{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,9]],"date-time":"2026-06-09T18:41:38Z","timestamp":1781030498040,"version":"3.54.1"},"reference-count":52,"publisher":"EDP Sciences","issue":"6","license":[{"start":{"date-parts":[[2024,12,16]],"date-time":"2024-12-16T00:00:00Z","timestamp":1734307200000},"content-version":"vor","delay-in-days":45,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["RAIRO-Oper. Res."],"accepted":{"date-parts":[[2024,11,2]]},"published-print":{"date-parts":[[2024,11]]},"abstract":"<jats:p>Let <jats:italic>G<\/jats:italic> = (<jats:italic>V<\/jats:italic> (<jats:italic>G<\/jats:italic>), <jats:italic>E<\/jats:italic>(<jats:italic>G<\/jats:italic>)) be a simple graph with vertex set <jats:italic>V<\/jats:italic> (<jats:italic>G<\/jats:italic>) and edge set <jats:italic>E<\/jats:italic>(<jats:italic>G<\/jats:italic>). Let <jats:italic>S<\/jats:italic> be a subset of <jats:italic>V<\/jats:italic> (<jats:italic>G<\/jats:italic>), and let <jats:italic>B<\/jats:italic>(<jats:italic>S<\/jats:italic>) be the set of neighbours of <jats:italic>S<\/jats:italic> in <jats:italic>V<\/jats:italic> (<jats:italic>G<\/jats:italic>)<jats:italic>\u2216S<\/jats:italic>. The differential <jats:italic>\u2202<\/jats:italic>(<jats:italic>S<\/jats:italic>) of <jats:italic>S<\/jats:italic> is the number <jats:italic>|B<\/jats:italic>(<jats:italic>S<\/jats:italic>)<jats:italic>|\u2212|S|<\/jats:italic>. The maximum value of <jats:italic>\u2202<\/jats:italic>(<jats:italic>S<\/jats:italic>) taken over all subsets <jats:italic>S<\/jats:italic> <jats:italic>\u2286<\/jats:italic> <jats:italic>V<\/jats:italic> (<jats:italic>G<\/jats:italic>) is the differential <jats:italic>\u2202<\/jats:italic>(<jats:italic>G<\/jats:italic>) of <jats:italic>G<\/jats:italic>. The graph R(<jats:italic>G<\/jats:italic>) is defined as the graph obtained from <jats:italic>G<\/jats:italic> by adding a new vertex <jats:italic>v<jats:sub>e<\/jats:sub><\/jats:italic> for each <jats:italic>e<\/jats:italic> <jats:italic>\u2208<\/jats:italic> <jats:italic>E<\/jats:italic>(<jats:italic>G<\/jats:italic>), and by joining <jats:italic>v<jats:sub>e<\/jats:sub><\/jats:italic> to the end vertices of <jats:italic>e<\/jats:italic>. In this paper we study the relationship between <jats:italic>\u2202<\/jats:italic>(<jats:italic>G<\/jats:italic>) and <jats:italic>\u2202<\/jats:italic>(R(<jats:italic>G<\/jats:italic>)), and give tight asymptotic bounds for <jats:italic>\u2202<\/jats:italic>(R(<jats:italic>G<\/jats:italic>)). We also exhibit some relationships between certain vertex sets of <jats:italic>G<\/jats:italic> and R(<jats:italic>G<\/jats:italic>) which involve well known graph theoretical parameters.<\/jats:p>","DOI":"10.1051\/ro\/2024212","type":"journal-article","created":{"date-parts":[[2024,11,5]],"date-time":"2024-11-05T08:46:38Z","timestamp":1730796398000},"page":"5467-5479","source":"Crossref","is-referenced-by-count":4,"title":["The differential on graph operator R(<i>G<\/i>)"],"prefix":"10.1051","volume":"58","author":[{"given":"Ludwin Ali","family":"Hern\u00e1ndez Basilio","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jes\u00fas Lea\u00f1os","family":"Mac\u00edas","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8909-1978","authenticated-orcid":false,"given":"Omar Rosario","family":"Cayetano","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jos\u00e9 Mar\u00eda","family":"Sigarreta Almira","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"250","published-online":{"date-parts":[[2024,12,16]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","first-page":"2171","DOI":"10.12988\/ijma.2015.54132","volume":"9","author":"Armada","year":"2015","journal-title":"Int. J. Math. Anal."},{"key":"R2","first-page":"319","volume":"103","author":"Basilio","year":"2017","journal-title":"Util. Math."},{"key":"R3","doi-asserted-by":"crossref","first-page":"205","DOI":"10.3390\/sym9100205","volume":"9","author":"Basilio","year":"2017","journal-title":"Symmetry"},{"key":"R4","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1007\/s10114-018-7307-3","volume":"35","author":"Basilio-Hern\u00e1ndez","year":"2019","journal-title":"Acta Math. Sin. English Ser."},{"key":"R5","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1007\/s00009-022-02236-7","volume":"20","author":"Bermudo","year":"2023","journal-title":"Mediterr. J. Math."},{"key":"R6","doi-asserted-by":"crossref","first-page":"3236","DOI":"10.1016\/j.disc.2012.07.021","volume":"312","author":"Bermudo","year":"2012","journal-title":"Discret. Math."},{"key":"R7","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1016\/j.dam.2012.11.013","volume":"165","author":"Bermudo","year":"2014","journal-title":"Discret. Appl. Math."},{"key":"R8","doi-asserted-by":"crossref","first-page":"155","DOI":"10.2298\/AADM140210003B","volume":"8","author":"Bermudo","year":"2014","journal-title":"Appl. Anal. Discret. Math."},{"key":"R9","doi-asserted-by":"crossref","first-page":"1124","DOI":"10.1080\/00207160.2014.941359","volume":"92","author":"Bermudo","year":"2015","journal-title":"Int. J. Comput. Math."},{"key":"R10","first-page":"257","volume":"97","author":"Bermudo","year":"2015","journal-title":"Util. Math."},{"key":"R11","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.dam.2018.03.043","volume":"263","author":"Bermudo","year":"2019","journal-title":"Discret. Appl. Math."},{"key":"R12","doi-asserted-by":"crossref","first-page":"1979","DOI":"10.2298\/FIL1607979B","volume":"30","author":"Bindusree","year":"2016","journal-title":"Filomat"},{"key":"R13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00009-021-01866-7","volume":"18","author":"Cabrera Mart\u00ednez","year":"2021","journal-title":"Mediterr. J. Math."},{"key":"R14","doi-asserted-by":"crossref","first-page":"327","DOI":"10.2989\/16073606.2020.1858992","volume":"45","author":"Cabrera Mart\u00ednez","year":"2022","journal-title":"Quaest. Math."},{"key":"R15","doi-asserted-by":"crossref","first-page":"290","DOI":"10.1016\/j.dam.2020.03.045","volume":"284","author":"Cabrera-Mart\u00ednez","year":"2020","journal-title":"Discret. Appl. Math."},{"key":"R16","doi-asserted-by":"crossref","first-page":"1036","DOI":"10.3390\/sym13061036","volume":"13","author":"Cabrera Mart\u00ednez","year":"2021","journal-title":"Symmetry"},{"key":"R17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00025-021-01473-8","volume":"76","author":"Cabrera Mart\u00ednez","year":"2021","journal-title":"Results Math."},{"key":"R18","doi-asserted-by":"crossref","first-page":"689","DOI":"10.2989\/16073606.2023.2252183","volume":"47","author":"Cabrera-Mart\u00ednez","year":"2023","journal-title":"Quaest. Math."},{"key":"R19","first-page":"4397","volume":"8","author":"Caga-anan","year":"2014","journal-title":"Appl. Math. Sci."},{"key":"R20","doi-asserted-by":"crossref","first-page":"210","DOI":"10.37236\/697","volume":"18","author":"Carballosa","year":"2011","journal-title":"Electron. J. Comb."},{"key":"R21","doi-asserted-by":"crossref","first-page":"1074","DOI":"10.3390\/math7111074","volume":"7","author":"Carballosa","year":"2019","journal-title":"Mathematics"},{"key":"R22","unstructured":"Cvetkocic D.M., Doob M. and Sachs H., Spectra of Graphs: Theory and Application. Vol. 10. Academic Press, New York (1980)."},{"key":"R23","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1016\/j.dam.2018.03.082","volume":"263","author":"Dettlaff","year":"2019","journal-title":"Discret. Appl. Math."},{"key":"R24","first-page":"93","volume":"79","author":"Farahani","year":"2017","journal-title":"Sci. Bull. Ser. B: Chem. Mater. Sci."},{"key":"R25","first-page":"2435","volume":"8","author":"Farahani","year":"2017","journal-title":"Int. J. Biochem. Biotech."},{"key":"R26","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/S0012-365X(01)00236-9","volume":"249","author":"Favaron","year":"2002","journal-title":"Discret. Math"},{"key":"R27","unstructured":"Goddard W. and Henning M.A., Generalised domination and independence in graphs. Congr. Numer. (1997) 161\u2013172."},{"key":"R28","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1007\/BF02854581","volume":"9","author":"Harary","year":"1960","journal-title":"Rend. Circ. Mat. Palermo"},{"key":"R29","unstructured":"Haynes T.W., Domination in Graphs: Volume 2: Advanced Topics. Routledge (2017)."},{"key":"R30","doi-asserted-by":"crossref","unstructured":"Haynes T.W., Hedetniemi S. and Slater P., Fundamentals of Domination in Graphs. CRC Press (2013).","DOI":"10.1201\/9781482246582"},{"key":"R31","doi-asserted-by":"crossref","first-page":"1249","DOI":"10.1051\/ro\/2020032","volume":"55","author":"Kanli","year":"2021","journal-title":"RAIRO-Oper. Res."},{"key":"R32","doi-asserted-by":"crossref","unstructured":"Kempe D., Kleinberg J. and Tardos E., Maximizing the spread of influence through a social network, in Proceedings of the Ninth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. ACM (2003) 137\u2013146.","DOI":"10.1145\/956750.956769"},{"key":"R33","doi-asserted-by":"crossref","unstructured":"Kempe D., Kleinberg J. and Tardos E., Influential nodes in a diffusion model for social networks, in International Colloquium on Automata, Languages, and Programming. Springer Berlin Heidelberg (2005) 1127\u20131138.","DOI":"10.1007\/11523468_91"},{"key":"R34","first-page":"75","volume":"50","author":"Krausz","year":"1943","journal-title":"Mat. Fiz. Lapok"},{"key":"R35","first-page":"43","volume":"69","author":"Mashburn","year":"2006","journal-title":"Util. Math."},{"key":"R36","doi-asserted-by":"crossref","first-page":"360","DOI":"10.3390\/sym10090360","volume":"10","author":"M\u00e9ndez-Berm\u00fadez","year":"2018","journal-title":"Symmetry"},{"key":"R37","doi-asserted-by":"crossref","first-page":"99","DOI":"10.47443\/dml.2022.179","volume":"11","author":"M\u00e9ndez-Berm\u00fadez","year":"2023","journal-title":"Discret. Math. Lett."},{"key":"R38","doi-asserted-by":"crossref","first-page":"2399","DOI":"10.3390\/sym13122399","volume":"13","author":"Michalski","year":"2021","journal-title":"Symmetry"},{"key":"R39","doi-asserted-by":"crossref","first-page":"381","DOI":"10.12732\/ijam.v32i3.2","volume":"32","author":"Natarajan","year":"2019","journal-title":"Int. J. Appl. Math."},{"key":"R40","doi-asserted-by":"crossref","first-page":"062305","DOI":"10.1103\/PhysRevE.102.062305","volume":"102","author":"Per\u00f3n","year":"2020","journal-title":"Phys. Rev. E"},{"key":"R41","unstructured":"Prisner E., Graph Dynamics. Vol. 338. CRC Press (1995)."},{"key":"R42","doi-asserted-by":"crossref","first-page":"129","DOI":"10.5556\/j.tkjm.41.2010.664","volume":"41","author":"Pushpam","year":"2010","journal-title":"Tamkang J. Math."},{"key":"R43","first-page":"1","volume":"3","author":"Ranjini","year":"2010","journal-title":"Int. J. Math. Comb."},{"key":"R44","first-page":"38","volume":"2","author":"Ranjini","year":"2020","journal-title":"Montes Taurus J. Pure Appl. Math."},{"key":"R45","doi-asserted-by":"crossref","first-page":"260","DOI":"10.1080\/10406638.2020.1726975","volume":"42","author":"Rashid","year":"2022","journal-title":"Polycycl. Aromat. Compd."},{"key":"R46","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1016\/j.dam.2021.03.015","volume":"300","author":"Samodivkin","year":"2021","journal-title":"Discret. Appl. Math."},{"key":"R47","first-page":"259","volume":"126","author":"Sigarreta","year":"2016","journal-title":"ARS Comb."},{"key":"R48","doi-asserted-by":"crossref","first-page":"241","DOI":"10.3390\/math9030241","volume":"9","author":"Sigarreta","year":"2021","journal-title":"Mathematics"},{"key":"R49","doi-asserted-by":"crossref","first-page":"197","DOI":"10.6028\/jres.082.019","volume":"82","author":"Slater","year":"1977","journal-title":"J. Res. Nat. Bureau Stand."},{"key":"R50","doi-asserted-by":"crossref","first-page":"290","DOI":"10.1016\/j.aml.2006.04.010","volume":"20","author":"Yan","year":"2007","journal-title":"Appl. Math. Lett."},{"key":"R51","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1137\/0403037","volume":"3","author":"Zhang","year":"1990","journal-title":"SIAM J. Discret. Math."},{"key":"R52","first-page":"251","volume":"101","author":"Zhang","year":"2011","journal-title":"ARS Comb."}],"container-title":["RAIRO - Operations Research"],"original-title":[],"link":[{"URL":"https:\/\/www.rairo-ro.org\/10.1051\/ro\/2024212\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,12,16]],"date-time":"2024-12-16T08:49:37Z","timestamp":1734338977000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.rairo-ro.org\/10.1051\/ro\/2024212"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11]]},"references-count":52,"journal-issue":{"issue":"6"},"alternative-id":["ro240267"],"URL":"https:\/\/doi.org\/10.1051\/ro\/2024212","relation":{},"ISSN":["0399-0559","2804-7303"],"issn-type":[{"value":"0399-0559","type":"print"},{"value":"2804-7303","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,11]]}}}