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For $ p \\geq 0 $, we define the positive and negative $ p $-energies of $ G $ by $$ \\mathcal{E}_p^+(G) = \\sum_{\\lambda_i > 0} \\lambda_i^p, \\quad \\mathcal{E}_p^-(G) = \\sum_{\\lambda_i < 0} |\\lambda_i|^p, $$ where $ \\lambda_1 \\geq \\cdots \\geq \\lambda_n $ are the eigenvalues of $A_G $. We prove that for all $ p \\geq 0 $, $$ \\chi"},"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":"2504.01295","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-04-02T01:59:22Z","cross_cats_sorted":[],"title_canon_sha256":"8eb082be1100fe565d661383ae867b01d2f4ccf378533c0a509de4c952547095","abstract_canon_sha256":"e6d2829faadfa79ec9b7d16bf5e8a2e849b60e96618e5527b74672e4a54aab46"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:09:10.740380Z","signature_b64":"2zxGwrP+d+3bgxDwLpUJf9iBulD0SOP/5RMoSygkZJ+Q4NvfFcb7nEj43+N57CDTbMbSAJS0UqMnkDBNU2F+Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3061ffee4ca46445dace73c45954ee058b4781b42379a59cb15875007667c685","last_reissued_at":"2026-07-05T11:09:10.739890Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:09:10.739890Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Spectral Lower Bound on Chromatic Numbers using $p$-Energy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Clive Elphick, Quanyu Tang, Shengtong Zhang","submitted_at":"2025-04-02T01:59:22Z","abstract_excerpt":"Let $A_G $ be the adjacency matrix of a simple graph $ G $, and let $ \\chi(G) $, $ \\chi_f(G) $, $ \\chi_q(G) $, $ \\xi(G) $ and $ \\xi_f(G) $ denote its chromatic number, fractional chromatic number, quantum chromatic number, orthogonal rank and projective rank, respectively. For $ p \\geq 0 $, we define the positive and negative $ p $-energies of $ G $ by $$ \\mathcal{E}_p^+(G) = \\sum_{\\lambda_i > 0} \\lambda_i^p, \\quad \\mathcal{E}_p^-(G) = \\sum_{\\lambda_i < 0} |\\lambda_i|^p, $$ where $ \\lambda_1 \\geq \\cdots \\geq \\lambda_n $ are the eigenvalues of $A_G $. We prove that for all $ p \\geq 0 $, $$ \\chi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.01295","kind":"arxiv","version":5},"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/2504.01295/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":"2504.01295","created_at":"2026-07-05T11:09:10.739948+00:00"},{"alias_kind":"arxiv_version","alias_value":"2504.01295v5","created_at":"2026-07-05T11:09:10.739948+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2504.01295","created_at":"2026-07-05T11:09:10.739948+00:00"},{"alias_kind":"pith_short_12","alias_value":"GBQ773SMURSE","created_at":"2026-07-05T11:09:10.739948+00:00"},{"alias_kind":"pith_short_16","alias_value":"GBQ773SMURSELWWO","created_at":"2026-07-05T11:09:10.739948+00:00"},{"alias_kind":"pith_short_8","alias_value":"GBQ773SM","created_at":"2026-07-05T11:09:10.739948+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2509.05814","citing_title":"A graph energy conjecture through the lenses of semidefinite programming","ref_index":9,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GBQ773SMURSELWWOOPCFSVHOAW","json":"https://pith.science/pith/GBQ773SMURSELWWOOPCFSVHOAW.json","graph_json":"https://pith.science/api/pith-number/GBQ773SMURSELWWOOPCFSVHOAW/graph.json","events_json":"https://pith.science/api/pith-number/GBQ773SMURSELWWOOPCFSVHOAW/events.json","paper":"https://pith.science/paper/GBQ773SM"},"agent_actions":{"view_html":"https://pith.science/pith/GBQ773SMURSELWWOOPCFSVHOAW","download_json":"https://pith.science/pith/GBQ773SMURSELWWOOPCFSVHOAW.json","view_paper":"https://pith.science/paper/GBQ773SM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2504.01295&json=true","fetch_graph":"https://pith.science/api/pith-number/GBQ773SMURSELWWOOPCFSVHOAW/graph.json","fetch_events":"https://pith.science/api/pith-number/GBQ773SMURSELWWOOPCFSVHOAW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GBQ773SMURSELWWOOPCFSVHOAW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GBQ773SMURSELWWOOPCFSVHOAW/action/storage_attestation","attest_author":"https://pith.science/pith/GBQ773SMURSELWWOOPCFSVHOAW/action/author_attestation","sign_citation":"https://pith.science/pith/GBQ773SMURSELWWOOPCFSVHOAW/action/citation_signature","submit_replication":"https://pith.science/pith/GBQ773SMURSELWWOOPCFSVHOAW/action/replication_record"}},"created_at":"2026-07-05T11:09:10.739948+00:00","updated_at":"2026-07-05T11:09:10.739948+00:00"}