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In this paper, we prove that $\\mu_2(G)\\geq \\frac{2m}{n}$ for almost all graphs. Moreover, we characterize the extremal graphs for any graphs. Finally, we provide the answer to Problem 3 in \\cite{KMT}, that is, the characterization of all graphs with $\\sigma=1$."},"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":"1711.06906","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-11-18T18:08:06Z","cross_cats_sorted":[],"title_canon_sha256":"5ca7c678f2b1c4d6586b6c5780c58a60a7cb9962f4c6f068167b628ed76bf870","abstract_canon_sha256":"55b79a713f1b52737b0af153211a6603c26907c9f0ecdb5fbf8a2b2ab27b81a8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:19:56.746286Z","signature_b64":"umGPu9ClwY/o5buNwJfUYR1LD7qQQRoB1q5gN5tCnaurU1VgvrysQhzi9NDiO9YkpJSUS+EGvU/T7yzfDNuKCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7dbee0bff269fd5ae686a7d5f39ab1ed2d74c0365d63e9f350061d959206369a","last_reissued_at":"2026-05-18T00:19:56.745549Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:19:56.745549Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Open problem on $\\sigma$-invariant","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kinkar Ch. 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