{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,20]],"date-time":"2026-05-20T19:50:16Z","timestamp":1779306616350,"version":"3.51.4"},"reference-count":27,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2018,8,24]],"date-time":"2018-08-24T00:00:00Z","timestamp":1535068800000},"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>A graph operator is a mapping     F : \u0393 \u2192  \u0393 \u2032     , where    \u0393    and     \u0393 \u2032     are families of graphs. The different kinds of graph operators are an important topic in Discrete Mathematics and its applications. The symmetry of this operations allows us to prove inequalities relating the hyperbolicity constants of a graph G and its graph operators: line graph,     \u039b ( G )    ; subdivision graph,     S ( G )    ; total graph,     T ( G )    ; and the operators     R ( G )     and     Q ( G )    . In particular, we get relationships such as     \u03b4 ( G ) \u2264 \u03b4 ( R ( G ) ) \u2264 \u03b4 ( G ) + 1 \/ 2    ,     \u03b4 ( \u039b ( G ) ) \u2264 \u03b4 ( Q ( G ) ) \u2264 \u03b4 ( \u039b ( G ) ) + 1 \/ 2    ,     \u03b4 ( S ( G ) ) \u2264 2 \u03b4 ( R ( G ) ) \u2264 \u03b4 ( S ( G ) ) + 1     and     \u03b4 ( R ( G ) ) \u2212 1 \/ 2 \u2264 \u03b4 ( \u039b ( G ) ) \u2264 5 \u03b4 ( R ( G ) ) + 5 \/ 2     for every graph which is not a tree. Moreover, we also derive some inequalities for the Gromov product and the Gromov product restricted to vertices.<\/jats:p>","DOI":"10.3390\/sym10090360","type":"journal-article","created":{"date-parts":[[2018,8,24]],"date-time":"2018-08-24T11:13:45Z","timestamp":1535109225000},"page":"360","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Hyperbolicity on Graph Operators"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1748-9901","authenticated-orcid":false,"given":"J. A.","family":"M\u00e9ndez-Berm\u00fadez","sequence":"first","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"}]},{"given":"Rosal\u00edo","family":"Reyes","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"}]},{"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"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0863-4695","authenticated-orcid":false,"given":"Jos\u00e9 M.","family":"Sigarreta","sequence":"additional","affiliation":[{"name":"Instituto de F\u00edsica, Benem\u00e9rita Universidad Aut\u00f3noma de Puebla, Apartado Postal J-48, Puebla 72570, Mexico"},{"name":"Facultad de Matem\u00e1ticas, Universidad Aut\u00f3noma de Guerrero, Carlos E. Adame No.54 Col. Garita, Acapulco Gro. 39650, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,8,24]]},"reference":[{"key":"ref_1","first-page":"75","article-title":"D\u00e9monstration nouvelle d\u2019un th\u00e9or\u00e8me de Whitney sur les r\u00e9seaux","volume":"50","author":"Krausz","year":"1943","journal-title":"Mat. Fiz. Lapok"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1007\/BF02854581","article-title":"Some properties of line digraphs","volume":"9","author":"Harary","year":"1960","journal-title":"Rend. Circ. Mat. Palermo"},{"key":"ref_3","unstructured":"Prisner, E. (1995). 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