{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:63EOZ3EQ3O6LOXP6KMB3MVGKFJ","short_pith_number":"pith:63EOZ3EQ","canonical_record":{"source":{"id":"2605.23737","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-22T15:13:35Z","cross_cats_sorted":[],"title_canon_sha256":"6951e8c3650cc8298c25c2c10299ebec9e2e9a9d4d7156217543e4782ef8ecd5","abstract_canon_sha256":"ef9863dad6c1b64f0e682cab296ddb3d10487ed2328dcf680ebd3adc9266ee8b"},"schema_version":"1.0"},"canonical_sha256":"f6c8ecec90dbbcb75dfe5303b654ca2a650064e218164ccccdb6c4ffca492c78","source":{"kind":"arxiv","id":"2605.23737","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.23737","created_at":"2026-05-25T02:02:29Z"},{"alias_kind":"arxiv_version","alias_value":"2605.23737v1","created_at":"2026-05-25T02:02:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.23737","created_at":"2026-05-25T02:02:29Z"},{"alias_kind":"pith_short_12","alias_value":"63EOZ3EQ3O6L","created_at":"2026-05-25T02:02:29Z"},{"alias_kind":"pith_short_16","alias_value":"63EOZ3EQ3O6LOXP6","created_at":"2026-05-25T02:02:29Z"},{"alias_kind":"pith_short_8","alias_value":"63EOZ3EQ","created_at":"2026-05-25T02:02:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:63EOZ3EQ3O6LOXP6KMB3MVGKFJ","target":"record","payload":{"canonical_record":{"source":{"id":"2605.23737","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-22T15:13:35Z","cross_cats_sorted":[],"title_canon_sha256":"6951e8c3650cc8298c25c2c10299ebec9e2e9a9d4d7156217543e4782ef8ecd5","abstract_canon_sha256":"ef9863dad6c1b64f0e682cab296ddb3d10487ed2328dcf680ebd3adc9266ee8b"},"schema_version":"1.0"},"canonical_sha256":"f6c8ecec90dbbcb75dfe5303b654ca2a650064e218164ccccdb6c4ffca492c78","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-25T02:02:29.257046Z","signature_b64":"7ImXzaMreTSkSU1bl4s47H7D2mt2I5/dS6U7Mwqdbuei9Q5aRu6g4xigQZxYu3rjD9vG7Bxhp0DlIgMxU0sxBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f6c8ecec90dbbcb75dfe5303b654ca2a650064e218164ccccdb6c4ffca492c78","last_reissued_at":"2026-05-25T02:02:29.256268Z","signature_status":"signed_v1","first_computed_at":"2026-05-25T02:02:29.256268Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.23737","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-25T02:02:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FLvbewuQzGG+yAxDhX7jjNeSRx/P0hlX4XA687Pjf9F/bGXtAurgYCdXDZZTMpFC/6tGkxAnXUPwSTHolF/kDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T17:30:47.212464Z"},"content_sha256":"6e4e524f0b72f7c794a5299c403e932c02621bc14f7475da4da1219640c458e6","schema_version":"1.0","event_id":"sha256:6e4e524f0b72f7c794a5299c403e932c02621bc14f7475da4da1219640c458e6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:63EOZ3EQ3O6LOXP6KMB3MVGKFJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Spectral radius and edge-disjoint connected factors of graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Wenqian Zhang, Xinying Tang","submitted_at":"2026-05-22T15:13:35Z","abstract_excerpt":"For a graph $G$, the spectral radius of $G$ is the largest eigenvalue of its adjacency matrix. A connected factor of $G$ is a connected spanning subgraph of $G$. For example, a spanning tree of $G$ is a 1-connected factor of $G$. Let $G$ be a graph of order $n$ with minimum degree $\\delta\\geq6$, where $n\\geq3\\delta$. In this paper, we give a sharp spectral radius condition for $G$ to contain $k$ edge-disjoint 2-connected factors and $\\left\\lfloor\\frac{\\delta-4k}{2}\\right\\rfloor$ edge-disjoint spanning trees, where $1\\leq k\\leq\\left\\lfloor\\frac{\\delta}{4}\\right\\rfloor$ is an integer."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.23737","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.23737/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-25T02:02:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1qlPZ0NwR3IrPzPYk1J299TnXBQWXjAzNoU97//DsDw3w22gcCHE4ut8ca/WfeejA4VhcLi2fDdRMNkFEw2sCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T17:30:47.212834Z"},"content_sha256":"79772f6f55d10a34cb0c739548fffc7bf20c1cbcb3727cbf30cd26221c158d29","schema_version":"1.0","event_id":"sha256:79772f6f55d10a34cb0c739548fffc7bf20c1cbcb3727cbf30cd26221c158d29"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/63EOZ3EQ3O6LOXP6KMB3MVGKFJ/bundle.json","state_url":"https://pith.science/pith/63EOZ3EQ3O6LOXP6KMB3MVGKFJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/63EOZ3EQ3O6LOXP6KMB3MVGKFJ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-01T17:30:47Z","links":{"resolver":"https://pith.science/pith/63EOZ3EQ3O6LOXP6KMB3MVGKFJ","bundle":"https://pith.science/pith/63EOZ3EQ3O6LOXP6KMB3MVGKFJ/bundle.json","state":"https://pith.science/pith/63EOZ3EQ3O6LOXP6KMB3MVGKFJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/63EOZ3EQ3O6LOXP6KMB3MVGKFJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:63EOZ3EQ3O6LOXP6KMB3MVGKFJ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"ef9863dad6c1b64f0e682cab296ddb3d10487ed2328dcf680ebd3adc9266ee8b","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-22T15:13:35Z","title_canon_sha256":"6951e8c3650cc8298c25c2c10299ebec9e2e9a9d4d7156217543e4782ef8ecd5"},"schema_version":"1.0","source":{"id":"2605.23737","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.23737","created_at":"2026-05-25T02:02:29Z"},{"alias_kind":"arxiv_version","alias_value":"2605.23737v1","created_at":"2026-05-25T02:02:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.23737","created_at":"2026-05-25T02:02:29Z"},{"alias_kind":"pith_short_12","alias_value":"63EOZ3EQ3O6L","created_at":"2026-05-25T02:02:29Z"},{"alias_kind":"pith_short_16","alias_value":"63EOZ3EQ3O6LOXP6","created_at":"2026-05-25T02:02:29Z"},{"alias_kind":"pith_short_8","alias_value":"63EOZ3EQ","created_at":"2026-05-25T02:02:29Z"}],"graph_snapshots":[{"event_id":"sha256:79772f6f55d10a34cb0c739548fffc7bf20c1cbcb3727cbf30cd26221c158d29","target":"graph","created_at":"2026-05-25T02:02:29Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2605.23737/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For a graph $G$, the spectral radius of $G$ is the largest eigenvalue of its adjacency matrix. A connected factor of $G$ is a connected spanning subgraph of $G$. For example, a spanning tree of $G$ is a 1-connected factor of $G$. Let $G$ be a graph of order $n$ with minimum degree $\\delta\\geq6$, where $n\\geq3\\delta$. In this paper, we give a sharp spectral radius condition for $G$ to contain $k$ edge-disjoint 2-connected factors and $\\left\\lfloor\\frac{\\delta-4k}{2}\\right\\rfloor$ edge-disjoint spanning trees, where $1\\leq k\\leq\\left\\lfloor\\frac{\\delta}{4}\\right\\rfloor$ is an integer.","authors_text":"Wenqian Zhang, Xinying Tang","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-22T15:13:35Z","title":"Spectral radius and edge-disjoint connected factors of graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.23737","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:6e4e524f0b72f7c794a5299c403e932c02621bc14f7475da4da1219640c458e6","target":"record","created_at":"2026-05-25T02:02:29Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"ef9863dad6c1b64f0e682cab296ddb3d10487ed2328dcf680ebd3adc9266ee8b","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-22T15:13:35Z","title_canon_sha256":"6951e8c3650cc8298c25c2c10299ebec9e2e9a9d4d7156217543e4782ef8ecd5"},"schema_version":"1.0","source":{"id":"2605.23737","kind":"arxiv","version":1}},"canonical_sha256":"f6c8ecec90dbbcb75dfe5303b654ca2a650064e218164ccccdb6c4ffca492c78","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f6c8ecec90dbbcb75dfe5303b654ca2a650064e218164ccccdb6c4ffca492c78","first_computed_at":"2026-05-25T02:02:29.256268Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-25T02:02:29.256268Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7ImXzaMreTSkSU1bl4s47H7D2mt2I5/dS6U7Mwqdbuei9Q5aRu6g4xigQZxYu3rjD9vG7Bxhp0DlIgMxU0sxBg==","signature_status":"signed_v1","signed_at":"2026-05-25T02:02:29.257046Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.23737","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6e4e524f0b72f7c794a5299c403e932c02621bc14f7475da4da1219640c458e6","sha256:79772f6f55d10a34cb0c739548fffc7bf20c1cbcb3727cbf30cd26221c158d29"],"state_sha256":"88ee81df8f2966a8c7f75a366b3d35be92af3c16248dc15b98b3a3a3eb97472a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bQZ60I1ObZ0KRI/7uTXIeC65hk1A9R+YNupUc19F9EqJGAUaEL+o5r2lxhrlWOmgWpIL3ihxeZw4SPUYWlRJCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T17:30:47.214829Z","bundle_sha256":"dd5931caa62c8792d9ce6be4ac4c69e14de8baa8542d08803db952aa902b4a0f"}}