"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2620-0150","authenticated-orcid":false,"given":"Edil D.","family":"Molina","sequence":"first","affiliation":[{"name":"Facultad de Matem\u00e1ticas, Universidad Aut\u00f3noma de Guerrero, Carlos E. Adame No.54 Col. Garita, 39650 Acalpulco Gro., Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2851-7442","authenticated-orcid":false,"given":"Jos\u00e9 M.","family":"Rodr\u00edguez","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Legan\u00e9s, Madrid, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jos\u00e9 L.","family":"S\u00e1nchez","sequence":"additional","affiliation":[{"name":"Facultad de Matem\u00e1ticas, Universidad Aut\u00f3noma de Guerrero, Carlos E. Adame No.54 Col. Garita, 39650 Acalpulco Gro., Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4352-5109","authenticated-orcid":false,"given":"Jos\u00e9 M.","family":"Sigarreta","sequence":"additional","affiliation":[{"name":"Facultad de Matem\u00e1ticas, Universidad Aut\u00f3noma de Guerrero, Carlos E. Adame No.54 Col. Garita, 39650 Acalpulco Gro., Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,5,12]]},"reference":[{"key":"ref_1","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_2","doi-asserted-by":"crossref","unstructured":"Devillers, J., and Balaban, A.T. (1999). Topological Indices and Related Descriptors in QSAR and QSPR, Gordon and Breach.","DOI":"10.1201\/9781482296945"},{"key":"ref_3","unstructured":"Karelson, M. (2000). Molecular Descriptors in QSAR\/QSPR, Wiley-Interscience."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Todeschini, R., and Consonni, V. (2000). Handbook of Molecular Descriptors, Wiley-VCH.","DOI":"10.1002\/9783527613106"},{"key":"ref_5","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."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"066102","DOI":"10.1103\/PhysRevE.82.066102","article-title":"Quantifying network heterogeneity","volume":"82","author":"Estrada","year":"2010","journal-title":"Phys Rev. E"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Pineda, J., Mart\u00ednez, C., Mendez, A., Mu\u00f1os, J., and Sigarreta, J.M. (2020). Application of Bipartite Networks to the Study of Water Quality. Sustainability, 12.","DOI":"10.3390\/su12125143"},{"key":"ref_8","first-page":"57","article-title":"Trees with the first smallest and largest generalized topological indices","volume":"50","author":"Li","year":"2004","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_9","first-page":"97","article-title":"On variable Zagreb indices","volume":"77","year":"2004","journal-title":"Croat. Chem. Acta"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"210","DOI":"10.1007\/s10910-009-9542-4","article-title":"On general sum-connectivity index","volume":"47","author":"Zhou","year":"2010","journal-title":"J. Math. Chem."},{"key":"ref_11","first-page":"243","article-title":"Bond Additive Modeling 1. Adriatic Indices","volume":"83","year":"2010","journal-title":"Croat. Chem. Acta"},{"key":"ref_12","first-page":"261","article-title":"Bond additive modeling 2. Mathematical properties of max\u2013min rodeg index","volume":"83","year":"2010","journal-title":"Croat. Chem. Acta"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"e25659","DOI":"10.1002\/qua.25659","article-title":"Comparative analysis of symmetric division deg index as potentially useful molecular descriptor","volume":"118","author":"Furtula","year":"2018","journal-title":"Int. J. Quantum Chem."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1369","DOI":"10.1007\/s10910-009-9520-x","article-title":"Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges","volume":"46","author":"Furtula","year":"2009","journal-title":"J. Math. Chem."},{"key":"ref_15","first-page":"619","article-title":"On geometric-arithmetic index of graphs","volume":"64","author":"Das","year":"2010","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_16","first-page":"595","article-title":"Survey on Geometric-Arithmetic Indices of Graphs","volume":"65","author":"Das","year":"2011","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2030","DOI":"10.1016\/j.dam.2011.06.020","article-title":"On first geometric-arithmetic index of graphs","volume":"159","author":"Das","year":"2011","journal-title":"Discrete Appl. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1865","DOI":"10.1007\/s10910-017-0811-3","article-title":"A new approximation to the geometric-arithmetic index","volume":"56","author":"Sigarreta","year":"2018","journal-title":"J. Math. Chem."},{"key":"ref_19","first-page":"33","article-title":"Some bounds on GA1 index of graphs","volume":"65","author":"Mogharrab","year":"2010","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_20","first-page":"142","article-title":"Spectral properties of geometric-arithmetic index","volume":"277","author":"Sigarreta","year":"2016","journal-title":"Appl. Math. Comput."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1199","DOI":"10.18514\/MMN.2015.1423","article-title":"Bounds for the geometric-arithmetic index of a graph","volume":"16","author":"Sigarreta","year":"2015","journal-title":"Miskolc Math. Notes"},{"key":"ref_22","first-page":"19","article-title":"Arithmetic-geometric indices of path graph","volume":"16","author":"Shegehall","year":"2015","journal-title":"J. Math. Comput. Sci."},{"key":"ref_23","first-page":"67","article-title":"Arithmetic-geometric indices of graphs with pendant vertices attached to the middle vertices of path","volume":"6","author":"Shegehall","year":"2015","journal-title":"J. Math. Comput. Sci."},{"key":"ref_24","first-page":"4341919","article-title":"Computation of new degree-based topological indices of graphene","volume":"2016","author":"Shegehall","year":"2016","journal-title":"J. Math."},{"key":"ref_25","first-page":"635","article-title":"On spectral radius and energy of arithmetic-geometric matrix of graphs","volume":"83","author":"Zheng","year":"2020","journal-title":"MACTH Commun. Math. Comput. Chem."},{"key":"ref_26","first-page":"651","article-title":"Arithmetic-geometric spectral radius and energy of graphs","volume":"83","author":"Guo","year":"2020","journal-title":"MACTH Commun. Math. Comput. Chem."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1016\/j.laa.2018.05.027","article-title":"Degree-based energies of graphs","volume":"554","author":"Das","year":"2018","journal-title":"Linear Algebra Appl."},{"key":"ref_28","first-page":"125706","article-title":"Arithmetic-geometric index and its relations with geometric-arithmetic index","volume":"391","author":"Popivoda","year":"2021","journal-title":"Appl. Math. Comput."},{"key":"ref_29","unstructured":"Carballosa, W., Granados, A., M\u00e9ndez-Berm\u00fadez, J.A., Pestana, D., and Portilla, A. Computational properties of the arithmetic-geometric index. Submitted."},{"key":"ref_30","first-page":"87","article-title":"On the arithmetic-geometric index of graphs","volume":"85","author":"Cui","year":"2021","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"49","DOI":"10.5937\/SPSUNP1801049M","article-title":"Upper bounds for arithmetic-geometric index of graphs","volume":"10","year":"2018","journal-title":"Sci. Publ. State Univ. Novi Pazar Ser. A Appl. Math. Inf. Mech."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Rodr\u00edguez, J.M., S\u00e1nchez, J.L., Sigarreta, J.M., and Tour\u00eds, E. (2021). Bounds on the arithmetic-geometric index. Symmetry, 13.","DOI":"10.3390\/sym13040689"},{"key":"ref_33","first-page":"511","article-title":"Certain topological indices and polynomials for the Isaac graphs","volume":"24","author":"Poojary","year":"2021","journal-title":"J. Discr. Math. Sci. Crypt."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Gutman, I. (2021). Relation between geometric-arithmetic and arithmetic-geometric indices. J. Math. Chem.","DOI":"10.1007\/s10910-021-01256-0"},{"key":"ref_35","first-page":"109","article-title":"On Upper Bounds for the Geometric\u2013Arithmetic Topological Index","volume":"80","year":"2018","journal-title":"MATCH Commun. Math. Comput. Chem."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/5\/857\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:59:32Z","timestamp":1760162372000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/5\/857"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,12]]},"references-count":35,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2021,5]]}},"alternative-id":["sym13050857"],"URL":"https:\/\/doi.org\/10.3390\/sym13050857","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,5,12]]}}}