{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:BFEY2E2IIXIKPC5TCPBAIUTWAL","short_pith_number":"pith:BFEY2E2I","canonical_record":{"source":{"id":"1510.05117","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-10-17T11:32:24Z","cross_cats_sorted":[],"title_canon_sha256":"705dda62bfc28e9f0a0db87d8f8ee5fcf09093c6e31fd81e729f64b758dedffe","abstract_canon_sha256":"7124bed7777e610947fa3bacc48d87ec881e4711315fa92195a5a6acc81a6a04"},"schema_version":"1.0"},"canonical_sha256":"09498d134845d0a78bb313c204527602c3e07579a64e1dd36ebe4361a5087b0c","source":{"kind":"arxiv","id":"1510.05117","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.05117","created_at":"2026-05-18T01:17:32Z"},{"alias_kind":"arxiv_version","alias_value":"1510.05117v3","created_at":"2026-05-18T01:17:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.05117","created_at":"2026-05-18T01:17:32Z"},{"alias_kind":"pith_short_12","alias_value":"BFEY2E2IIXIK","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_16","alias_value":"BFEY2E2IIXIKPC5T","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_8","alias_value":"BFEY2E2I","created_at":"2026-05-18T12:29:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:BFEY2E2IIXIKPC5TCPBAIUTWAL","target":"record","payload":{"canonical_record":{"source":{"id":"1510.05117","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-10-17T11:32:24Z","cross_cats_sorted":[],"title_canon_sha256":"705dda62bfc28e9f0a0db87d8f8ee5fcf09093c6e31fd81e729f64b758dedffe","abstract_canon_sha256":"7124bed7777e610947fa3bacc48d87ec881e4711315fa92195a5a6acc81a6a04"},"schema_version":"1.0"},"canonical_sha256":"09498d134845d0a78bb313c204527602c3e07579a64e1dd36ebe4361a5087b0c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:17:32.830238Z","signature_b64":"lTOATY9tF+Sd32o0G1UoIsemd/hi/kpOqMhxyMc9XfItwfiAuxTw3y7w1fgoWr76mHAKJDmWG1O/lJyJ2zUAAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"09498d134845d0a78bb313c204527602c3e07579a64e1dd36ebe4361a5087b0c","last_reissued_at":"2026-05-18T01:17:32.829626Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:17:32.829626Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1510.05117","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:17:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FKBmystUmxx3h1FYj7v5wDanUdpOtZm+QyoxuullVuTTIksBstYheRyKh8k/0E7mqYWcSCsNOWbXBW4/5NyBDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T15:27:41.471577Z"},"content_sha256":"591e213e86f424961a3f472e771e483e7f81dc1494771781efa6775d01e3d87c","schema_version":"1.0","event_id":"sha256:591e213e86f424961a3f472e771e483e7f81dc1494771781efa6775d01e3d87c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:BFEY2E2IIXIKPC5TCPBAIUTWAL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Proof of a conjecture on `plateaux' phenomenon of graph Laplacian eigenvalues","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ebrahim Ghorbani","submitted_at":"2015-10-17T11:32:24Z","abstract_excerpt":"Let $G$ be a simple graph. A pendant path of $G$ is a path such that one of its end vertices has degree $1$, the other end has degree $\\ge3$, and all the internal vertices have degree $2$. Let $p_k(G)$ be the number of pendant paths of length $k$ of $G$, and $q_k(G)$ be the number of vertices with degree $\\ge3$ which are an end vertex of some pendant paths of length $k$. Motivated by the problem of characterizing dendritic trees, N. Saito and E. Woei conjectured that any graph $G$ has some Laplacian eigenvalue with multiplicity at least $p_k(G)-q_k(G)$. We prove a more general result for both "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.05117","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:17:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eOKugISrxXsrYz/0edtzGhfYuEHZh/dsaCA8iG90MT/X/siqXe1aCG8MIJoiojCwieFk02BLgtIa0HeJOSIGCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T15:27:41.472022Z"},"content_sha256":"ac47515a8c5759308834324e116f423fd6d5d973c80393f781d25ade3bed0125","schema_version":"1.0","event_id":"sha256:ac47515a8c5759308834324e116f423fd6d5d973c80393f781d25ade3bed0125"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BFEY2E2IIXIKPC5TCPBAIUTWAL/bundle.json","state_url":"https://pith.science/pith/BFEY2E2IIXIKPC5TCPBAIUTWAL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BFEY2E2IIXIKPC5TCPBAIUTWAL/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":"2026-06-09T15:27:41Z","links":{"resolver":"https://pith.science/pith/BFEY2E2IIXIKPC5TCPBAIUTWAL","bundle":"https://pith.science/pith/BFEY2E2IIXIKPC5TCPBAIUTWAL/bundle.json","state":"https://pith.science/pith/BFEY2E2IIXIKPC5TCPBAIUTWAL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BFEY2E2IIXIKPC5TCPBAIUTWAL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:BFEY2E2IIXIKPC5TCPBAIUTWAL","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":"7124bed7777e610947fa3bacc48d87ec881e4711315fa92195a5a6acc81a6a04","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-10-17T11:32:24Z","title_canon_sha256":"705dda62bfc28e9f0a0db87d8f8ee5fcf09093c6e31fd81e729f64b758dedffe"},"schema_version":"1.0","source":{"id":"1510.05117","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.05117","created_at":"2026-05-18T01:17:32Z"},{"alias_kind":"arxiv_version","alias_value":"1510.05117v3","created_at":"2026-05-18T01:17:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.05117","created_at":"2026-05-18T01:17:32Z"},{"alias_kind":"pith_short_12","alias_value":"BFEY2E2IIXIK","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_16","alias_value":"BFEY2E2IIXIKPC5T","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_8","alias_value":"BFEY2E2I","created_at":"2026-05-18T12:29:14Z"}],"graph_snapshots":[{"event_id":"sha256:ac47515a8c5759308834324e116f423fd6d5d973c80393f781d25ade3bed0125","target":"graph","created_at":"2026-05-18T01:17:32Z","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":"Let $G$ be a simple graph. A pendant path of $G$ is a path such that one of its end vertices has degree $1$, the other end has degree $\\ge3$, and all the internal vertices have degree $2$. Let $p_k(G)$ be the number of pendant paths of length $k$ of $G$, and $q_k(G)$ be the number of vertices with degree $\\ge3$ which are an end vertex of some pendant paths of length $k$. Motivated by the problem of characterizing dendritic trees, N. Saito and E. Woei conjectured that any graph $G$ has some Laplacian eigenvalue with multiplicity at least $p_k(G)-q_k(G)$. We prove a more general result for both ","authors_text":"Ebrahim Ghorbani","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-10-17T11:32:24Z","title":"Proof of a conjecture on `plateaux' phenomenon of graph Laplacian eigenvalues"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.05117","kind":"arxiv","version":3},"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":"sha256:591e213e86f424961a3f472e771e483e7f81dc1494771781efa6775d01e3d87c","target":"record","created_at":"2026-05-18T01:17:32Z","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":"7124bed7777e610947fa3bacc48d87ec881e4711315fa92195a5a6acc81a6a04","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-10-17T11:32:24Z","title_canon_sha256":"705dda62bfc28e9f0a0db87d8f8ee5fcf09093c6e31fd81e729f64b758dedffe"},"schema_version":"1.0","source":{"id":"1510.05117","kind":"arxiv","version":3}},"canonical_sha256":"09498d134845d0a78bb313c204527602c3e07579a64e1dd36ebe4361a5087b0c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"09498d134845d0a78bb313c204527602c3e07579a64e1dd36ebe4361a5087b0c","first_computed_at":"2026-05-18T01:17:32.829626Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:17:32.829626Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lTOATY9tF+Sd32o0G1UoIsemd/hi/kpOqMhxyMc9XfItwfiAuxTw3y7w1fgoWr76mHAKJDmWG1O/lJyJ2zUAAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:17:32.830238Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.05117","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:591e213e86f424961a3f472e771e483e7f81dc1494771781efa6775d01e3d87c","sha256:ac47515a8c5759308834324e116f423fd6d5d973c80393f781d25ade3bed0125"],"state_sha256":"7cf56d515fa0668835eecd8659be1e4586acb4960f3d785d2601f2abcf63e14c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NMMgHgipbnPiajZjpe2VwC7vh0TEAmmp+xXV3KrEzFBTF8qf1N+pxbc6LbG6nVK1rHDe569JF9WikFISQ+z3Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T15:27:41.474889Z","bundle_sha256":"797feb1cf6ca0099bfb5052395af6279891076e6ac7d1e480c18543a17181756"}}