{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:QZNAAM6W5QNW3SWB36Z46WTGZ7","short_pith_number":"pith:QZNAAM6W","canonical_record":{"source":{"id":"1905.01921","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2019-05-06T10:51:34Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"ebbeba0cc3ee79456681060fdb265c10673bab0b1cf14b58008b3308e1810a44","abstract_canon_sha256":"d1e694838437448c9dc4695697b4d0c118ce64ae2e2f18d12aaebc11eadce8e1"},"schema_version":"1.0"},"canonical_sha256":"865a0033d6ec1b6dcac1dfb3cf5a66cfeed53a9223b178191c3111b3ea530391","source":{"kind":"arxiv","id":"1905.01921","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1905.01921","created_at":"2026-05-17T23:46:57Z"},{"alias_kind":"arxiv_version","alias_value":"1905.01921v1","created_at":"2026-05-17T23:46:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1905.01921","created_at":"2026-05-17T23:46:57Z"},{"alias_kind":"pith_short_12","alias_value":"QZNAAM6W5QNW","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_16","alias_value":"QZNAAM6W5QNW3SWB","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_8","alias_value":"QZNAAM6W","created_at":"2026-05-18T12:33:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:QZNAAM6W5QNW3SWB36Z46WTGZ7","target":"record","payload":{"canonical_record":{"source":{"id":"1905.01921","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2019-05-06T10:51:34Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"ebbeba0cc3ee79456681060fdb265c10673bab0b1cf14b58008b3308e1810a44","abstract_canon_sha256":"d1e694838437448c9dc4695697b4d0c118ce64ae2e2f18d12aaebc11eadce8e1"},"schema_version":"1.0"},"canonical_sha256":"865a0033d6ec1b6dcac1dfb3cf5a66cfeed53a9223b178191c3111b3ea530391","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:46:57.018028Z","signature_b64":"lys4/F/EZU2DcN0Na9bANId47zzQHleiM7XuptrKcxJM8LL6DYdG5hAJ2Sd9QmlxJd0MngAxqDqKHOu3R8BgDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"865a0033d6ec1b6dcac1dfb3cf5a66cfeed53a9223b178191c3111b3ea530391","last_reissued_at":"2026-05-17T23:46:57.017434Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:46:57.017434Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1905.01921","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":"2026-05-17T23:46:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IcaoE1Gb4T2jzWr3899UHP2BFYV2Kr9hB+CWS6mMDWlfJdXOco4+1aZUKLNP94ZZKzlDavZoOePM3VOCbfJGCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T14:55:36.675900Z"},"content_sha256":"a8429844e99f7cf900b1a9b546d6be95fe9535bd4438a51ed1bcd8e583b88de6","schema_version":"1.0","event_id":"sha256:a8429844e99f7cf900b1a9b546d6be95fe9535bd4438a51ed1bcd8e583b88de6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:QZNAAM6W5QNW3SWB36Z46WTGZ7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Nonsingular (Vertex-Weighted) Block Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Abraham Berman, Cheng Zheng, Naomi Shaked-Monderer, Ranveer Singh","submitted_at":"2019-05-06T10:51:34Z","abstract_excerpt":"A graph $G$ is \\emph{nonsingular (singular)} if its adjacency matrix $A(G)$ is nonsingular (singular). In this article, we consider the nonsingularity of block graphs, i.e., graphs in which every block is a clique. Extending the problem, we characterize nonsingular vertex-weighted block graphs in terms of reduced vertex-weighted graphs resulting after successive deletion and contraction of pendant blocks. Special cases where nonsingularity of block graphs may be directly determined are discussed."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.01921","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":""},"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":"2026-05-17T23:46:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"F2f4DYN8tYl1w2/2iiUF1W0T/7oUc1QG0Xh4yuzP+cbqfnUC4ZDBhYAqaHxTMpiDQIAtKOT9l0BUt31gmssqDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T14:55:36.676247Z"},"content_sha256":"4a3d25dfd269b8355bc4826569bbe1c2607cf9bda6ceb89591d7b8127415ffd4","schema_version":"1.0","event_id":"sha256:4a3d25dfd269b8355bc4826569bbe1c2607cf9bda6ceb89591d7b8127415ffd4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QZNAAM6W5QNW3SWB36Z46WTGZ7/bundle.json","state_url":"https://pith.science/pith/QZNAAM6W5QNW3SWB36Z46WTGZ7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QZNAAM6W5QNW3SWB36Z46WTGZ7/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":"2026-06-23T14:55:36Z","links":{"resolver":"https://pith.science/pith/QZNAAM6W5QNW3SWB36Z46WTGZ7","bundle":"https://pith.science/pith/QZNAAM6W5QNW3SWB36Z46WTGZ7/bundle.json","state":"https://pith.science/pith/QZNAAM6W5QNW3SWB36Z46WTGZ7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QZNAAM6W5QNW3SWB36Z46WTGZ7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:QZNAAM6W5QNW3SWB36Z46WTGZ7","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":"d1e694838437448c9dc4695697b4d0c118ce64ae2e2f18d12aaebc11eadce8e1","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2019-05-06T10:51:34Z","title_canon_sha256":"ebbeba0cc3ee79456681060fdb265c10673bab0b1cf14b58008b3308e1810a44"},"schema_version":"1.0","source":{"id":"1905.01921","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1905.01921","created_at":"2026-05-17T23:46:57Z"},{"alias_kind":"arxiv_version","alias_value":"1905.01921v1","created_at":"2026-05-17T23:46:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1905.01921","created_at":"2026-05-17T23:46:57Z"},{"alias_kind":"pith_short_12","alias_value":"QZNAAM6W5QNW","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_16","alias_value":"QZNAAM6W5QNW3SWB","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_8","alias_value":"QZNAAM6W","created_at":"2026-05-18T12:33:27Z"}],"graph_snapshots":[{"event_id":"sha256:4a3d25dfd269b8355bc4826569bbe1c2607cf9bda6ceb89591d7b8127415ffd4","target":"graph","created_at":"2026-05-17T23:46:57Z","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":"A graph $G$ is \\emph{nonsingular (singular)} if its adjacency matrix $A(G)$ is nonsingular (singular). In this article, we consider the nonsingularity of block graphs, i.e., graphs in which every block is a clique. Extending the problem, we characterize nonsingular vertex-weighted block graphs in terms of reduced vertex-weighted graphs resulting after successive deletion and contraction of pendant blocks. Special cases where nonsingularity of block graphs may be directly determined are discussed.","authors_text":"Abraham Berman, Cheng Zheng, Naomi Shaked-Monderer, Ranveer Singh","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2019-05-06T10:51:34Z","title":"Nonsingular (Vertex-Weighted) Block Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.01921","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:a8429844e99f7cf900b1a9b546d6be95fe9535bd4438a51ed1bcd8e583b88de6","target":"record","created_at":"2026-05-17T23:46:57Z","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":"d1e694838437448c9dc4695697b4d0c118ce64ae2e2f18d12aaebc11eadce8e1","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2019-05-06T10:51:34Z","title_canon_sha256":"ebbeba0cc3ee79456681060fdb265c10673bab0b1cf14b58008b3308e1810a44"},"schema_version":"1.0","source":{"id":"1905.01921","kind":"arxiv","version":1}},"canonical_sha256":"865a0033d6ec1b6dcac1dfb3cf5a66cfeed53a9223b178191c3111b3ea530391","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"865a0033d6ec1b6dcac1dfb3cf5a66cfeed53a9223b178191c3111b3ea530391","first_computed_at":"2026-05-17T23:46:57.017434Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:46:57.017434Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lys4/F/EZU2DcN0Na9bANId47zzQHleiM7XuptrKcxJM8LL6DYdG5hAJ2Sd9QmlxJd0MngAxqDqKHOu3R8BgDg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:46:57.018028Z","signed_message":"canonical_sha256_bytes"},"source_id":"1905.01921","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a8429844e99f7cf900b1a9b546d6be95fe9535bd4438a51ed1bcd8e583b88de6","sha256:4a3d25dfd269b8355bc4826569bbe1c2607cf9bda6ceb89591d7b8127415ffd4"],"state_sha256":"232615344fdd601a4fb877b3b18136010058bb30a0edd9096cd49f60dacc8046"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GGXf1xxPZyLvkOoMqiL0QiSVL2JhPTtYbiQ/qn/UjGXWZzh3ZgIby4EvPK0t1wJXgL440BmIR70AgW5Q8ZMGAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T14:55:36.678136Z","bundle_sha256":"e186c352c5f15ed26809aa4871fc1a2bdfc3dff969f9edd62a93c7cab894d085"}}