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We first give a short algebraic proof of this result. For a vertex $v\\in V(G)$, let \\(c(v)\\) denote the order of the largest clique of \\(G\\) containing \\(v\\). Our main result is the following vertex localized bound that refines the result of Abreu and Nikiforov: \\[\n  q(G) \\leq\n  2\\sum_{v\\in V(G)}\\left(1-\\frac{1}{c(v)}\\r"},"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":"2605.23283","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-22T06:51:14Z","cross_cats_sorted":[],"title_canon_sha256":"172f6af3dc209ee2858e0b28219b0ab1a7817da9246c679e878703b5c1e843e0","abstract_canon_sha256":"651e32786198f542576b3e62d5eed2f7d6dcc80d4a29cefbef3bd2faa2384850"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-25T02:01:47.208011Z","signature_b64":"bTWJtWSEvmH3Yqh3v4MlRXw/ZF3E54tw6d9Us7DjNXLWam6MfxL7v1Odp0su9AvrJ9/CS//J3qaGoSsmWjdYCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"417fc6b2556704e8d8ce87537c495e00a37829b29ee2096781ba9795fa348273","last_reissued_at":"2026-05-25T02:01:47.207440Z","signature_status":"signed_v1","first_computed_at":"2026-05-25T02:01:47.207440Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Localized Tur\\'{a}n-type inequalities for $Q$-index","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hitesh Kumar, M. Rajesh Kannan, Shivaramakrishna Pragada","submitted_at":"2026-05-22T06:51:14Z","abstract_excerpt":"For a connected graph \\(G\\), let $q(G)$ denote the $Q$-index of $G$, i.e., the largest eigenvalue of its signless Laplacian matrix. Abreu and Nikiforov (2013) showed that \\[\n  q(G) \\leq 2n\\left(1-\\frac{1}{\\omega(G)}\\right), \\] where $\\omega(G)$ denotes the clique number of $G$. We first give a short algebraic proof of this result. For a vertex $v\\in V(G)$, let \\(c(v)\\) denote the order of the largest clique of \\(G\\) containing \\(v\\). Our main result is the following vertex localized bound that refines the result of Abreu and Nikiforov: \\[\n  q(G) \\leq\n  2\\sum_{v\\in V(G)}\\left(1-\\frac{1}{c(v)}\\r"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.23283","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/2605.23283/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":"2605.23283","created_at":"2026-05-25T02:01:47.207546+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.23283v1","created_at":"2026-05-25T02:01:47.207546+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.23283","created_at":"2026-05-25T02:01:47.207546+00:00"},{"alias_kind":"pith_short_12","alias_value":"IF74NMSVM4CO","created_at":"2026-05-25T02:01:47.207546+00:00"},{"alias_kind":"pith_short_16","alias_value":"IF74NMSVM4CORWGO","created_at":"2026-05-25T02:01:47.207546+00:00"},{"alias_kind":"pith_short_8","alias_value":"IF74NMSV","created_at":"2026-05-25T02:01:47.207546+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/IF74NMSVM4CORWGOQ5JXYSK6AC","json":"https://pith.science/pith/IF74NMSVM4CORWGOQ5JXYSK6AC.json","graph_json":"https://pith.science/api/pith-number/IF74NMSVM4CORWGOQ5JXYSK6AC/graph.json","events_json":"https://pith.science/api/pith-number/IF74NMSVM4CORWGOQ5JXYSK6AC/events.json","paper":"https://pith.science/paper/IF74NMSV"},"agent_actions":{"view_html":"https://pith.science/pith/IF74NMSVM4CORWGOQ5JXYSK6AC","download_json":"https://pith.science/pith/IF74NMSVM4CORWGOQ5JXYSK6AC.json","view_paper":"https://pith.science/paper/IF74NMSV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.23283&json=true","fetch_graph":"https://pith.science/api/pith-number/IF74NMSVM4CORWGOQ5JXYSK6AC/graph.json","fetch_events":"https://pith.science/api/pith-number/IF74NMSVM4CORWGOQ5JXYSK6AC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IF74NMSVM4CORWGOQ5JXYSK6AC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IF74NMSVM4CORWGOQ5JXYSK6AC/action/storage_attestation","attest_author":"https://pith.science/pith/IF74NMSVM4CORWGOQ5JXYSK6AC/action/author_attestation","sign_citation":"https://pith.science/pith/IF74NMSVM4CORWGOQ5JXYSK6AC/action/citation_signature","submit_replication":"https://pith.science/pith/IF74NMSVM4CORWGOQ5JXYSK6AC/action/replication_record"}},"created_at":"2026-05-25T02:01:47.207546+00:00","updated_at":"2026-05-25T02:01:47.207546+00:00"}