{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T14:16:34Z","timestamp":1753884994366,"version":"3.41.2"},"reference-count":11,"publisher":"World Scientific Pub Co Pte Ltd","issue":"06","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Math. Algorithm. Appl."],"published-print":{"date-parts":[[2023,8]]},"abstract":"<jats:p> The order two element graph of a group [Formula: see text] is the simple undirected graph whose vertex set consists of all elements of [Formula: see text] and two distinct vertices [Formula: see text] are adjacent if and only if either [Formula: see text], where [Formula: see text] is the identity element of [Formula: see text]. We denote this graph by [Formula: see text]. In this paper, we identify those commutative groups [Formula: see text] for which the graph [Formula: see text] is connected. We also characterize the structures of the graph [Formula: see text] for any positive integer [Formula: see text]. Moreover, we consider the symmetric group [Formula: see text], the dihedral group [Formula: see text] for any positive integer [Formula: see text] and for these groups, we look into the structure and other graph-theoretic properties of the corresponding graphs. Finally, we give a list of finite groups of order [Formula: see text] with their corresponding order two element graph structures. <\/jats:p>","DOI":"10.1142\/s1793830922501361","type":"journal-article","created":{"date-parts":[[2022,8,17]],"date-time":"2022-08-17T04:19:30Z","timestamp":1660709970000},"source":"Crossref","is-referenced-by-count":0,"title":["Order two element graph over a group"],"prefix":"10.1142","volume":"15","author":[{"given":"S.","family":"Pradhan","sequence":"first","affiliation":[{"name":"Department of Mathematics, Nagar College, Murshidabad 742159, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3955-9464","authenticated-orcid":false,"given":"S.","family":"Kar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Jadavpur University, Kolkata 700032, India"}]},{"given":"B.","family":"Biswas","sequence":"additional","affiliation":[{"name":"Department of Science and Humanities, Ranaghat Government Polytechnic, Nadia 741201, India"}]}],"member":"219","published-online":{"date-parts":[[2022,9,30]]},"reference":[{"key":"S1793830922501361BIB001","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1007\/978-1-4419-6990-3_2","volume-title":"Commutative Algebra: Noetherian and Non-Noetherian Perspectives","author":"Anderson D. 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