{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T00:59:52Z","timestamp":1760057992422,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,3,4]],"date-time":"2025-03-04T00:00:00Z","timestamp":1741046400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Symmetry in non-planar graphs is a fundamental concept that enhances understanding, simplifies analyses, and has practical implications in diverse fields such as science, engineering, and mathematics. A total \u03ba-labeling for a graph G\u03bd\u02c7 is composed of two labeling: one is an edge labeling \u03a5e:E(G\u03bd\u02c7)\u2192{1,2,3,\u2026,\u03bae} and the other is a vertex labeling \u03a5v:V(G\u03bd\u02c7)\u2192{0,2,4,\u2026,2\u03bav}, where \u03ba=max{\u03bae,2\u03bav}. The weight of an edge under reflexive labeling is defined as wt(pq)=\u03a5v(p)+\u03a5e(pq)+\u03a5v(q)\u2200e=pq. The total \u03ba\u2212labeling is said to be an edge-irregular reflexive \u03ba\u2212labeling, if for every two edges e\u03b4 and ej, the weights are distinct. The lowest value of \u03ba for which the graph G\u03bd\u02c7 has an irregular reflexive edge \u03ba\u2212labeling is called the reflexive edge strength of G\u03bd\u02c7, denoted as res(G\u03bd\u02c7). res(G\u03bd\u02c7) captures irregularity while preserving reflexivity by quantifying edge-weight variability in reflexive graphs. In this article, we are interested in determining the tight lower bound for non-planar prisms, cross prisms, cross-particle modified graphs, and cross-particle glowing graphs under reflexive labeling.<\/jats:p>","DOI":"10.3390\/sym17030386","type":"journal-article","created":{"date-parts":[[2025,3,4]],"date-time":"2025-03-04T05:41:51Z","timestamp":1741066911000},"page":"386","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Edge-Irregular Reflexive Strength of Non-Planar Graphs"],"prefix":"10.3390","volume":"17","author":[{"given":"Suleman","family":"Khan","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, Institute of Southern Punjab, Multan 32100, Pakistan"}]},{"given":"Muhammad Waseem","family":"Akram","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Institute of Southern Punjab, Multan 32100, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5228-1073","authenticated-orcid":false,"given":"Umar","family":"Ishtiaq","sequence":"additional","affiliation":[{"name":"Office of Research, Innovation and Commercialization, University of Management and Technology, Lahore 54770, Pakistan"}]},{"given":"Mubariz","family":"Garayev","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, Riyadh P.O. Box 2455, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8042-1806","authenticated-orcid":false,"given":"Ioan-Lucian","family":"Popa","sequence":"additional","affiliation":[{"name":"Department of Computing, Mathematics and Electronics, \u201c1 Decembrie 1918\u201d University of Alba Iulia, 510009 Alba Iulia, Romania"},{"name":"Faculty of Mathematics and Computer Science, Transilvania University of Brasov, Iuliu Maniu Street 50, 500091 Brasov, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Bondy, J.A., and Murty, U.S.R. (2008). Graph Theory, Springer.","DOI":"10.1007\/978-1-84628-970-5"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1145\/7531.7533","article-title":"On the minimal synchronism needed for distributed consensus","volume":"34","author":"Dolev","year":"1987","journal-title":"J. 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Comb."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/3\/386\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:46:52Z","timestamp":1760028412000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/3\/386"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,4]]},"references-count":18,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,3]]}},"alternative-id":["sym17030386"],"URL":"https:\/\/doi.org\/10.3390\/sym17030386","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,3,4]]}}}