{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T02:39:25Z","timestamp":1747190365150,"version":"3.40.5"},"reference-count":15,"publisher":"Wiley","license":[{"start":{"date-parts":[[2020,10,1]],"date-time":"2020-10-01T00:00:00Z","timestamp":1601510400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"CGANT Research Group\u2014the University of Jember"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2020,10,1]]},"abstract":"<jats:p>A total <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mi>k<\/mml:mi><\/mml:math>-labeling is a function <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:msub><mml:mrow><mml:mi>f<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>e<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:math> from the edge set to first natural number <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:msub><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>e<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:math> and a function <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:msub><mml:mrow><mml:mi>f<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>v<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:math> from the vertex set to non negative even number up to <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mn>2<\/mml:mn><mml:msub><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>v<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:math>, where <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mi>k<\/mml:mi><mml:mo>=<\/mml:mo><mml:mtext>max<\/mml:mtext><mml:mfenced open=\"{\" close=\"}\" separators=\"|\"><mml:mrow><mml:msub><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>e<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>,<\/mml:mo><mml:mn>2<\/mml:mn><mml:msub><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>v<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfenced><\/mml:math>. A <jats:italic>vertex irregular reflexive<\/jats:italic><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mi>k<\/mml:mi><\/mml:math><jats:italic>-labeling<\/jats:italic> of a simple, undirected, and finite graph <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:mi>G<\/mml:mi><\/mml:math> is total <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\"><mml:mi>k<\/mml:mi><\/mml:math>-labeling, if for every two different vertices <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\"><mml:mi>x<\/mml:mi><\/mml:math> and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M12\"><mml:msup><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo>\u2032<\/mml:mo><\/mml:mrow><\/mml:msup><\/mml:math> of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M13\"><mml:mi>G<\/mml:mi><\/mml:math>, <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M14\"><mml:mi>w<\/mml:mi><mml:mi>t<\/mml:mi><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>\u2260<\/mml:mo><mml:mi>w<\/mml:mi><mml:mi>t<\/mml:mi><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:msup><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo>\u2032<\/mml:mo><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:mfenced><\/mml:math>, where <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M15\"><mml:mi>w<\/mml:mi><mml:mi>t<\/mml:mi><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>=<\/mml:mo><mml:msub><mml:mrow><mml:mi>f<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>v<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>+<\/mml:mo><mml:msub><mml:mi mathvariant=\"normal\">\u03a3<\/mml:mi><mml:mrow><mml:mi>x<\/mml:mi><mml:mi>y<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:mi>E<\/mml:mi><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:msub><mml:msub><mml:mrow><mml:mi>f<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>e<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>x<\/mml:mi><mml:mi>y<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math>. The minimum <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M16\"><mml:mi>k<\/mml:mi><\/mml:math> for graph <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M17\"><mml:mi>G<\/mml:mi><\/mml:math> which has a vertex irregular reflexive <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M18\"><mml:mi>k<\/mml:mi><\/mml:math>-labeling is called the reflexive vertex strength of the graph <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M19\"><mml:mi>G<\/mml:mi><\/mml:math>, denoted by <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M20\"><mml:mtext>rvs<\/mml:mtext><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math>. In this paper, we determined the exact value of the reflexive vertex strength of any graph with pendant vertex which is useful to analyse the reflexive vertex strength on sunlet graph, helm graph, subdivided star graph, and broom graph.<\/jats:p>","DOI":"10.1155\/2020\/7812812","type":"journal-article","created":{"date-parts":[[2020,10,3]],"date-time":"2020-10-03T12:23:22Z","timestamp":1601727802000},"page":"1-8","source":"Crossref","is-referenced-by-count":5,"title":["On the Construction of the Reflexive Vertex <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mi>k<\/mml:mi><\/mml:math>-Labeling of Any Graph with Pendant Vertex"],"prefix":"10.1155","volume":"2020","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7508-0760","authenticated-orcid":true,"given":"I. H.","family":"Agustin","sequence":"first","affiliation":[{"name":"CGANT-University of Jember, Jember, Indonesia"},{"name":"Department of Mathematics, Airlangga University, Surabaya, Indonesia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2292-8443","authenticated-orcid":true,"given":"M. I.","family":"Utoyo","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Airlangga University, Surabaya, Indonesia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0575-3039","authenticated-orcid":true,"family":"Dafik","sequence":"additional","affiliation":[{"name":"CGANT-University of Jember, Jember, Indonesia"},{"name":"Department of Mathematics Education, University of Jember, Jember, Indonesia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"M.","family":"Venkatachalam","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Kongunadu Arts and Science College, Coimbatore, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"family":"Surahmat","sequence":"additional","affiliation":[{"name":"Department of Mathematics Education, Universitas Islam Malang, Malang, Indonesia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"year":"2016","key":"1"},{"year":"2001","key":"2"},{"key":"3","doi-asserted-by":"crossref","first-page":"1378","DOI":"10.1016\/j.disc.2005.11.075","volume":"307","year":"2007","journal-title":"Discrete Mathematics"},{"journal-title":"AKCE International Journal of Graphs and Combinatorics","year":"2018","key":"4"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1016\/j.akcej.2017.10.003"},{"key":"6","first-page":"129","volume":"112","year":"2013","journal-title":"Ars Combinatoria"},{"key":"7","first-page":"1","volume":"2020","year":"2020","journal-title":"ASCM Published By Jangjeon Mathematical Society"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1088\/1742-6596\/1211\/1\/012011"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2010.06.041"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.5614\/j.math.fund.sci.2015.47.3.6"},{"key":"11","first-page":"1","volume":"2012","year":"2011","journal-title":"The Electronic Journal of Combinatorics"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1016\/j.procs.2015.12.085"},{"key":"13","doi-asserted-by":"publisher","DOI":"10.5614\/ejgta.2018.6.2.5"},{"key":"14","doi-asserted-by":"publisher","DOI":"10.1088\/1742-6596\/855\/1\/012004"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.1088\/1742-6596\/1211\/1\/012005"}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2020\/7812812.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2020\/7812812.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2020\/7812812.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,10,3]],"date-time":"2020-10-03T15:20:18Z","timestamp":1601738418000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/ijmms\/2020\/7812812\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10,1]]},"references-count":15,"alternative-id":["7812812","7812812"],"URL":"https:\/\/doi.org\/10.1155\/2020\/7812812","relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"type":"print","value":"0161-1712"},{"type":"electronic","value":"1687-0425"}],"subject":[],"published":{"date-parts":[[2020,10,1]]}}}