{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,8]],"date-time":"2026-03-08T17:17:50Z","timestamp":1772990270370,"version":"3.50.1"},"reference-count":56,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2025,3,6]],"date-time":"2025-03-06T00:00:00Z","timestamp":1741219200000},"content-version":"vor","delay-in-days":64,"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"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2025,1]]},"abstract":"<jats:p>\n                    This study examines the hub number of ring graphs and investigates their behavior under operations such as union, intersection, and join. Different findings for this parameter are found for a variety of types of ring graphs, such as commutative ring graphs, path ring graphs, complete ring graphs, cycle ring graphs, and star ring graphs, for which the hub number\n                    <jats:italic>h<\/jats:italic>\n                    (RG) is calculated. Bounds for\n                    <jats:italic>h<\/jats:italic>\n                    (RG) are also computed, as are the connections between\n                    <jats:italic>h<\/jats:italic>\n                    (RG) and other known ring graph characteristics. Moreover, we studied the hub number of some restricted hyper\u2010totient graphs. The results of this study advance our knowledge of the features and structural attributes of ring graphs, which could benefit many domains involving network analysis and complex systems. Understanding the hub structures and ring graph operations can lead to more efficient algorithms and deeper insights into network design, data clustering, and communication systems.\n                  <\/jats:p>","DOI":"10.1155\/ijmm\/5522259","type":"journal-article","created":{"date-parts":[[2025,3,12]],"date-time":"2025-03-12T01:36:22Z","timestamp":1741743382000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the Hub Number of Ring Graphs and Their Behavior Under Graph Operations"],"prefix":"10.1155","volume":"2025","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6252-8968","authenticated-orcid":false,"given":"Mohammed","family":"Alsharafi","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4881-6493","authenticated-orcid":false,"given":"Haifa","family":"Ahmed","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Saad","family":"Tobaili","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2025,3,6]]},"reference":[{"key":"e_1_2_11_1_2","doi-asserted-by":"publisher","DOI":"10.15407\/dopovidi2022.04.025"},{"key":"e_1_2_11_2_2","doi-asserted-by":"publisher","DOI":"10.1109\/CCWC47524.2020.9031224"},{"key":"e_1_2_11_3_2","first-page":"1","article-title":"Graph Algebras","author":"Piotr M.","year":"2019","journal-title":"arXiv: Rings and Algebras"},{"key":"e_1_2_11_4_2","article-title":"Algebraic Property Graphs","author":"Joshua S.","year":"2019","journal-title":"arXiv: Database"},{"key":"e_1_2_11_5_2","doi-asserted-by":"publisher","DOI":"10.1109\/JPROC.2018.2821924"},{"key":"e_1_2_11_6_2","doi-asserted-by":"crossref","unstructured":"ZadehL. 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