{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:41:01Z","timestamp":1760186461617,"version":"build-2065373602"},"reference-count":16,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2019,1,22]],"date-time":"2019-01-22T00:00:00Z","timestamp":1548115200000},"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>The main aim of the paper is to give the crossing number of the join product     G +  D n      for the disconnected graph G of order five consisting of one isolated vertex and of one vertex incident with some vertex of the three-cycle, and     D n     consists of n isolated vertices. In the proofs, the idea of the new representation of the minimum numbers of crossings between two different subgraphs that do not cross the edges of the graph G by the graph of configurations     G D     in the considered drawing D of     G +  D n      will be used. Finally, by adding some edges to the graph G, we are able to obtain the crossing numbers of the join product with the discrete graph     D n     and with the path     P n     on n vertices for three other graphs.<\/jats:p>","DOI":"10.3390\/sym11020123","type":"journal-article","created":{"date-parts":[[2019,1,24]],"date-time":"2019-01-24T03:52:32Z","timestamp":1548301952000},"page":"123","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Determining Crossing Number of Join of the Discrete Graph with Two Symmetric Graphs of Order Five"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2837-8879","authenticated-orcid":false,"given":"Michal","family":"Sta\u0161","sequence":"first","affiliation":[{"name":"Faculty of Electrical Engineering and Informatics, Technical University of Ko\u0161ice, 040 01 Ko\u0161ice, Slovakia"}]}],"member":"1968","published-online":{"date-parts":[[2019,1,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1016\/j.endm.2007.01.049","article-title":"The join of graphs and crossing numbers","volume":"28","year":"2007","journal-title":"Electron. 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Proceedings of the Aplimat 2018: 17th Conference on Applied Mathematics, Bratislava, Slovak."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1017\/S000497271800059X","article-title":"Determining crossing numbers of graphs of order six using cyclic permutations","volume":"98","year":"2018","journal-title":"Bull. Aust. Math. Soc."},{"key":"ref_11","first-page":"32","article-title":"Minimum crossings in join of graphs with paths and cycles","volume":"12","author":"Valo","year":"2012","journal-title":"Acta Electrotech. Inform."},{"key":"ref_12","first-page":"1","article-title":"On the join products of two special graphs on five vertices with the path and the cycle","volume":"6","year":"2018","journal-title":"Math. Model. Geom."},{"key":"ref_13","first-page":"29","article-title":"The optimal drawing of K5,n","volume":"21","author":"Medina","year":"2014","journal-title":"Electron. J. Comb."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"143","DOI":"10.37193\/CJM.2018.02.03","article-title":"Cyclic permutations and crossing numbers of join products of symmetric graph of order six","volume":"34","year":"2018","journal-title":"Carpathian J. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3","DOI":"10.15546\/aeei-2018-0001","article-title":"Software solution of the algorithm of the cyclic-order graph","volume":"18","year":"2018","journal-title":"Acta Electrotech. Inform."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"657","DOI":"10.1002\/jgt.3190170602","article-title":"Cyclic-order graphs and Zarankiewicz\u2019s crossing number conjecture","volume":"17","author":"Woodall","year":"1993","journal-title":"J. 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