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For every real $\\alpha\\in\\left[ 0,1\\right] $, write $A_{\\alpha}\\left( G\\right) $ for the matrix \\[ A_{\\alpha}\\left( G\\right) =\\alpha D\\left( G\\right) +(1-\\alpha)A\\left( G\\right) . \\] Let $\\alpha_{0}\\left( G\\right) $ be the smallest $\\alpha$ for which $A_{\\alpha}(G)$ is positive semidefinite. It is known that $\\alpha_{0}\\left( G\\right) \\leq1/2$. The main results of this paper are:\n  (1) if $G$ is $d$-regular then \\[ \\alpha_{0}=\\frac{-\\lambda_{\\min}(A(G))}{d-\\lambda_{\\min}(A(G))}, \\] wher"},"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":"1611.01818","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-06T18:27:26Z","cross_cats_sorted":[],"title_canon_sha256":"173280cba1ee5f73848c99a4d9e126ae9694f3152cbbeaa725fbd1976674e882","abstract_canon_sha256":"d5c8314b725cbe55cfd89752f56f50625b4ada9e98377a62fa2dcd66868b3afb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:00:04.414780Z","signature_b64":"h6Jd5vbsjJllFX1vWWJYfuiDdIyKYxXt7QgcQSydSWoo6hrsOdjXdWCswqEDC8F9cE1tSe3DBBsT5TmhCdT1BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"461522f5641226123e9cc208ed1872d469fc6e688a836e49195f80d1d238779e","last_reissued_at":"2026-05-18T01:00:04.414227Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:00:04.414227Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A note on the positive semidefinitness of $A_\\alpha (G)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Oscar Rojo, Vladimir Nikiforov","submitted_at":"2016-11-06T18:27:26Z","abstract_excerpt":"Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For every real $\\alpha\\in\\left[ 0,1\\right] $, write $A_{\\alpha}\\left( G\\right) $ for the matrix \\[ A_{\\alpha}\\left( G\\right) =\\alpha D\\left( G\\right) +(1-\\alpha)A\\left( G\\right) . \\] Let $\\alpha_{0}\\left( G\\right) $ be the smallest $\\alpha$ for which $A_{\\alpha}(G)$ is positive semidefinite. It is known that $\\alpha_{0}\\left( G\\right) \\leq1/2$. The main results of this paper are:\n  (1) if $G$ is $d$-regular then \\[ \\alpha_{0}=\\frac{-\\lambda_{\\min}(A(G))}{d-\\lambda_{\\min}(A(G))}, \\] wher"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.01818","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":"1611.01818","created_at":"2026-05-18T01:00:04.414308+00:00"},{"alias_kind":"arxiv_version","alias_value":"1611.01818v1","created_at":"2026-05-18T01:00:04.414308+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.01818","created_at":"2026-05-18T01:00:04.414308+00:00"},{"alias_kind":"pith_short_12","alias_value":"IYKSF5LECITB","created_at":"2026-05-18T12:30:22.444734+00:00"},{"alias_kind":"pith_short_16","alias_value":"IYKSF5LECITBEPU4","created_at":"2026-05-18T12:30:22.444734+00:00"},{"alias_kind":"pith_short_8","alias_value":"IYKSF5LE","created_at":"2026-05-18T12:30:22.444734+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/IYKSF5LECITBEPU4YIEO2GDS2R","json":"https://pith.science/pith/IYKSF5LECITBEPU4YIEO2GDS2R.json","graph_json":"https://pith.science/api/pith-number/IYKSF5LECITBEPU4YIEO2GDS2R/graph.json","events_json":"https://pith.science/api/pith-number/IYKSF5LECITBEPU4YIEO2GDS2R/events.json","paper":"https://pith.science/paper/IYKSF5LE"},"agent_actions":{"view_html":"https://pith.science/pith/IYKSF5LECITBEPU4YIEO2GDS2R","download_json":"https://pith.science/pith/IYKSF5LECITBEPU4YIEO2GDS2R.json","view_paper":"https://pith.science/paper/IYKSF5LE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1611.01818&json=true","fetch_graph":"https://pith.science/api/pith-number/IYKSF5LECITBEPU4YIEO2GDS2R/graph.json","fetch_events":"https://pith.science/api/pith-number/IYKSF5LECITBEPU4YIEO2GDS2R/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IYKSF5LECITBEPU4YIEO2GDS2R/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IYKSF5LECITBEPU4YIEO2GDS2R/action/storage_attestation","attest_author":"https://pith.science/pith/IYKSF5LECITBEPU4YIEO2GDS2R/action/author_attestation","sign_citation":"https://pith.science/pith/IYKSF5LECITBEPU4YIEO2GDS2R/action/citation_signature","submit_replication":"https://pith.science/pith/IYKSF5LECITBEPU4YIEO2GDS2R/action/replication_record"}},"created_at":"2026-05-18T01:00:04.414308+00:00","updated_at":"2026-05-18T01:00:04.414308+00:00"}