{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T03:41:06Z","timestamp":1776397266469,"version":"3.51.2"},"reference-count":14,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T00:00:00Z","timestamp":1762473600000},"content-version":"vor","delay-in-days":310,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T00:00:00Z","timestamp":1735689600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"funder":[{"DOI":"10.13039\/100023491","name":"Department of Mathematics","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100023491","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2025,1]]},"abstract":"\n The join of graphs\n G<\/jats:italic>\n and\n H<\/jats:italic>\n , denoted by\n G<\/jats:italic>\n +\n H<\/jats:italic>\n , is the graph obtained from the disjoint union of\n G<\/jats:italic>\n and\n H<\/jats:italic>\n by joining each vertex in\n G<\/jats:italic>\n to each vertex in\n H<\/jats:italic>\n . An edge\n u<\/jats:italic>\n w<\/jats:italic>\n is said to be subdivided if\n u<\/jats:italic>\n w<\/jats:italic>\n is replaced by the path\n P<\/jats:italic>\n :\n u<\/jats:italic>\n v<\/jats:italic>\n w<\/jats:italic>\n , where\n v<\/jats:italic>\n is the new vertex. A graph obtained by subdividing each edge of a graph\n G<\/jats:italic>\n is called the subdivision of\n G<\/jats:italic>\n , and is denoted by\n S<\/jats:italic>\n (\n G<\/jats:italic>\n ). In this paper, we present results on the gracefulness of some join graphs, namely,\n T<\/jats:italic>\n +\n P<\/jats:italic>\n \n n<\/jats:italic>\n <\/jats:sub>\n ,\n T<\/jats:italic>\n +\n P<\/jats:italic>\n \n n<\/jats:italic>\n <\/jats:sub>\n (\n m<\/jats:italic>\n ),\n C<\/jats:italic>\n 4<\/jats:sub>\n +\n O<\/jats:italic>\n \n p<\/jats:italic>\n <\/jats:sub>\n ,\n P<\/jats:italic>\n \n s<\/jats:italic>\n <\/jats:sub>\n (\n m<\/jats:italic>\n ) +\n P<\/jats:italic>\n \n t<\/jats:italic>\n <\/jats:sub>\n (\n n<\/jats:italic>\n ) for some\n m<\/jats:italic>\n ,\n n<\/jats:italic>\n ,\n s<\/jats:italic>\n ,\n t<\/jats:italic>\n and\n T<\/jats:italic>\n is a graceful tree. We also show that the subdivision graph\n S<\/jats:italic>\n (\n K<\/jats:italic>\n \n m<\/jats:italic>\n ,\n n<\/jats:italic>\n <\/jats:sub>\n ) of complete bipartite graph\n K<\/jats:italic>\n \n m<\/jats:italic>\n ,\n n<\/jats:italic>\n <\/jats:sub>\n is graceful.\n <\/jats:p>","DOI":"10.1155\/jama\/5533881","type":"journal-article","created":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T10:21:20Z","timestamp":1762510880000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Graceful Labeling of some Join Graphs and the Subdivision of Complete Bipartite Graphs"],"prefix":"10.1155","volume":"2025","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5952-7606","authenticated-orcid":false,"given":"A.","family":"Panpa","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"P.","family":"Chaiprasert","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"C.","family":"Tisklang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2025,11,7]]},"reference":[{"key":"e_1_2_11_1_2","doi-asserted-by":"publisher","DOI":"10.1155\/2021\/9958077"},{"key":"e_1_2_11_2_2","first-page":"349","volume-title":"Theory of Graphs (Internat. Symposium, Rome, July 1966)","author":"Rosa A.","year":"1967"},{"key":"e_1_2_11_3_2","doi-asserted-by":"publisher","DOI":"10.1137\/0601045"},{"key":"e_1_2_11_4_2","first-page":"201","article-title":"Cordial Graphs: A Weaker Version of Graceful and Harmonious Graphs","volume":"23","author":"Cahit I.","year":"1987","journal-title":"Ars Combinatoria"},{"key":"e_1_2_11_5_2","doi-asserted-by":"publisher","DOI":"10.37236\/11668"},{"key":"e_1_2_11_6_2","doi-asserted-by":"publisher","DOI":"10.1109\/PROC.1977.10517"},{"key":"e_1_2_11_7_2","unstructured":"RingelG. Problem 25 in Theory of Graphs and its Applications Proc. Symposium Smolenice 1963 1964."},{"key":"e_1_2_11_8_2","volume-title":"On Graceful Graphs: sum of Graphs, Research Report 78","author":"Koh K. M.","year":"1979"},{"key":"e_1_2_11_9_2","doi-asserted-by":"publisher","DOI":"10.1016\/0315-0860(87)90064-4"},{"key":"e_1_2_11_10_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.akcej.2015.11.013"},{"key":"e_1_2_11_11_2","first-page":"18","volume-title":"Proceedings One day Combinatorics Conference, Research Notes in Mathematics","author":"Bermond J. C.","year":"1979"},{"key":"e_1_2_11_12_2","first-page":"283","article-title":"Gracefulness of n cone Cm\u2228Knc","volume":"66","author":"Bhat-Nayak V. N.","year":"2003","journal-title":"Ars Combinatoria"},{"key":"e_1_2_11_13_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(97)00069-1"},{"key":"e_1_2_11_14_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.endm.2016.05.011"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/jama\/5533881","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/full-xml\/10.1155\/jama\/5533881","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/jama\/5533881","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T02:42:20Z","timestamp":1776393740000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/jama\/5533881"}},"subtitle":[],"editor":[{"given":"Aliaa","family":"Burqan","sequence":"additional","affiliation":[],"role":[{"role":"editor","vocabulary":"crossref"}]}],"short-title":[],"issued":{"date-parts":[[2025,1]]},"references-count":14,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2025,1]]}},"alternative-id":["10.1155\/jama\/5533881"],"URL":"https:\/\/doi.org\/10.1155\/jama\/5533881","archive":["Portico"],"relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"value":"1110-757X","type":"print"},{"value":"1687-0042","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1]]},"assertion":[{"value":"2025-01-29","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2025-09-30","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2025-11-07","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"5533881"}}