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In this paper we obtain some fundamental characterizations of the normal subgroup based power graph. We show some relation between the graph $\\Gamma_H(G)$ and the power graph $\\Gamma(\\frac{G}{H})$. We show that $\\Gamma_H(G)$ is complete if and only of $\\frac{G}{H}$ is cyclic group of order $1$ or $p^m$, where"},"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":"1601.04431","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-18T09:05:06Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"6996a2b37d0115da881ab5037701574bc1c473483256fac0107a4342f30cc5c5","abstract_canon_sha256":"7ebedaf792634ed51df6113d506835ca71e3690a0c312d19084614de8a177420"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:45.071971Z","signature_b64":"sEJqu/s0mD1O7diQDuzHkiIlMbV3vimMRoL6wQ4xOc5VRZ1MwhTZj/VHDtiDlOlXivFONUg1AIfH8BKkDl9MCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"357bf7c1e61111391028225327bf55c589583793f5d2e813740a368b97c93d10","last_reissued_at":"2026-05-18T01:22:45.071486Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:45.071486Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Normal Subgroup Based Power Graph of a finite Group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.CO","authors_text":"A. 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