{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T10:42:10Z","timestamp":1740134530342,"version":"3.37.3"},"reference-count":11,"publisher":"Wiley","license":[{"start":{"date-parts":[[2018,6,6]],"date-time":"2018-06-06T00:00:00Z","timestamp":1528243200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2018,6,6]]},"abstract":"<jats:p>Let <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mi>G<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>V<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>E<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math> be a graph with the vertex set <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:mi>V<\/mml:mi><\/mml:mrow><\/mml:math> and the edge set <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mrow><mml:mi>E<\/mml:mi><\/mml:mrow><\/mml:math>, respectively. By a graph <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mi>G<\/mml:mi><mml:mo>=<\/mml:mo><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>V<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>E<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math> we mean a finite undirected graph with neither loops nor multiple edges. The number of vertices of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:math> is called order of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:math> and it is denoted by <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mrow><mml:mi>p<\/mml:mi><\/mml:mrow><\/mml:math>. Let <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:math> be a <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>p<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>q<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math> graph. A super mean graph on <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:math> is an injection <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\"><mml:mi>f<\/mml:mi><mml:mo>:<\/mml:mo><mml:mi>V<\/mml:mi><mml:mo>\u2192<\/mml:mo><mml:mo stretchy=\"false\">{<\/mml:mo><mml:mn fontstyle=\"italic\">1,2<\/mml:mn><mml:mo>,<\/mml:mo><mml:mn fontstyle=\"italic\">3<\/mml:mn><mml:mo>\u2026<\/mml:mo><mml:mo>,<\/mml:mo><mml:mi>p<\/mml:mi><mml:mo>+<\/mml:mo><mml:mi>q<\/mml:mi><mml:mo stretchy=\"false\">}<\/mml:mo><\/mml:math> such that, for each edge <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M12\"><mml:mi>e<\/mml:mi><mml:mo>=<\/mml:mo><mml:mi>u<\/mml:mi><mml:mi>v<\/mml:mi><\/mml:math> in <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M13\"><mml:mrow><mml:mi>E<\/mml:mi><\/mml:mrow><\/mml:math> labeled by <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M14\"><mml:msup><mml:mrow><mml:mi>f<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo>\u204e<\/mml:mo><\/mml:mrow><\/mml:msup><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>e<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>=<\/mml:mo><mml:mfenced open=\"\u2308\" close=\"\u2309\" separators=\"|\"><mml:mrow><mml:mrow><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>f<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>u<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>+<\/mml:mo><mml:mi>f<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>v<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:mfenced><\/mml:mrow><mml:mo>\/<\/mml:mo><mml:mrow><mml:mn fontstyle=\"italic\">2<\/mml:mn><\/mml:mrow><\/mml:mrow><\/mml:mrow><\/mml:mfenced><\/mml:math>, the set <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M15\"><mml:mi>f<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>V<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>\u222a<\/mml:mo><mml:mo stretchy=\"false\">{<\/mml:mo><mml:msup><mml:mrow><mml:mi>f<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo>\u204e<\/mml:mo><\/mml:mrow><\/mml:msup><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>e<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>:<\/mml:mo><mml:mi>e<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:mi>E<\/mml:mi><mml:mo stretchy=\"false\">}<\/mml:mo><\/mml:math> forms <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M16\"><mml:mrow><mml:mfenced open=\"{\" close=\"}\" separators=\"|\"><mml:mrow><mml:mn fontstyle=\"italic\">1,2<\/mml:mn><mml:mo>,<\/mml:mo><mml:mn fontstyle=\"italic\">3<\/mml:mn><mml:mo>,<\/mml:mo><mml:mo>\u2026<\/mml:mo><mml:mo>,<\/mml:mo><mml:mi>p<\/mml:mi><mml:mo>+<\/mml:mo><mml:mi>q<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:math>. A graph which admits super mean labeling is called super mean graph. The<jats:italic> total graph <\/jats:italic><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M17\"><mml:mi>T<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>G<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math><jats:italic> of <\/jats:italic><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M18\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:math> is the graph with the vertex set <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M19\"><mml:mi>V<\/mml:mi><mml:mo>\u222a<\/mml:mo><mml:mi>E<\/mml:mi><\/mml:math> and two vertices are adjacent whenever they are either adjacent or incident in <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M20\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:math>. We have showed that graphs <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M21\"><mml:mi>T<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msub><mml:mrow><mml:mi>P<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math> and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M22\"><mml:mi>T<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfenced><\/mml:math> are super mean, where <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M23\"><mml:mrow><mml:msub><mml:mrow><mml:mi>P<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math> is a path on <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M24\"><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:math> vertices and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M25\"><mml:mrow><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math> is a cycle on <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M26\"><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:math> vertices.<\/jats:p>","DOI":"10.1155\/2018\/9250424","type":"journal-article","created":{"date-parts":[[2018,6,6]],"date-time":"2018-06-06T19:32:19Z","timestamp":1528313539000},"page":"1-5","source":"Crossref","is-referenced-by-count":3,"title":["On Super Mean Labeling for Total Graph of Path and Cycle"],"prefix":"10.1155","volume":"2018","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4071-840X","authenticated-orcid":true,"given":"Nur","family":"Inayah","sequence":"first","affiliation":[{"name":"Mathematics Department, Faculty of Sciences and Technology, State Islamic University of Syarif Hidayatullah, Jakarta, Indonesia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"I. Wayan","family":"Sudarsana","sequence":"additional","affiliation":[{"name":"Combinatorial and Applied Mathematics Research Group (CAMRG), Department of Mathematics, Faculty of Mathematics and Natural Sciences, Tadulako University, Palu, Indonesia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Selvy","family":"Musdalifah","sequence":"additional","affiliation":[{"name":"Combinatorial and Applied Mathematics Research Group (CAMRG), Department of Mathematics, Faculty of Mathematics and Natural Sciences, Tadulako University, Palu, Indonesia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nurhasanah","family":"Daeng Mangesa","sequence":"additional","affiliation":[{"name":"Combinatorial and Applied Mathematics Research Group (CAMRG), Department of Mathematics, Faculty of Mathematics and Natural Sciences, Tadulako University, Palu, Indonesia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"2","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0067364"},{"issue":"7-8","key":"8","first-page":"210","volume":"26","year":"2003","journal-title":"National Academy of Science Letters"},{"issue":"1","key":"11","first-page":"143","volume":"25","year":"2006","journal-title":"Bulletin of Pure & Applied Sciences"},{"key":"9","first-page":"103","volume":"22e","year":"2003","journal-title":"Bulletin of Pure & Applied Sciences- Mathematics and Statistics"},{"key":"10","first-page":"29","volume":"58","year":"2003","journal-title":"Pure and Applied Mathematics Journal"},{"key":"3","first-page":"83","volume":"1","year":"2012","journal-title":"International Journal of Mathematical Combinatorics"},{"issue":"1","key":"4","first-page":"103","volume":"6","year":"2009","journal-title":"AKCE International Journal of Graphs and Combinatorics"},{"issue":"1","key":"5","first-page":"53","volume":"46","year":"2010","journal-title":"SUT Journal of Mathematics"},{"key":"6","first-page":"101","volume":"96","year":"2015","journal-title":"Utilitas Mathematica"},{"key":"7","first-page":"65","volume":"112","year":"2013","journal-title":"Ars Combinatoria"},{"journal-title":"The Electronic Journal of Combinatorics","year":"2016","key":"1"}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2018\/9250424.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2018\/9250424.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2018\/9250424.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2018,6,6]],"date-time":"2018-06-06T19:32:22Z","timestamp":1528313542000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/ijmms\/2018\/9250424\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,6,6]]},"references-count":11,"alternative-id":["9250424","9250424"],"URL":"https:\/\/doi.org\/10.1155\/2018\/9250424","relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"type":"print","value":"0161-1712"},{"type":"electronic","value":"1687-0425"}],"subject":[],"published":{"date-parts":[[2018,6,6]]}}}