{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,5]],"date-time":"2026-03-05T06:47:17Z","timestamp":1772693237860,"version":"3.50.1"},"reference-count":30,"publisher":"World Scientific Pub Co Pte Lt","issue":"03","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Math. Algorithm. Appl."],"published-print":{"date-parts":[[2016,9]]},"abstract":"<jats:p> The rings considered in this paper are commutative with identity which are not integral domains. Recall that an ideal [Formula: see text] of a ring [Formula: see text] is called an annihilating ideal if there exists [Formula: see text] such that [Formula: see text]. As in [M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl. 10(4) (2011) 727\u2013739], for any ring [Formula: see text], we denote by [Formula: see text] the set of all annihilating ideals of [Formula: see text] and by [Formula: see text] the set of all nonzero annihilating ideals of [Formula: see text]. Let [Formula: see text] be a ring. In [S. Visweswaran and H. D. Patel, A graph associated with the set of all nonzero annihilating ideals of a commutative ring, Discrete Math. Algorithm Appl. 6(4) (2014), Article ID: 1450047, 22pp], we introduced and studied the properties of a graph, denoted by [Formula: see text], which is an undirected simple graph whose vertex set is [Formula: see text] and distinct elements [Formula: see text] are joined by an edge in this graph if and only if [Formula: see text]. The aim of this paper is to study the interplay between the ring theoretic properties of a ring [Formula: see text] and the graph theoretic properties of [Formula: see text], where [Formula: see text] is the complement of [Formula: see text]. In this paper, we first determine when [Formula: see text] is connected and also determine its diameter when it is connected. We next discuss the girth of [Formula: see text] and study regarding the cliques of [Formula: see text]. Moreover, it is shown that [Formula: see text] is complemented if and only if [Formula: see text] is reduced. <\/jats:p>","DOI":"10.1142\/s1793830916500439","type":"journal-article","created":{"date-parts":[[2016,4,26]],"date-time":"2016-04-26T08:37:36Z","timestamp":1461659856000},"page":"1650043","source":"Crossref","is-referenced-by-count":7,"title":["On the complement of a graph associated with the set of all nonzero annihilating ideals of a commutative ring"],"prefix":"10.1142","volume":"08","author":[{"given":"S.","family":"Visweswaran","sequence":"first","affiliation":[{"name":"Department of Mathematics, Saurashtra University, Rajkot 360 005, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Patat","family":"Sarman","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Saurashtra University, Rajkot 360 005, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2016,8]]},"reference":[{"key":"S1793830916500439BIB001","doi-asserted-by":"publisher","DOI":"10.4134\/JKMS.2012.49.1.085"},{"key":"S1793830916500439BIB002","doi-asserted-by":"publisher","DOI":"10.1016\/j.jpaa.2009.03.013"},{"key":"S1793830916500439BIB003","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2011.10.020"},{"key":"S1793830916500439BIB004","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-6990-3_2"},{"key":"S1793830916500439BIB005","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2008.06.028"},{"key":"S1793830916500439BIB006","doi-asserted-by":"publisher","DOI":"10.1142\/S0219498812500740"},{"key":"S1793830916500439BIB007","doi-asserted-by":"publisher","DOI":"10.1142\/S021949881250212X"},{"key":"S1793830916500439BIB008","doi-asserted-by":"publisher","DOI":"10.1080\/00927872.2014.897198"},{"key":"S1793830916500439BIB009","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-4049(02)00250-5"},{"key":"S1793830916500439BIB010","doi-asserted-by":"publisher","DOI":"10.1006\/jabr.1998.7840"},{"key":"S1793830916500439BIB011","doi-asserted-by":"publisher","DOI":"10.1080\/00927872.2012.678956"},{"key":"S1793830916500439BIB012","volume-title":"Introduction to Commutative Algebra","author":"Atiyah M. 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