{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T20:38:42Z","timestamp":1774643922600,"version":"3.50.1"},"reference-count":40,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2018,10,1]],"date-time":"2018-10-01T00:00:00Z","timestamp":1538352000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100008530","name":"European Regional Development Fund","doi-asserted-by":"publisher","award":["MTM2016-78227-C2-1-P and MTM2017-90584-REDT"],"award-info":[{"award-number":["MTM2016-78227-C2-1-P and MTM2017-90584-REDT"]}],"id":[{"id":"10.13039\/501100008530","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Some years ago, the harmonic polynomial was introduced to study the harmonic topological index. Here, using this polynomial, we obtain several properties of the harmonic index of many classical symmetric operations of graphs: Cartesian product, corona product, join, Cartesian sum and lexicographic product. Some upper and lower bounds for the harmonic indices of these operations of graphs, in terms of related indices, are derived from known bounds on the integral of a product on nonnegative convex functions. Besides, we provide an algorithm that computes the harmonic polynomial with complexity     O (  n 2  )    .<\/jats:p>","DOI":"10.3390\/sym10100456","type":"journal-article","created":{"date-parts":[[2018,10,2]],"date-time":"2018-10-02T08:23:50Z","timestamp":1538468630000},"page":"456","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Harmonic Index and Harmonic Polynomial on Graph Operations"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1429-3644","authenticated-orcid":false,"given":"Juan C.","family":"Hern\u00e1ndez-G\u00f3mez","sequence":"first","affiliation":[{"name":"Facultad de Matem\u00e1ticas, Universidad Aut\u00f3noma de Guerrero, Carlos E. Adame No.54 Col. Garita, Acalpulco Gro. 39650, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1748-9901","authenticated-orcid":false,"given":"J. A.","family":"M\u00e9ndez-Berm\u00fadez","sequence":"additional","affiliation":[{"name":"Instituto de F\u00edsica, Benem\u00e9rita Universidad Aut\u00f3noma de Puebla, Apartado Postal J-48, Puebla 72570, 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, Legan\u00e9s, 28911 Madrid, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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, Acalpulco Gro. 39650, Mexico"},{"name":"Instituto de F\u00edsica, Benem\u00e9rita Universidad Aut\u00f3noma de Puebla, Apartado Postal J-48, Puebla 72570, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,10,1]]},"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","first-page":"6609","DOI":"10.1021\/ja00856a001","article-title":"On characterization of molecular branching","volume":"97","year":"1975","journal-title":"J. Am. Chem. Soc."},{"key":"ref_3","unstructured":"Gutman, I., and Furtula, B. (2008). Recent Results in the Theory of Randi\u0107 Index, University Kragujevac."},{"key":"ref_4","unstructured":"Li, X., and Gutman, I. (2006). 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