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In this paper, we proved for any integer $r\\geq 2$, and any real $p\\geq 1$, and any $r$-uniform hypergraph $H$ with $m={s\\choose r}$ edges (for some real $s\\geq r-1$), we have $$\\lambda_p(H)\\leq \\frac{rm}{s^{r/p}}.$$ The equality holds if"},"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":"1803.08653","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-03-23T04:22:35Z","cross_cats_sorted":[],"title_canon_sha256":"ab3db3770b281d3c24ea59b86f551bec030de36deb5535a343177d8e1b406279","abstract_canon_sha256":"7c73dc02bf61934294e44a0896e719a89ddf3a65abfdcbae7e6ed8ecdbef64e4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:20:18.951660Z","signature_b64":"9pG6KWFU5yWIBWSdLnrz1PlTwtcEwnfMxF7zh/PE1u76cwM4v01pSurt610mUEv2V0r52VZwEpWAQaWatGV/Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"64c36b0503689daf34701a0c1d2cdde7f1009b832903b7fc04a91ce1daf6a394","last_reissued_at":"2026-05-18T00:20:18.950929Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:20:18.950929Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The maximum $p$-Spectral Radius of Hypergraphs with $m$ Edges","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Linyuan Lu","submitted_at":"2018-03-23T04:22:35Z","abstract_excerpt":"For $r\\geq 2$ and $p\\geq 1$, the $p$-spectral radius of an $r$-uniform hypergraph $H=(V,E)$ on $n$ vertices is defined to be $$\\rho_p(H)=\\max_{{\\bf x}\\in \\mathbb{R}^n: \\|{\\bf x}\\|_p=1}r \\cdot \\!\\!\\!\\! \\sum_{\\{i_1,i_2,\\ldots, i_r\\}\\in E(H)} x_{i_1}x_{i_2}\\cdots x_{i_r},$$ where the maximum is taken over all ${\\bf x\\in \\mathbb{R}^n}$ with the $p$-norm equals 1. In this paper, we proved for any integer $r\\geq 2$, and any real $p\\geq 1$, and any $r$-uniform hypergraph $H$ with $m={s\\choose r}$ edges (for some real $s\\geq r-1$), we have $$\\lambda_p(H)\\leq \\frac{rm}{s^{r/p}}.$$ The equality holds if"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.08653","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":"1803.08653","created_at":"2026-05-18T00:20:18.951037+00:00"},{"alias_kind":"arxiv_version","alias_value":"1803.08653v1","created_at":"2026-05-18T00:20:18.951037+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.08653","created_at":"2026-05-18T00:20:18.951037+00:00"},{"alias_kind":"pith_short_12","alias_value":"MTBWWBIDNCO2","created_at":"2026-05-18T12:32:40.477152+00:00"},{"alias_kind":"pith_short_16","alias_value":"MTBWWBIDNCO26NDQ","created_at":"2026-05-18T12:32:40.477152+00:00"},{"alias_kind":"pith_short_8","alias_value":"MTBWWBID","created_at":"2026-05-18T12:32:40.477152+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/MTBWWBIDNCO26NDQDIGB2LG547","json":"https://pith.science/pith/MTBWWBIDNCO26NDQDIGB2LG547.json","graph_json":"https://pith.science/api/pith-number/MTBWWBIDNCO26NDQDIGB2LG547/graph.json","events_json":"https://pith.science/api/pith-number/MTBWWBIDNCO26NDQDIGB2LG547/events.json","paper":"https://pith.science/paper/MTBWWBID"},"agent_actions":{"view_html":"https://pith.science/pith/MTBWWBIDNCO26NDQDIGB2LG547","download_json":"https://pith.science/pith/MTBWWBIDNCO26NDQDIGB2LG547.json","view_paper":"https://pith.science/paper/MTBWWBID","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1803.08653&json=true","fetch_graph":"https://pith.science/api/pith-number/MTBWWBIDNCO26NDQDIGB2LG547/graph.json","fetch_events":"https://pith.science/api/pith-number/MTBWWBIDNCO26NDQDIGB2LG547/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MTBWWBIDNCO26NDQDIGB2LG547/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MTBWWBIDNCO26NDQDIGB2LG547/action/storage_attestation","attest_author":"https://pith.science/pith/MTBWWBIDNCO26NDQDIGB2LG547/action/author_attestation","sign_citation":"https://pith.science/pith/MTBWWBIDNCO26NDQDIGB2LG547/action/citation_signature","submit_replication":"https://pith.science/pith/MTBWWBIDNCO26NDQDIGB2LG547/action/replication_record"}},"created_at":"2026-05-18T00:20:18.951037+00:00","updated_at":"2026-05-18T00:20:18.951037+00:00"}