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We describe the arithmetical structures on graph $G$ with a cut vertex $v$ in terms of the arithmetical structures on their blocks. More precisely, if $G_1,\\ldots,G_s$ are the induced subgraphs of $G$ obtained from each of the connected components of $G-v$ by adding the vertex $v$ and their incident edges, then the arithmetical structures on $G$"},"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":"1606.03726","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-06-12T15:08:26Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"e0305c8545d063e8ab61eeee0f881a2941f5c5d892b570b7623ae7e982474297","abstract_canon_sha256":"ec1775dea2f26a3616dbaab0198baab91853f6c777abdd017ce7ba1f05c59e93"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:32.790817Z","signature_b64":"fwlazDz3TxViSRBK7cTMQ1dr9SO12dHT53UsvE4GgJulribrgC91nXQ3cr6u3WotPwTdNSLqX0zHTP82kHdzBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0263574eebe2e02941314dd1d35ed23f1689f7145f72c4ea60b892ad33daa16b","last_reissued_at":"2026-05-18T00:42:32.790172Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:32.790172Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Arithmetical structures on graphs with connectivity one","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Carlos E. Valencia, Hugo Corrales","submitted_at":"2016-06-12T15:08:26Z","abstract_excerpt":"Given a graph $G$, an arithmetical structure on $G$ is a pair of positive integer vectors $({\\bf d},{\\bf r})$ such that $\\mathrm{gcd}({\\bf r}_v\\, | \\,v\\in V(G))=1$ and \\[ (\\mathrm{diag}({\\bf d})-A){\\bf r}=0, \\] where $A$ is the adjacency matrix of $G$. We describe the arithmetical structures on graph $G$ with a cut vertex $v$ in terms of the arithmetical structures on their blocks. More precisely, if $G_1,\\ldots,G_s$ are the induced subgraphs of $G$ obtained from each of the connected components of $G-v$ by adding the vertex $v$ and their incident edges, then the arithmetical structures on $G$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.03726","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1606.03726","created_at":"2026-05-18T00:42:32.790264+00:00"},{"alias_kind":"arxiv_version","alias_value":"1606.03726v3","created_at":"2026-05-18T00:42:32.790264+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.03726","created_at":"2026-05-18T00:42:32.790264+00:00"},{"alias_kind":"pith_short_12","alias_value":"AJRVOTXL4LQC","created_at":"2026-05-18T12:30:07.202191+00:00"},{"alias_kind":"pith_short_16","alias_value":"AJRVOTXL4LQCSQJR","created_at":"2026-05-18T12:30:07.202191+00:00"},{"alias_kind":"pith_short_8","alias_value":"AJRVOTXL","created_at":"2026-05-18T12:30:07.202191+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/AJRVOTXL4LQCSQJRJXI5GXWSH4","json":"https://pith.science/pith/AJRVOTXL4LQCSQJRJXI5GXWSH4.json","graph_json":"https://pith.science/api/pith-number/AJRVOTXL4LQCSQJRJXI5GXWSH4/graph.json","events_json":"https://pith.science/api/pith-number/AJRVOTXL4LQCSQJRJXI5GXWSH4/events.json","paper":"https://pith.science/paper/AJRVOTXL"},"agent_actions":{"view_html":"https://pith.science/pith/AJRVOTXL4LQCSQJRJXI5GXWSH4","download_json":"https://pith.science/pith/AJRVOTXL4LQCSQJRJXI5GXWSH4.json","view_paper":"https://pith.science/paper/AJRVOTXL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1606.03726&json=true","fetch_graph":"https://pith.science/api/pith-number/AJRVOTXL4LQCSQJRJXI5GXWSH4/graph.json","fetch_events":"https://pith.science/api/pith-number/AJRVOTXL4LQCSQJRJXI5GXWSH4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AJRVOTXL4LQCSQJRJXI5GXWSH4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AJRVOTXL4LQCSQJRJXI5GXWSH4/action/storage_attestation","attest_author":"https://pith.science/pith/AJRVOTXL4LQCSQJRJXI5GXWSH4/action/author_attestation","sign_citation":"https://pith.science/pith/AJRVOTXL4LQCSQJRJXI5GXWSH4/action/citation_signature","submit_replication":"https://pith.science/pith/AJRVOTXL4LQCSQJRJXI5GXWSH4/action/replication_record"}},"created_at":"2026-05-18T00:42:32.790264+00:00","updated_at":"2026-05-18T00:42:32.790264+00:00"}