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We prove that for any connected graph $G$ of order at least 3, $s_g(G)=n$ if $n\\neq 4k+2$ and $s_g(G)\\leq n+1$ otherwise, except the case of some infinite family of stars."},"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":"1205.2569","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-05-11T16:42:33Z","cross_cats_sorted":[],"title_canon_sha256":"17b5109419b0b0d9da12af55893a3cf58cfeca10a53776a9658266118e8b1086","abstract_canon_sha256":"1b3f05f86c5a8cc4f8a508db57afb1f1dbccf6de41038b1af15f2d62bde88096"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:56:00.193244Z","signature_b64":"6PQhnHjrweXgyE97kkJyQwY0SILk/Vq4rp4SOv2bljlt429eRGRXaeU7bsZDwoagt45XkbR6LngRz0j+zCaUCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"94219a95614005c80e57bd7f4af1af993c85d62c259a1404c68fa1e5efdbaa62","last_reissued_at":"2026-05-18T01:56:00.192781Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:56:00.192781Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Group Irregularity Strength of Connected Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Marcin Anholcer, Martin Milanic, Sylwia Cichacz","submitted_at":"2012-05-11T16:42:33Z","abstract_excerpt":"We investigate the group irregularity strength ($s_g(G)$) of graphs, i.e. the smallest value of $s$ such that taking any Abelian group $\\gr$ of order $s$, there exists a function $f:E(G)\\rightarrow \\gr$ such that the sums of edge labels at every vertex are distinct. 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