{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T03:35:53Z","timestamp":1777520153560,"version":"3.51.4"},"reference-count":11,"publisher":"Wiley","license":[{"start":{"date-parts":[[2018,8,1]],"date-time":"2018-08-01T00:00:00Z","timestamp":1533081600000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100007280","name":"Srinakharinwirot University","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100007280","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2018,8,1]]},"abstract":"<jats:p>For a set <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mi>W<\/mml:mi><mml:mo>=<\/mml:mo><mml:mfenced open=\"{\" close=\"}\" separators=\"|\"><mml:mrow><mml:msub><mml:mrow><mml:mi>w<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mi>w<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>,<\/mml:mo><mml:mo>\u2026<\/mml:mo><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mi>w<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfenced><\/mml:math> of vertices and a vertex <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:mi>v<\/mml:mi><\/mml:mrow><\/mml:math> of a connected graph <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:math>, the multirepresentation of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mrow><mml:mi>v<\/mml:mi><\/mml:mrow><\/mml:math> with respect to <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mrow><mml:mi>W<\/mml:mi><\/mml:mrow><\/mml:math> is the <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:math>-multiset <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mi>m<\/mml:mi><mml:mi>r<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>v<\/mml:mi><mml:mrow><mml:mo>\u2223<\/mml:mo><\/mml:mrow><mml:mi>W<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>=<\/mml:mo><mml:mfenced open=\"{\" close=\"}\" separators=\"|\"><mml:mrow><mml:mi>d<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>v<\/mml:mi><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mi>w<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfenced><mml:mo>,<\/mml:mo><mml:mi>d<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>v<\/mml:mi><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mi>w<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">2<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfenced><mml:mo>,<\/mml:mo><mml:mo>\u2026<\/mml:mo><mml:mo>,<\/mml:mo><mml:mi>d<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>v<\/mml:mi><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mi>w<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mfenced><mml:mo>,<\/mml:mo><\/mml:math> where <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mi>d<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>v<\/mml:mi><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mi>w<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>i<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math> is the distance between the vertices <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:mrow><mml:mi>v<\/mml:mi><\/mml:mrow><\/mml:math> and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\"><mml:mrow><mml:msub><mml:mrow><mml:mi>w<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>i<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math> for <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\"><mml:mi>i<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn mathvariant=\"normal\">1,2<\/mml:mn><mml:mo>,<\/mml:mo><mml:mo>\u2026<\/mml:mo><mml:mo>,<\/mml:mo><mml:mi>k<\/mml:mi><\/mml:math>. The set <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M12\"><mml:mrow><mml:mi>W<\/mml:mi><\/mml:mrow><\/mml:math> is a multiresolving set of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M13\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:math> if every two distinct vertices of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M14\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:math> have distinct multirepresentations with respect to <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M15\"><mml:mrow><mml:mi>W<\/mml:mi><\/mml:mrow><\/mml:math>. The minimum cardinality of a multiresolving set of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M16\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:math> is the multidimension <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M17\"><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">d<\/mml:mi><mml:mi mathvariant=\"normal\">i<\/mml:mi><mml:mi mathvariant=\"normal\">m<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>M<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>G<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math> of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M18\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:math>. It is shown that, for every pair <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M19\"><mml:mi>k<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>n<\/mml:mi><\/mml:math> of integers with <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M20\"><mml:mi>k<\/mml:mi><mml:mo>\u2265<\/mml:mo><mml:mn mathvariant=\"normal\">3<\/mml:mn><\/mml:math> and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M21\"><mml:mi>n<\/mml:mi><mml:mo>\u2265<\/mml:mo><mml:mn mathvariant=\"normal\">3<\/mml:mn><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>k<\/mml:mi><mml:mo>-<\/mml:mo><mml:mn mathvariant=\"normal\">1<\/mml:mn><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>, there is a connected graph <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M22\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:math> of order <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M23\"><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:math> with <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M24\"><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">d<\/mml:mi><mml:mi mathvariant=\"normal\">i<\/mml:mi><mml:mi mathvariant=\"normal\">m<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>M<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>G<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>=<\/mml:mo><mml:mi>k<\/mml:mi><\/mml:math>. For a multiset <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M25\"><mml:mo stretchy=\"false\">{<\/mml:mo><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>,<\/mml:mo><mml:mo>\u2026<\/mml:mo><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">}<\/mml:mo><\/mml:math> and an integer <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M26\"><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><\/mml:math>, we define <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M27\"><mml:mo stretchy=\"false\">{<\/mml:mo><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>,<\/mml:mo><mml:mo>\u2026<\/mml:mo><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">}<\/mml:mo><mml:mo>+<\/mml:mo><mml:mfenced open=\"{\" close=\"}\" separators=\"|\"><mml:mrow><mml:mi>c<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>c<\/mml:mi><mml:mo>,<\/mml:mo><mml:mo>\u2026<\/mml:mo><mml:mo>,<\/mml:mo><mml:mi>c<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>=<\/mml:mo><mml:mfenced open=\"{\" close=\"}\" separators=\"|\"><mml:mrow><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>+<\/mml:mo><mml:mi>c<\/mml:mi><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>+<\/mml:mo><mml:mi>c<\/mml:mi><mml:mo>,<\/mml:mo><mml:mo>\u2026<\/mml:mo><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>+<\/mml:mo><mml:mi>c<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math>. A multisimilar equivalence relation <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M28\"><mml:mrow><mml:msub><mml:mrow><mml:mi>R<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>W<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math> on <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M29\"><mml:mi>V<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>G<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math> with respect to <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M30\"><mml:mrow><mml:mi>W<\/mml:mi><\/mml:mrow><\/mml:math> is defined by <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M31\"><mml:mi>u<\/mml:mi><mml:mo>\u2009<\/mml:mo><mml:mo>\u2009<\/mml:mo><mml:msub><mml:mrow><mml:mi>R<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>W<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>\u2009<\/mml:mo><mml:mo>\u2009<\/mml:mo><mml:mi>v<\/mml:mi><\/mml:math> if <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M32\"><mml:mi>m<\/mml:mi><mml:mi>r<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>u<\/mml:mi><mml:mrow><mml:mo>\u2223<\/mml:mo><\/mml:mrow><mml:mi>W<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>=<\/mml:mo><mml:mi>m<\/mml:mi><mml:mi>r<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>v<\/mml:mi><mml:mrow><mml:mo>\u2223<\/mml:mo><\/mml:mrow><mml:mi>W<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>+<\/mml:mo><mml:mfenced open=\"{\" close=\"}\" separators=\"|\"><mml:mrow><mml:msub><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>W<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>u<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>v<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>W<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>u<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>v<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>,<\/mml:mo><mml:mo>\u2026<\/mml:mo><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>W<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>u<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>v<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mfenced><\/mml:math> for some integer <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M33\"><mml:msub><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>W<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>u<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>v<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>. We study the relationship between the elements in multirepresentations of vertices that belong to the same multisimilar equivalence class and also establish the upper bound for the cardinality of a multisimilar equivalence class. Moreover, a multiresolving set with prescribed multisimilar equivalence classes is presented.<\/jats:p>","DOI":"10.1155\/2018\/8978193","type":"journal-article","created":{"date-parts":[[2018,8,1]],"date-time":"2018-08-01T19:45:21Z","timestamp":1533152721000},"page":"1-6","source":"Crossref","is-referenced-by-count":9,"title":["The Multiresolving Sets of Graphs with Prescribed Multisimilar Equivalence Classes"],"prefix":"10.1155","volume":"2018","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7725-2984","authenticated-orcid":true,"given":"Varanoot","family":"Khemmani","sequence":"first","affiliation":[{"name":"Department of Mathematics, Srinakharinwirot University, Sukhumvit 23, Bangkok 10110, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6438-0343","authenticated-orcid":true,"given":"Supachoke","family":"Isariyapalakul","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Srinakharinwirot University, Sukhumvit 23, Bangkok 10110, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"1","doi-asserted-by":"publisher","DOI":"10.1016\/S0166-218X(00)00198-0"},{"key":"2","first-page":"191","volume":"2","year":"1976","journal-title":"Ars Combinatoria"},{"key":"10","first-page":"549","volume":"14","year":"1975","journal-title":"Congressus Numerantium"},{"issue":"4","key":"11","first-page":"445","volume":"22","year":"1988","journal-title":"Journal of Mathematical and Physical Sciences"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1016\/S1873-9776(98)80014-X"},{"key":"3","year":"1981"},{"key":"4","year":"1982"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-0208(08)72964-5"},{"key":"7","year":"1994"},{"key":"8","first-page":"193","volume":"1","year":"2009","journal-title":"Academic SWU"},{"key":"9","journal-title":"Discrete Applied Mathematics"}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2018\/8978193.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2018\/8978193.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2018\/8978193.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2018,8,1]],"date-time":"2018-08-01T19:45:23Z","timestamp":1533152723000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/ijmms\/2018\/8978193\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,8,1]]},"references-count":11,"alternative-id":["8978193","8978193"],"URL":"https:\/\/doi.org\/10.1155\/2018\/8978193","relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"value":"0161-1712","type":"print"},{"value":"1687-0425","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,8,1]]}}}