{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,4]],"date-time":"2026-05-04T19:10:44Z","timestamp":1777921844302,"version":"3.51.4"},"reference-count":26,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2023,12,5]],"date-time":"2023-12-05T00:00:00Z","timestamp":1701734400000},"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>Given n\u22656, D={1,2,\u2026,\u230an2\u230b}, and a generating set S\u2286D, the circulant graph Cn(S) has Zn as a vertex set in which two distinct vertices i and j are adjacent if and only if |i\u2212j|n\u2208S, where |x|n=min(|x|,n\u2212|x|) is the circular distance modulo n. In this paper, we determine the L(2,1)-labeling number of Cn(D\u2216X), referred to as \u03bb(Cn(D\u2216X)), for X={\u230an2\u230b}, X={a}, X={a,b}, and in the general case when |X|&lt;\u230an2\u230b\u2212\u2308n4\u2309, where a,b\u2208D. Furthermore, we demonstrate that for all n\u22656 and any given set S, \u03bb(Cn(S))=n+gcd(n,S\u00af)\u22122 if and only if gcd(n,S\u00af)\u22652, and \u03bb(Cn(S))\u2264n\u22121 if and only if gcd(n,S\u00af)=1. Additionally, we establish that when the diameter of Cn(S) equals 2, \u03bb(Cn(S))=n\u22121. This observation motivated us to investigate the properties of S that lead to a diameter of Cn(S) equal to 2. Then, we introduce a highly distinctive family, denoted as An, that generates a large number of generating sets. For each value of n, we acquire a circulant graph Cn(An) with a diameter of 2, \u03bb(Cn(An))=n\u22121, and various additional interesting properties.<\/jats:p>","DOI":"10.3390\/sym15122160","type":"journal-article","created":{"date-parts":[[2023,12,5]],"date-time":"2023-12-05T08:09:17Z","timestamp":1701763757000},"page":"2160","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Labeling Circulant Graphs: Distance Two Condition"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1779-1163","authenticated-orcid":false,"given":"Laila","family":"Loudiki","sequence":"first","affiliation":[{"name":"Polydisciplinary Faculty of Safi, Cadi Ayyad University, B.P. 4162, Sidi Bouzid, Safi 46000, Morocco"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ez-Zobair","family":"Bidine","sequence":"additional","affiliation":[{"name":"Faculty of Sciences and Technologies of Settat, Hassan First, University of Settat, Road to Casablanca, Km 3,5 B.P. 577, Settat 26000, Morocco"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mustapha","family":"Kchikech","sequence":"additional","affiliation":[{"name":"Polydisciplinary Faculty of Safi, Cadi Ayyad University, B.P. 4162, Sidi Bouzid, Safi 46000, Morocco"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,12,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"57","DOI":"10.21136\/MB.2002.133978","article-title":"Radio antipodal colorings of graphs","volume":"127","author":"Chartrand","year":"2002","journal-title":"Math. 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