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The nil-graph of ideals of $R$ is defined as the graph $\\mathbb{AG}_N(R)$ whose vertex set is $\\{I:\\ (0)\\neq I\\lhd R$ and there exists a non-trivial ideal $J$ such that $IJ\\subseteq {\\rm Nil}(R)\\}$ and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ\\subseteq {\\rm Nil}(R)$. Here, we study conditions under which $\\mathbb{AG}_N(R)$ is complete or bipartite. Also, the independence number of $\\mathbb{AG}_N(R)$ is determined, where $R$ is a reduced ring. Finally, we classify Arti"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1611.03730","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-11-10T08:24:25Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"78d81e30ff16a77c86153af56d719592086f1305b82353717e44655a14a6c546","abstract_canon_sha256":"82572f98bda87580c5632b97f042a6ac0641355030660bf325c4c43e93ef134c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:59:33.511132Z","signature_b64":"SEDFyiglZcDIavMeGCb6ATkyFu0i/cXfgfYGUEVBys7AJZm8fyCsGTZdjOsLUr8i8U6VaifmK41e3V6wIz+GDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ca53d112259dd0898dab14ab82a018fa988a551119b15e2eec03c346a2cb5350","last_reissued_at":"2026-05-18T00:59:33.510508Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:59:33.510508Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some Properties of the Nil-Graphs of Ideals of Commutative Rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"F. Shaveisi, R. Nikandish","submitted_at":"2016-11-10T08:24:25Z","abstract_excerpt":"Let $R$ be a commutative ring with identity and ${\\rm Nil}(R)$ be the set of nilpotent elements of $R$. The nil-graph of ideals of $R$ is defined as the graph $\\mathbb{AG}_N(R)$ whose vertex set is $\\{I:\\ (0)\\neq I\\lhd R$ and there exists a non-trivial ideal $J$ such that $IJ\\subseteq {\\rm Nil}(R)\\}$ and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ\\subseteq {\\rm Nil}(R)$. Here, we study conditions under which $\\mathbb{AG}_N(R)$ is complete or bipartite. Also, the independence number of $\\mathbb{AG}_N(R)$ is determined, where $R$ is a reduced ring. 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