{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:YQF2PNCY2TFCIIDD5RHS74QIK3","short_pith_number":"pith:YQF2PNCY","schema_version":"1.0","canonical_sha256":"c40ba7b458d4ca242063ec4f2ff20856f470ce842529e09a5ddbce4b2df2e3b5","source":{"kind":"arxiv","id":"1104.4280","version":1},"attestation_state":"computed","paper":{"title":"On the ordering of trees by the Laplacian coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aleksandar Ili\\' c","submitted_at":"2011-04-21T15:02:04Z","abstract_excerpt":"We generalize the results from [X.-D. Zhang, X.-P. Lv, Y.-H. Chen, \\textit{Ordering trees by the Laplacian coefficients}, Linear Algebra Appl. (2009), doi:10.1016/j.laa.2009.04.018] on the partial ordering of trees with given diameter. For two $n$-vertex trees $T_1$ and $T_2$, if $c_k (T_1) \\leqslant c_k (T_2)$ holds for all Laplacian coefficients $c_k$, $k = 0, 1, ..., n$, we say that $T_1$ is dominated by $T_2$ and write $T_1 \\preceq_c T_2$. We proved that among $n$ vertex trees with fixed diameter $d$, the caterpillar $C_{n, d}$ has minimal Laplacian coefficients $c_k$, $k = 0, 1,..., n$. T"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1104.4280","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-04-21T15:02:04Z","cross_cats_sorted":[],"title_canon_sha256":"04d7d49f4023d35ea932c6b92443ab9c630eedfd216117fba51cfc798aebcfbe","abstract_canon_sha256":"187c0d2bd4c820624a041cf5a9a0f671dcd551e132345fb0f6ecddf546836720"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:22:01.503374Z","signature_b64":"duRUEVB6bsSH6BUbDMtlAde6lXshjkTMMy2XIBPTlXLKC59p237p5pEMp5d9rWSZdQKO5V9+CoEeRApOrxp3Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c40ba7b458d4ca242063ec4f2ff20856f470ce842529e09a5ddbce4b2df2e3b5","last_reissued_at":"2026-05-18T02:22:01.502810Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:22:01.502810Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the ordering of trees by the Laplacian coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aleksandar Ili\\' c","submitted_at":"2011-04-21T15:02:04Z","abstract_excerpt":"We generalize the results from [X.-D. Zhang, X.-P. Lv, Y.-H. Chen, \\textit{Ordering trees by the Laplacian coefficients}, Linear Algebra Appl. (2009), doi:10.1016/j.laa.2009.04.018] on the partial ordering of trees with given diameter. For two $n$-vertex trees $T_1$ and $T_2$, if $c_k (T_1) \\leqslant c_k (T_2)$ holds for all Laplacian coefficients $c_k$, $k = 0, 1, ..., n$, we say that $T_1$ is dominated by $T_2$ and write $T_1 \\preceq_c T_2$. We proved that among $n$ vertex trees with fixed diameter $d$, the caterpillar $C_{n, d}$ has minimal Laplacian coefficients $c_k$, $k = 0, 1,..., n$. T"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.4280","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1104.4280","created_at":"2026-05-18T02:22:01.502897+00:00"},{"alias_kind":"arxiv_version","alias_value":"1104.4280v1","created_at":"2026-05-18T02:22:01.502897+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.4280","created_at":"2026-05-18T02:22:01.502897+00:00"},{"alias_kind":"pith_short_12","alias_value":"YQF2PNCY2TFC","created_at":"2026-05-18T12:26:47.523578+00:00"},{"alias_kind":"pith_short_16","alias_value":"YQF2PNCY2TFCIIDD","created_at":"2026-05-18T12:26:47.523578+00:00"},{"alias_kind":"pith_short_8","alias_value":"YQF2PNCY","created_at":"2026-05-18T12:26:47.523578+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/YQF2PNCY2TFCIIDD5RHS74QIK3","json":"https://pith.science/pith/YQF2PNCY2TFCIIDD5RHS74QIK3.json","graph_json":"https://pith.science/api/pith-number/YQF2PNCY2TFCIIDD5RHS74QIK3/graph.json","events_json":"https://pith.science/api/pith-number/YQF2PNCY2TFCIIDD5RHS74QIK3/events.json","paper":"https://pith.science/paper/YQF2PNCY"},"agent_actions":{"view_html":"https://pith.science/pith/YQF2PNCY2TFCIIDD5RHS74QIK3","download_json":"https://pith.science/pith/YQF2PNCY2TFCIIDD5RHS74QIK3.json","view_paper":"https://pith.science/paper/YQF2PNCY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1104.4280&json=true","fetch_graph":"https://pith.science/api/pith-number/YQF2PNCY2TFCIIDD5RHS74QIK3/graph.json","fetch_events":"https://pith.science/api/pith-number/YQF2PNCY2TFCIIDD5RHS74QIK3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YQF2PNCY2TFCIIDD5RHS74QIK3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YQF2PNCY2TFCIIDD5RHS74QIK3/action/storage_attestation","attest_author":"https://pith.science/pith/YQF2PNCY2TFCIIDD5RHS74QIK3/action/author_attestation","sign_citation":"https://pith.science/pith/YQF2PNCY2TFCIIDD5RHS74QIK3/action/citation_signature","submit_replication":"https://pith.science/pith/YQF2PNCY2TFCIIDD5RHS74QIK3/action/replication_record"}},"created_at":"2026-05-18T02:22:01.502897+00:00","updated_at":"2026-05-18T02:22:01.502897+00:00"}