{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T06:18:08Z","timestamp":1777702688642,"version":"3.51.4"},"reference-count":34,"publisher":"SAGE Publications","issue":"2","license":[{"start":{"date-parts":[[2018,8,27]],"date-time":"2018-08-27T00:00:00Z","timestamp":1535328000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Journal of Intelligent & Fuzzy Systems"],"published-print":{"date-parts":[[2019,3,16]]},"abstract":"\n For a graph\n G<\/jats:italic>\n = (\n V, E<\/jats:italic>\n ) the\n L<\/jats:italic>\n (3, 1, 1)-labeling is a mapping\n \u03bc<\/jats:italic>\n from the vertex set\n V<\/jats:italic>\n to the set of non-negative integers {0, 1, 2, \u2026} such that |\n \u03bc<\/jats:italic>\n (\n x<\/jats:italic>\n ) -\n \u03bc<\/jats:italic>\n (\n y<\/jats:italic>\n ) |\u22653 if\n d<\/jats:italic>\n (\n x, y<\/jats:italic>\n ) =3 and |\n \u03bc<\/jats:italic>\n (\n x<\/jats:italic>\n ) -\n \u03bc<\/jats:italic>\n (\n y<\/jats:italic>\n ) |\u22651 if\n d<\/jats:italic>\n (\n x, y<\/jats:italic>\n ) =2 or 3, where\n d<\/jats:italic>\n (\n x, y<\/jats:italic>\n ) is the distance between the vertices\n x<\/jats:italic>\n and\n y<\/jats:italic>\n .\n \u03bb<\/jats:italic>\n 3,1,1<\/jats:sub>\n (\n G<\/jats:italic>\n ) represents the\n L<\/jats:italic>\n (3, 1, 1)-labeling number of the graph\n G<\/jats:italic>\n and it is the largest non-negative integer used to label the graph\n G<\/jats:italic>\n .\n <\/jats:p>\n \n In the present article we have studied\n L<\/jats:italic>\n (3, 1, 1)-labeling of squares of some simple graphs, namely square of paths\n \n \n (<\/mml:mo>\n \n P<\/mml:mi>\n n<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msubsup>\n )<\/mml:mo>\n <\/mml:math>\n <\/jats:inline-formula>\n , square of complete graphs\n \n \n (<\/mml:mo>\n \n K<\/mml:mi>\n n<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msubsup>\n )<\/mml:mo>\n <\/mml:math>\n <\/jats:inline-formula>\n and square of complete bipartite graphs\n \n \n (<\/mml:mo>\n \n K<\/mml:mi>\n \n p<\/mml:mi>\n ,<\/mml:mo>\n q<\/mml:mi>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msubsup>\n )<\/mml:mo>\n <\/mml:math>\n <\/jats:inline-formula>\n and we obtained good results for these graph classes. The results produced in this article are unique and exact. This is the first result about\n L<\/jats:italic>\n (3, 1, 1)-labeling for these graph classes.\n <\/jats:p>","DOI":"10.3233\/jifs-172195","type":"journal-article","created":{"date-parts":[[2018,8,28]],"date-time":"2018-08-28T12:59:29Z","timestamp":1535461169000},"page":"1917-1925","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":10,"title":["L<\/i>\n (3, 1, 1)-labeling numbers of square of paths, complete graphs and complete bipartite graphs"],"prefix":"10.1177","volume":"36","author":[{"given":"Sk","family":"Amanathulla","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sankar","family":"Sahoo","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Madhumangal","family":"Pal","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2018,8,27]]},"reference":[{"key":"e_1_3_1_2_2","doi-asserted-by":"publisher","DOI":"10.1002\/net.20154"},{"key":"e_1_3_1_3_2","doi-asserted-by":"publisher","DOI":"10.11650\/twjm\/1500406480"},{"key":"e_1_3_1_4_2","unstructured":"ClippertonJ. 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