{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,8]],"date-time":"2026-02-08T02:46:44Z","timestamp":1770518804263,"version":"3.49.0"},"reference-count":17,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,10,20]],"date-time":"2021-10-20T00:00:00Z","timestamp":1634688000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Let G=(V,E) be a graph, and let \u03b2\u2208R. Motivated by a service coverage maximization problem with limited resources, we study the \u03b2-differential of G. The \u03b2-differential of G, denoted by \u2202\u03b2(G), is defined as \u2202\u03b2(G):=max{|B(S)|\u2212\u03b2|S|suchthatS\u2286V}. The case in which \u03b2=1 is known as the differential of G, and hence \u2202\u03b2(G) can be considered as a generalization of the differential \u2202(G) of G. In this paper, upper and lower bounds for \u2202\u03b2(G) are given in terms of its order |G|, minimum degree \u03b4(G), maximum degree \u0394(G), among other invariants of G. Likewise, the \u03b2-differential for graphs with heavy vertices is studied, extending the set of applications that this concept can have.<\/jats:p>","DOI":"10.3390\/axioms10040265","type":"journal-article","created":{"date-parts":[[2021,10,20]],"date-time":"2021-10-20T07:05:57Z","timestamp":1634713557000},"page":"265","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["General Properties on Differential Sets of a Graph"],"prefix":"10.3390","volume":"10","author":[{"given":"Ludwin A.","family":"Basilio","sequence":"first","affiliation":[{"name":"Faculty of Mathematics, Autonomous University of Guerrero, Carlos E. Adame 5, Col. La Garita, Acapulco 39650, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sergio","family":"Bermudo","sequence":"additional","affiliation":[{"name":"Department of Economy, Quantitative Methods and Economic History, Pablo de Olavide University, Carretera de Utrera Km. 1, 41013 Sevilla, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1429-3644","authenticated-orcid":false,"given":"Juan C.","family":"Hern\u00e1ndez-G\u00f3mez","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics, Autonomous University of Guerrero, Carlos E. Adame 5, Col. La Garita, Acapulco 39650, 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":"Faculty of Mathematics, Autonomous University of Guerrero, Carlos E. Adame 5, Col. La Garita, Acapulco 39650, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,10,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1016\/j.dam.2012.11.013","article-title":"Computing the differential of a graph: Hardness, approximability and exact algorithms","volume":"165","author":"Bermudo","year":"2014","journal-title":"Discret. Appl. Math."},{"key":"ref_2","first-page":"43","article-title":"Differentials in graphs","volume":"69","author":"Mashburn","year":"2006","journal-title":"Util. Math."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Abreu-Blaya, R., Bermudo, S., Rodr\u00edguez, J.M., and Tour\u00eds, E. (2021). Topological Indices and f-Polynomials on Some Graph Products. 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