{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:M5CVK62YHD6X4OEO2LAR43DPW6","short_pith_number":"pith:M5CVK62Y","canonical_record":{"source":{"id":"1201.3221","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-01-16T11:36:47Z","cross_cats_sorted":[],"title_canon_sha256":"472a00bde673a211be3b39d865d0df244ffcd1cafcb86220ccc76d9e65bd4428","abstract_canon_sha256":"c80de11950ae477e583e71dbf519231fc717dc8270bf9371d9dc559e6cbec622"},"schema_version":"1.0"},"canonical_sha256":"6745557b5838fd7e388ed2c11e6c6fb78e322085ad085777be790039ec887d21","source":{"kind":"arxiv","id":"1201.3221","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.3221","created_at":"2026-05-18T03:00:50Z"},{"alias_kind":"arxiv_version","alias_value":"1201.3221v4","created_at":"2026-05-18T03:00:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.3221","created_at":"2026-05-18T03:00:50Z"},{"alias_kind":"pith_short_12","alias_value":"M5CVK62YHD6X","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_16","alias_value":"M5CVK62YHD6X4OEO","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_8","alias_value":"M5CVK62Y","created_at":"2026-05-18T12:27:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:M5CVK62YHD6X4OEO2LAR43DPW6","target":"record","payload":{"canonical_record":{"source":{"id":"1201.3221","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-01-16T11:36:47Z","cross_cats_sorted":[],"title_canon_sha256":"472a00bde673a211be3b39d865d0df244ffcd1cafcb86220ccc76d9e65bd4428","abstract_canon_sha256":"c80de11950ae477e583e71dbf519231fc717dc8270bf9371d9dc559e6cbec622"},"schema_version":"1.0"},"canonical_sha256":"6745557b5838fd7e388ed2c11e6c6fb78e322085ad085777be790039ec887d21","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:00:50.019782Z","signature_b64":"zsQYx6Fb0fi46IUsTo3yv3V9ZipeCv1iTUvDES+gtvIHt5P9wtA9fOCm+53MM4z+/0ZrbWgWSPq75AnS86bgDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6745557b5838fd7e388ed2c11e6c6fb78e322085ad085777be790039ec887d21","last_reissued_at":"2026-05-18T03:00:50.019115Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:00:50.019115Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1201.3221","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:00:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"p2oRdPcYnFSKqfJx5xo3HGwa8ww1aSqDW8B/9taVHvJH52feOjiSIL9FLv00VoYBFovspjETmhcwrYrCAFUVCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T15:31:23.539560Z"},"content_sha256":"3e226aeae02456a026d1f6f000f73eb7316ae0adad9f6530ecc5cd4ad819bc20","schema_version":"1.0","event_id":"sha256:3e226aeae02456a026d1f6f000f73eb7316ae0adad9f6530ecc5cd4ad819bc20"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:M5CVK62YHD6X4OEO2LAR43DPW6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Spanning trees and even integer eigenvalues of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ebrahim Ghorbani","submitted_at":"2012-01-16T11:36:47Z","abstract_excerpt":"For a graph $G$, let $L(G)$ and $Q(G)$ be the Laplacian and signless Laplacian matrices of $G$, respectively, and $\\tau(G)$ be the number of spanning trees of $G$. We prove that if $G$ has an odd number of vertices and $\\tau(G)$ is not divisible by $4$, then (i) $L(G)$ has no even integer eigenvalue, (ii) $Q(G)$ has no integer eigenvalue $\\lambda\\equiv2\\pmod4$, and (iii) $Q(G)$ has at most one eigenvalue $\\lambda\\equiv0\\pmod4$ and such an eigenvalue is simple. As a consequence, we extend previous results by Gutman and Sciriha and by Bapat on the nullity of adjacency matrices of the line graphs"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.3221","kind":"arxiv","version":4},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:00:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kQQ+DrKTozK9iqryCajWF5yZk9VVqbamU9Hso4UqljJtF5Zdw90eHKvCn82UD9uawRhaBdkw4YgMDTEXqClXCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T15:31:23.539947Z"},"content_sha256":"373046a54f9d7297c07978765ec3dcc0b60465963351fb797077b4d1bf77706d","schema_version":"1.0","event_id":"sha256:373046a54f9d7297c07978765ec3dcc0b60465963351fb797077b4d1bf77706d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/M5CVK62YHD6X4OEO2LAR43DPW6/bundle.json","state_url":"https://pith.science/pith/M5CVK62YHD6X4OEO2LAR43DPW6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/M5CVK62YHD6X4OEO2LAR43DPW6/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":"2026-06-09T15:31:23Z","links":{"resolver":"https://pith.science/pith/M5CVK62YHD6X4OEO2LAR43DPW6","bundle":"https://pith.science/pith/M5CVK62YHD6X4OEO2LAR43DPW6/bundle.json","state":"https://pith.science/pith/M5CVK62YHD6X4OEO2LAR43DPW6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/M5CVK62YHD6X4OEO2LAR43DPW6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:M5CVK62YHD6X4OEO2LAR43DPW6","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":"c80de11950ae477e583e71dbf519231fc717dc8270bf9371d9dc559e6cbec622","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-01-16T11:36:47Z","title_canon_sha256":"472a00bde673a211be3b39d865d0df244ffcd1cafcb86220ccc76d9e65bd4428"},"schema_version":"1.0","source":{"id":"1201.3221","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.3221","created_at":"2026-05-18T03:00:50Z"},{"alias_kind":"arxiv_version","alias_value":"1201.3221v4","created_at":"2026-05-18T03:00:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.3221","created_at":"2026-05-18T03:00:50Z"},{"alias_kind":"pith_short_12","alias_value":"M5CVK62YHD6X","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_16","alias_value":"M5CVK62YHD6X4OEO","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_8","alias_value":"M5CVK62Y","created_at":"2026-05-18T12:27:14Z"}],"graph_snapshots":[{"event_id":"sha256:373046a54f9d7297c07978765ec3dcc0b60465963351fb797077b4d1bf77706d","target":"graph","created_at":"2026-05-18T03:00:50Z","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"},"paper":{"abstract_excerpt":"For a graph $G$, let $L(G)$ and $Q(G)$ be the Laplacian and signless Laplacian matrices of $G$, respectively, and $\\tau(G)$ be the number of spanning trees of $G$. We prove that if $G$ has an odd number of vertices and $\\tau(G)$ is not divisible by $4$, then (i) $L(G)$ has no even integer eigenvalue, (ii) $Q(G)$ has no integer eigenvalue $\\lambda\\equiv2\\pmod4$, and (iii) $Q(G)$ has at most one eigenvalue $\\lambda\\equiv0\\pmod4$ and such an eigenvalue is simple. As a consequence, we extend previous results by Gutman and Sciriha and by Bapat on the nullity of adjacency matrices of the line graphs","authors_text":"Ebrahim Ghorbani","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-01-16T11:36:47Z","title":"Spanning trees and even integer eigenvalues of graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.3221","kind":"arxiv","version":4},"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":"sha256:3e226aeae02456a026d1f6f000f73eb7316ae0adad9f6530ecc5cd4ad819bc20","target":"record","created_at":"2026-05-18T03:00:50Z","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":"c80de11950ae477e583e71dbf519231fc717dc8270bf9371d9dc559e6cbec622","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-01-16T11:36:47Z","title_canon_sha256":"472a00bde673a211be3b39d865d0df244ffcd1cafcb86220ccc76d9e65bd4428"},"schema_version":"1.0","source":{"id":"1201.3221","kind":"arxiv","version":4}},"canonical_sha256":"6745557b5838fd7e388ed2c11e6c6fb78e322085ad085777be790039ec887d21","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6745557b5838fd7e388ed2c11e6c6fb78e322085ad085777be790039ec887d21","first_computed_at":"2026-05-18T03:00:50.019115Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:00:50.019115Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zsQYx6Fb0fi46IUsTo3yv3V9ZipeCv1iTUvDES+gtvIHt5P9wtA9fOCm+53MM4z+/0ZrbWgWSPq75AnS86bgDw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:00:50.019782Z","signed_message":"canonical_sha256_bytes"},"source_id":"1201.3221","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3e226aeae02456a026d1f6f000f73eb7316ae0adad9f6530ecc5cd4ad819bc20","sha256:373046a54f9d7297c07978765ec3dcc0b60465963351fb797077b4d1bf77706d"],"state_sha256":"5a9f596afe1154272dd75326753d73a20317a3e2185582948c4800c93e026074"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Za2IwfAQkT6UHqLKt3ELIYndCTrfLkTygVv3ypF8YFVkBquZsxXvBVCdF0dxBtu1zElv3lUlYSG7RX8h7bBqBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T15:31:23.541922Z","bundle_sha256":"e9d9afbb8c281a6c142b51bf60b440ecc39697e7695e8376ce59a9c5530db56c"}}