{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:JJLEU6XO5TU2I2NCW4YGJ3EQSI","short_pith_number":"pith:JJLEU6XO","canonical_record":{"source":{"id":"2606.06945","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-05T06:17:01Z","cross_cats_sorted":[],"title_canon_sha256":"089ae0ac82462fdffd26d90a0c369716186de8f0b3ca0f7e6accb8a73a61200e","abstract_canon_sha256":"562e43b0449d96142767a2206af0483416585911765aa40a74011c38bd176bac"},"schema_version":"1.0"},"canonical_sha256":"4a564a7aeeece9a469a2b73064ec90921a34cdb8aadf3822e6a1113ea2e2486f","source":{"kind":"arxiv","id":"2606.06945","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.06945","created_at":"2026-06-08T01:04:37Z"},{"alias_kind":"arxiv_version","alias_value":"2606.06945v1","created_at":"2026-06-08T01:04:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.06945","created_at":"2026-06-08T01:04:37Z"},{"alias_kind":"pith_short_12","alias_value":"JJLEU6XO5TU2","created_at":"2026-06-08T01:04:37Z"},{"alias_kind":"pith_short_16","alias_value":"JJLEU6XO5TU2I2NC","created_at":"2026-06-08T01:04:37Z"},{"alias_kind":"pith_short_8","alias_value":"JJLEU6XO","created_at":"2026-06-08T01:04:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:JJLEU6XO5TU2I2NCW4YGJ3EQSI","target":"record","payload":{"canonical_record":{"source":{"id":"2606.06945","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-05T06:17:01Z","cross_cats_sorted":[],"title_canon_sha256":"089ae0ac82462fdffd26d90a0c369716186de8f0b3ca0f7e6accb8a73a61200e","abstract_canon_sha256":"562e43b0449d96142767a2206af0483416585911765aa40a74011c38bd176bac"},"schema_version":"1.0"},"canonical_sha256":"4a564a7aeeece9a469a2b73064ec90921a34cdb8aadf3822e6a1113ea2e2486f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-08T01:04:37.332143Z","signature_b64":"RjL5rwvWVN/zaRM/QKET+1OAiuZXwOdkTEgGnt8ROxGkVZx+wXqnC5FpHczfKE+SonvPOarRWA/v+a0ZHociCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4a564a7aeeece9a469a2b73064ec90921a34cdb8aadf3822e6a1113ea2e2486f","last_reissued_at":"2026-06-08T01:04:37.331233Z","signature_status":"signed_v1","first_computed_at":"2026-06-08T01:04:37.331233Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.06945","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":"2026-06-08T01:04:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LoVUgDCUeUJefw+bsYvbt+D3d6oc4iuch9Ipmk+3ka+Age3l+YC6qSkZj2v91OwDFOy1u2ZQ8iBnQKl8+azJDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T08:14:23.169337Z"},"content_sha256":"32222b7646b1e40cf94f4d05af01c5ddcd061a476593c81203ef22853e892155","schema_version":"1.0","event_id":"sha256:32222b7646b1e40cf94f4d05af01c5ddcd061a476593c81203ef22853e892155"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:JJLEU6XO5TU2I2NCW4YGJ3EQSI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On a distance Laplacian analog of Brouwer's conjecture for several classes of graphs","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Silin Huang","submitted_at":"2026-06-05T06:17:01Z","abstract_excerpt":"Zhou et al. (2025) proposed a distance Laplacian analog of Brouwer's conjecture on partial sums of Laplacian eigenvalues, asserting that for any connected graph $G$, $\\sum_{i=1}^r \\partial_i^L(G)\\le W(G)+\\binom{r+2}{3},$ where $\\partial_i^L(G)$ are the eigenvalues of the distance Laplacian matrix and $W(G)$ is the Wiener index. We prove this inequality for three broad classes of graphs, thereby improving and extending existing results. First, we prove that all connected graphs of diameter at most $D$ satisfy the inequality once the order $n$ satisfies $n\\ge\\lceil\\frac49(D+1)^3\\rceil$. Second, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.06945","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/2606.06945/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":"2026-06-08T01:04:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3yjhxV5PtNLvbZ/tGLauvG5p8Y6hj6Z4qicFXXIS+ExvgpKO37+LOytBKEBUfb1awR450FC3Cb1rQxhnkdiCAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T08:14:23.169716Z"},"content_sha256":"14a54af73d95bd4938a6bb636923d6fa52e6d13fdee3b087a7336b937babaebf","schema_version":"1.0","event_id":"sha256:14a54af73d95bd4938a6bb636923d6fa52e6d13fdee3b087a7336b937babaebf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JJLEU6XO5TU2I2NCW4YGJ3EQSI/bundle.json","state_url":"https://pith.science/pith/JJLEU6XO5TU2I2NCW4YGJ3EQSI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JJLEU6XO5TU2I2NCW4YGJ3EQSI/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":"2026-06-20T08:14:23Z","links":{"resolver":"https://pith.science/pith/JJLEU6XO5TU2I2NCW4YGJ3EQSI","bundle":"https://pith.science/pith/JJLEU6XO5TU2I2NCW4YGJ3EQSI/bundle.json","state":"https://pith.science/pith/JJLEU6XO5TU2I2NCW4YGJ3EQSI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JJLEU6XO5TU2I2NCW4YGJ3EQSI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:JJLEU6XO5TU2I2NCW4YGJ3EQSI","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":"562e43b0449d96142767a2206af0483416585911765aa40a74011c38bd176bac","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-05T06:17:01Z","title_canon_sha256":"089ae0ac82462fdffd26d90a0c369716186de8f0b3ca0f7e6accb8a73a61200e"},"schema_version":"1.0","source":{"id":"2606.06945","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.06945","created_at":"2026-06-08T01:04:37Z"},{"alias_kind":"arxiv_version","alias_value":"2606.06945v1","created_at":"2026-06-08T01:04:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.06945","created_at":"2026-06-08T01:04:37Z"},{"alias_kind":"pith_short_12","alias_value":"JJLEU6XO5TU2","created_at":"2026-06-08T01:04:37Z"},{"alias_kind":"pith_short_16","alias_value":"JJLEU6XO5TU2I2NC","created_at":"2026-06-08T01:04:37Z"},{"alias_kind":"pith_short_8","alias_value":"JJLEU6XO","created_at":"2026-06-08T01:04:37Z"}],"graph_snapshots":[{"event_id":"sha256:14a54af73d95bd4938a6bb636923d6fa52e6d13fdee3b087a7336b937babaebf","target":"graph","created_at":"2026-06-08T01:04:37Z","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/2606.06945/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Zhou et al. (2025) proposed a distance Laplacian analog of Brouwer's conjecture on partial sums of Laplacian eigenvalues, asserting that for any connected graph $G$, $\\sum_{i=1}^r \\partial_i^L(G)\\le W(G)+\\binom{r+2}{3},$ where $\\partial_i^L(G)$ are the eigenvalues of the distance Laplacian matrix and $W(G)$ is the Wiener index. We prove this inequality for three broad classes of graphs, thereby improving and extending existing results. First, we prove that all connected graphs of diameter at most $D$ satisfy the inequality once the order $n$ satisfies $n\\ge\\lceil\\frac49(D+1)^3\\rceil$. Second, ","authors_text":"Silin Huang","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-05T06:17:01Z","title":"On a distance Laplacian analog of Brouwer's conjecture for several classes of graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.06945","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":"sha256:32222b7646b1e40cf94f4d05af01c5ddcd061a476593c81203ef22853e892155","target":"record","created_at":"2026-06-08T01:04:37Z","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":"562e43b0449d96142767a2206af0483416585911765aa40a74011c38bd176bac","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-05T06:17:01Z","title_canon_sha256":"089ae0ac82462fdffd26d90a0c369716186de8f0b3ca0f7e6accb8a73a61200e"},"schema_version":"1.0","source":{"id":"2606.06945","kind":"arxiv","version":1}},"canonical_sha256":"4a564a7aeeece9a469a2b73064ec90921a34cdb8aadf3822e6a1113ea2e2486f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4a564a7aeeece9a469a2b73064ec90921a34cdb8aadf3822e6a1113ea2e2486f","first_computed_at":"2026-06-08T01:04:37.331233Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-08T01:04:37.331233Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RjL5rwvWVN/zaRM/QKET+1OAiuZXwOdkTEgGnt8ROxGkVZx+wXqnC5FpHczfKE+SonvPOarRWA/v+a0ZHociCQ==","signature_status":"signed_v1","signed_at":"2026-06-08T01:04:37.332143Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.06945","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:32222b7646b1e40cf94f4d05af01c5ddcd061a476593c81203ef22853e892155","sha256:14a54af73d95bd4938a6bb636923d6fa52e6d13fdee3b087a7336b937babaebf"],"state_sha256":"136b206c1955681a9e9c8b8b23cc649aa8a9b1fe08398455ed20f39a5b00d3ca"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"E/0mwT5WQ+XF3JgQbt4rZry787ypX4fw5+YUmwnk5GNYghoZDnHLOiKYcmmwl7LqfkjyHdwe/jQI96TnKt5uDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T08:14:23.171715Z","bundle_sha256":"177da0f78bbbb40d7bd50e33ad6976684de2ca4c7742350364f054e713f3135e"}}