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More generally, what remains true when the exponent $2$ is replaced by a constant less than $2$? We study the natural power-law version of this question for all $1<p\\le2$. For $1<p\\le 2$, define \\[\n  d_p(G)=\\max_{\\varnothing\\ne S\\subseteq V(G)}\\frac{e(G[S])}{|S|^p}. \\] We determine the sharp asymptotic upper bound for $\\lambda(G)$ in terms of "},"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":"2606.08913","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-08T01:26:29Z","cross_cats_sorted":[],"title_canon_sha256":"947fad88db043a3a587b4587ce1e4e721de82a296825635a5bcb05758955164a","abstract_canon_sha256":"321a6c4e46863c155cb7bc839ee689cb2a9388c032d1b16d9d79edf6f48916b4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-09T02:07:46.933528Z","signature_b64":"gElIsMmGl2FNluDl7mlH/lTECKi4lFgIYY3Cvlmaqt88IYjmY53YLvP2s7K6pvmxeoHOvX3iqGbqk3Z4wHqwCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"23885db17a1755db8199f08f6eeb66f493d86fde54f291365f44f8dbd33c4cab","last_reissued_at":"2026-06-09T02:07:46.932142Z","signature_status":"signed_v1","first_computed_at":"2026-06-09T02:07:46.932142Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sharp Bounds for Guiduli-Type Hereditary Spectral Problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dongxiu Cai, Jiasheng Zeng, Xiao-Dong Zhang","submitted_at":"2026-06-08T01:26:29Z","abstract_excerpt":"Guiduli asked in 1996 the following problem concerning the maximum spectral radius of a graph under hereditary density constraints. 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For $1<p\\le 2$, define \\[\n  d_p(G)=\\max_{\\varnothing\\ne S\\subseteq V(G)}\\frac{e(G[S])}{|S|^p}. \\] We determine the sharp asymptotic upper bound for $\\lambda(G)$ in terms of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.08913","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/2606.08913/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2606.08913","created_at":"2026-06-09T02:07:46.932751+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.08913v1","created_at":"2026-06-09T02:07:46.932751+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.08913","created_at":"2026-06-09T02:07:46.932751+00:00"},{"alias_kind":"pith_short_12","alias_value":"EOEF3ML2C5K5","created_at":"2026-06-09T02:07:46.932751+00:00"},{"alias_kind":"pith_short_16","alias_value":"EOEF3ML2C5K5XAMZ","created_at":"2026-06-09T02:07:46.932751+00:00"},{"alias_kind":"pith_short_8","alias_value":"EOEF3ML2","created_at":"2026-06-09T02:07:46.932751+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/EOEF3ML2C5K5XAMZ6CHW523G6S","json":"https://pith.science/pith/EOEF3ML2C5K5XAMZ6CHW523G6S.json","graph_json":"https://pith.science/api/pith-number/EOEF3ML2C5K5XAMZ6CHW523G6S/graph.json","events_json":"https://pith.science/api/pith-number/EOEF3ML2C5K5XAMZ6CHW523G6S/events.json","paper":"https://pith.science/paper/EOEF3ML2"},"agent_actions":{"view_html":"https://pith.science/pith/EOEF3ML2C5K5XAMZ6CHW523G6S","download_json":"https://pith.science/pith/EOEF3ML2C5K5XAMZ6CHW523G6S.json","view_paper":"https://pith.science/paper/EOEF3ML2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.08913&json=true","fetch_graph":"https://pith.science/api/pith-number/EOEF3ML2C5K5XAMZ6CHW523G6S/graph.json","fetch_events":"https://pith.science/api/pith-number/EOEF3ML2C5K5XAMZ6CHW523G6S/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EOEF3ML2C5K5XAMZ6CHW523G6S/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EOEF3ML2C5K5XAMZ6CHW523G6S/action/storage_attestation","attest_author":"https://pith.science/pith/EOEF3ML2C5K5XAMZ6CHW523G6S/action/author_attestation","sign_citation":"https://pith.science/pith/EOEF3ML2C5K5XAMZ6CHW523G6S/action/citation_signature","submit_replication":"https://pith.science/pith/EOEF3ML2C5K5XAMZ6CHW523G6S/action/replication_record"}},"created_at":"2026-06-09T02:07:46.932751+00:00","updated_at":"2026-06-09T02:07:46.932751+00:00"}