{"id":"https://openalex.org/W4389336642","doi":"https://doi.org/10.3390/sym15122160","title":"Labeling Circulant Graphs: Distance Two Condition","display_name":"Labeling Circulant Graphs: Distance Two Condition","publication_year":2023,"publication_date":"2023-12-05","ids":{"openalex":"https://openalex.org/W4389336642","doi":"https://doi.org/10.3390/sym15122160"},"language":"en","primary_location":{"id":"doi:10.3390/sym15122160","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15122160","pdf_url":"https://www.mdpi.com/2073-8994/15/12/2160/pdf?version=1701764657","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/15/12/2160/pdf?version=1701764657","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5063653504","display_name":"Laila Loudiki","orcid":"https://orcid.org/0000-0003-1779-1163"},"institutions":[{"id":"https://openalex.org/I119856527","display_name":"Cadi Ayyad University","ror":"https://ror.org/04xf6nm78","country_code":"MA","type":"education","lineage":["https://openalex.org/I119856527"]}],"countries":["MA"],"is_corresponding":true,"raw_author_name":"Laila Loudiki","raw_affiliation_strings":["Polydisciplinary Faculty of Safi, Cadi Ayyad University, B.P. 4162, Sidi Bouzid, Safi 46000, Morocco"],"raw_orcid":"https://orcid.org/0000-0003-1779-1163","affiliations":[{"raw_affiliation_string":"Polydisciplinary Faculty of Safi, Cadi Ayyad University, B.P. 4162, Sidi Bouzid, Safi 46000, Morocco","institution_ids":["https://openalex.org/I119856527"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5092734569","display_name":"Ez-Zobair Bidine","orcid":null},"institutions":[{"id":"https://openalex.org/I4210145365","display_name":"Universit\u00e9 Hassan 1er","ror":"https://ror.org/03cdvht47","country_code":"MA","type":"education","lineage":["https://openalex.org/I4210145365"]}],"countries":["MA"],"is_corresponding":false,"raw_author_name":"Ez-Zobair Bidine","raw_affiliation_strings":["Faculty of Sciences and Technologies of Settat, Hassan First, University of Settat, Road to Casablanca, Km 3,5 B.P. 577, Settat 26000, Morocco"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Faculty of Sciences and Technologies of Settat, Hassan First, University of Settat, Road to Casablanca, Km 3,5 B.P. 577, Settat 26000, Morocco","institution_ids":["https://openalex.org/I4210145365"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5081277685","display_name":"Mustapha Kchikech","orcid":"https://orcid.org/0000-0002-1919-8606"},"institutions":[{"id":"https://openalex.org/I119856527","display_name":"Cadi Ayyad University","ror":"https://ror.org/04xf6nm78","country_code":"MA","type":"education","lineage":["https://openalex.org/I119856527"]}],"countries":["MA"],"is_corresponding":false,"raw_author_name":"Mustapha Kchikech","raw_affiliation_strings":["Polydisciplinary Faculty of Safi, Cadi Ayyad University, B.P. 4162, Sidi Bouzid, Safi 46000, Morocco"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Polydisciplinary Faculty of Safi, Cadi Ayyad University, B.P. 4162, Sidi Bouzid, Safi 46000, Morocco","institution_ids":["https://openalex.org/I119856527"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5063653504"],"corresponding_institution_ids":["https://openalex.org/I119856527"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.3705,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.66430284,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":98},"biblio":{"volume":"15","issue":"12","first_page":"2160","last_page":"2160"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11797","display_name":"graph theory and CDMA systems","score":0.9976000189781189,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8758549094200134},{"id":"https://openalex.org/keywords/circulant-matrix","display_name":"Circulant matrix","score":0.8597661256790161},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.675108790397644},{"id":"https://openalex.org/keywords/modulo","display_name":"Modulo","score":0.6384483575820923},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.6326199769973755},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.517479658126831},{"id":"https://openalex.org/keywords/circulant-graph","display_name":"Circulant graph","score":0.42923545837402344},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.37208646535873413},{"id":"https://openalex.org/keywords/line-graph","display_name":"Line graph","score":0.09163525700569153},{"id":"https://openalex.org/keywords/graph-power","display_name":"Graph power","score":0.06295228004455566}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8758549094200134},{"id":"https://openalex.org/C115973184","wikidata":"https://www.wikidata.org/wiki/Q245457","display_name":"Circulant matrix","level":2,"score":0.8597661256790161},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.675108790397644},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.6384483575820923},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.6326199769973755},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.517479658126831},{"id":"https://openalex.org/C82190004","wikidata":"https://www.wikidata.org/wiki/Q5121633","display_name":"Circulant graph","level":5,"score":0.42923545837402344},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.37208646535873413},{"id":"https://openalex.org/C203776342","wikidata":"https://www.wikidata.org/wiki/Q1378376","display_name":"Line graph","level":3,"score":0.09163525700569153},{"id":"https://openalex.org/C149530733","wikidata":"https://www.wikidata.org/wiki/Q5597091","display_name":"Graph power","level":4,"score":0.06295228004455566}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/sym15122160","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15122160","pdf_url":"https://www.mdpi.com/2073-8994/15/12/2160/pdf?version=1701764657","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:31787e2a7bcd442eac9cb211d5b4c266","is_oa":false,"landing_page_url":"https://doaj.org/article/31787e2a7bcd442eac9cb211d5b4c266","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 15, Iss 12, p 2160 (2023)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/sym15122160","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15122160","pdf_url":"https://www.mdpi.com/2073-8994/15/12/2160/pdf?version=1701764657","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4389336642.pdf","grobid_xml":"https://content.openalex.org/works/W4389336642.grobid-xml"},"referenced_works_count":26,"referenced_works":["https://openalex.org/W1970388956","https://openalex.org/W1971367602","https://openalex.org/W1974517402","https://openalex.org/W1983030532","https://openalex.org/W1992815391","https://openalex.org/W1998152720","https://openalex.org/W2015267805","https://openalex.org/W2028341678","https://openalex.org/W2060681532","https://openalex.org/W2074820146","https://openalex.org/W2076534374","https://openalex.org/W2084287981","https://openalex.org/W2085421834","https://openalex.org/W2094613978","https://openalex.org/W2116418681","https://openalex.org/W2126408028","https://openalex.org/W2133003613","https://openalex.org/W2148410650","https://openalex.org/W2161972193","https://openalex.org/W2474966596","https://openalex.org/W2563363393","https://openalex.org/W2605843308","https://openalex.org/W2741497699","https://openalex.org/W2801125030","https://openalex.org/W6751316938","https://openalex.org/W6801076681"],"related_works":["https://openalex.org/W4378765291","https://openalex.org/W2349424204","https://openalex.org/W2962912586","https://openalex.org/W4401009353","https://openalex.org/W2116405749","https://openalex.org/W2884647475","https://openalex.org/W4285120246","https://openalex.org/W2003567067","https://openalex.org/W2942592462","https://openalex.org/W2359095061"],"abstract_inverted_index":{"Given":[0],"n\u22656,":[1],"D={1,2,\u2026,\u230an2\u230b},":[2],"and":[3,24,29,59,75,82,86,89,160],"a":[4,15,118,128,137,150,155],"generating":[5,141],"set":[6,17,78],"S\u2286D,":[7],"the":[8,36,46,61,98,111],"circulant":[9,151],"graph":[10,152],"Cn(S)":[11,101,121],"has":[12],"Zn":[13],"as":[14,53,133],"vertex":[16],"in":[18,60],"which":[19],"two":[20],"distinct":[21],"vertices":[22],"i":[23],"j":[25],"are":[26],"adjacent":[27],"if":[28,31,81,84,88,91],"only":[30,83,90],"|i\u2212j|n\u2208S,":[32],"where":[33,66],"|x|n=min(|x|,n\u2212|x|)":[34],"is":[35],"circular":[37],"distance":[38],"modulo":[39],"n.":[40],"In":[41],"this":[42],"paper,":[43],"we":[44,69,94,126,148],"determine":[45],"L(2,1)-labeling":[47],"number":[48,139],"of":[49,100,113,120,140,146,157],"Cn(D\u2216X),":[50],"referred":[51],"to":[52,109,117,123],"\u03bb(Cn(D\u2216X)),":[54],"for":[55,72],"X={\u230an2\u230b},":[56],"X={a},":[57],"X={a,b},":[58],"general":[62],"case":[63],"when":[64,97],"|X|&lt;\u230an2\u230b\u2212\u2308n4\u2309,":[65],"a,b\u2208D.":[67],"Furthermore,":[68],"demonstrate":[70],"that":[71,96,115,135],"all":[73],"n\u22656":[74],"any":[76],"given":[77],"S,":[79],"\u03bb(Cn(S))=n+gcd(n,S\u00af)\u22122":[80],"gcd(n,S\u00af)\u22652,":[85],"\u03bb(Cn(S))\u2264n\u22121":[87],"gcd(n,S\u00af)=1.":[92],"Additionally,":[93],"establish":[95],"diameter":[99,119,156],"equals":[102],"2,":[103,158],"\u03bb(Cn(S))=n\u22121.":[104],"This":[105],"observation":[106],"motivated":[107],"us":[108],"investigate":[110],"properties":[112],"S":[114],"lead":[116],"equal":[122],"2.":[124],"Then,":[125],"introduce":[127],"highly":[129],"distinctive":[130],"family,":[131],"denoted":[132],"An,":[134],"generates":[136],"large":[138],"sets.":[142],"For":[143],"each":[144],"value":[145],"n,":[147],"acquire":[149],"Cn(An)":[153],"with":[154],"\u03bb(Cn(An))=n\u22121,":[159],"various":[161],"additional":[162],"interesting":[163],"properties.":[164]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":1}],"updated_date":"2026-06-15T08:34:33.830935","created_date":"2025-10-10T00:00:00"}
