{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:37:28Z","timestamp":1760233048113,"version":"build-2065373602"},"reference-count":14,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,12,16]],"date-time":"2022-12-16T00:00:00Z","timestamp":1671148800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100005357","name":"Slovak Research and Development Agency","doi-asserted-by":"publisher","award":["APVV-19-0153","VEGA 1\/0243\/23"],"award-info":[{"award-number":["APVV-19-0153","VEGA 1\/0243\/23"]}],"id":[{"id":"10.13039\/501100005357","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A k-labeling from the vertex set of a simple graph G=(V,E) to a set of integers {1,2,\u2026,k} is defined to be a modular edge irregular if, for every couple of distinct edges, their modular edge weights are distinct. The modular edge weight is the remainder of the division of the sum of end vertex labels by modulo |E(G)|. The modular edge irregularity strength of a graph is known as the maximal vertex label k, minimized over all modular edge irregular k-labelings of the graph. In this paper we describe labeling schemes with symmetrical distribution of even and odd edge weights and investigate the existence of (modular) edge irregular labelings of joins of paths and cycles with isolated vertices. We estimate the bounds of the (modular) edge irregularity strength for the join graphs Pn+Km\u00af and Cn+Km\u00af and determine the corresponding exact value of the (modular) edge irregularity strength for some fan graphs and wheel graphs in order to prove the sharpness of the presented bounds.<\/jats:p>","DOI":"10.3390\/sym14122671","type":"journal-article","created":{"date-parts":[[2022,12,19]],"date-time":"2022-12-19T06:58:29Z","timestamp":1671433109000},"page":"2671","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Modular Version of Edge Irregularity Strength for Fan and Wheel Graphs"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9600-2497","authenticated-orcid":false,"given":"Debi Oktia","family":"Haryeni","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Military Mathematics and Natural Sciences, The Republic of Indonesia Defense University, IPSC Area, Sentul, Bogor 16810, Indonesia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8927-6043","authenticated-orcid":false,"given":"Zata Yumni","family":"Awanis","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Mataram, Jalan Majapahit No. 62, Mataram 83125, Indonesia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5758-0347","authenticated-orcid":false,"given":"Martin","family":"Ba\u010da","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Informatics, Technical University, 042 00 Ko\u0161ice, Slovakia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8432-9836","authenticated-orcid":false,"given":"Andrea","family":"Semani\u010dov\u00e1-Fe\u0148ov\u010d\u00edkov\u00e1","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Informatics, Technical University, 042 00 Ko\u0161ice, Slovakia"},{"name":"Division of Mathematics, Saveetha School of Engineering, SIMATS, Chennai 602 105, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,16]]},"reference":[{"key":"ref_1","first-page":"607","article-title":"On edge irregularity strength of graphs","volume":"243","author":"Ahmad","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_2","first-page":"187","article-title":"Irregular networks","volume":"64","author":"Chartrand","year":"1988","journal-title":"Congr. Numer."},{"key":"ref_3","first-page":"155","article-title":"On edge irregularity strength of Toeplitz graphs","volume":"78","author":"Ahmad","year":"2016","journal-title":"U.P.B. Sci. Bull. Ser. A"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"414","DOI":"10.1016\/j.akcej.2018.06.011","article-title":"On the edge irregularity strength of grid graphs","volume":"17","author":"Tarawneh","year":"2020","journal-title":"AKCE Int. J. Graphs Comb."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2724","DOI":"10.3934\/math.2021166","article-title":"On the edge irregularity strength for some classes of plane graphs","volume":"6","author":"Tarawneh","year":"2021","journal-title":"AIMS Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1475","DOI":"10.3934\/math.2023074","article-title":"Modular edge irregularity strength of graphs","volume":"8","author":"Koam","year":"2023","journal-title":"AIMS Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"435","DOI":"10.5614\/ejgta.2020.8.2.19","article-title":"Modular irregularity strength of graphs","volume":"8","author":"Muthugurupackiam","year":"2020","journal-title":"Electron. J. Graph Theory Appl."},{"key":"ref_8","first-page":"1132","article-title":"Modular irregularity strength of graphs","volume":"9","author":"Muthugurupackiam","year":"2018","journal-title":"J. Comput. Math. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Ba\u010da, M., Kim\u00e1kov\u00e1, Z., Lascs\u00e1kov\u00e1, M., and Semani\u010dov\u00e1-Fe\u0148ov\u010d\u00edkov\u00e1, A. (2021). The irregularity and modular irregularity strength of fan graphs. Symmetry, 13.","DOI":"10.3390\/sym13040605"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"4746609","DOI":"10.1155\/2021\/4746609","article-title":"Modular irregular labeling on double-star and friendship graphs","volume":"2021","author":"Sugeng","year":"2021","journal-title":"J. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"53","DOI":"10.30598\/tensorvol2iss2pp53-58","article-title":"Modular irregularity strength of triangular book graph","volume":"2","author":"Tilukay","year":"2021","journal-title":"Tensor-Pure Appl. Math. J."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Nisa, I.C. (2022). Modular irregular labeling on complete graphs. Daya-Mat.-J. Inov. Pendidik. Mat., 10.","DOI":"10.26858\/jdm.v10i3.37426"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"201","DOI":"10.5614\/ejgta.2018.6.1.15","article-title":"Computing the edge irregularity strengths of chain graphs and the join of two graphs","volume":"6","author":"Ahmad","year":"2018","journal-title":"Electron. J. Graph Theory Appl."},{"key":"ref_14","unstructured":"Hartsfield, N., and Ringel, G. (1990). Pearls in Graph Theory: A Comprehensive Introduction, Academic Press."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/12\/2671\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:42:55Z","timestamp":1760146975000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/12\/2671"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,16]]},"references-count":14,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["sym14122671"],"URL":"https:\/\/doi.org\/10.3390\/sym14122671","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,12,16]]}}}