{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:VBGJ5UWNISBTIRQMOTNE5OT5E6","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"ddc28c39ddb1bb383695e72f6e59bb3bcb0860cfbd2d968d5ed4d9484b10a79e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-02T15:23:20Z","title_canon_sha256":"42b4d2cfc278dc118c1ae8712e2f95749f5327b2df03b246e428a22893a8ad0a"},"schema_version":"1.0","source":{"id":"1507.00625","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.00625","created_at":"2026-05-18T01:37:24Z"},{"alias_kind":"arxiv_version","alias_value":"1507.00625v1","created_at":"2026-05-18T01:37:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.00625","created_at":"2026-05-18T01:37:24Z"},{"alias_kind":"pith_short_12","alias_value":"VBGJ5UWNISBT","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"VBGJ5UWNISBTIRQM","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"VBGJ5UWN","created_at":"2026-05-18T12:29:44Z"}],"graph_snapshots":[{"event_id":"sha256:1f00b9263f70dd349301046eeb776c3b555e470da2558bf5611cffb0689f5b5a","target":"graph","created_at":"2026-05-18T01:37:24Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"This note presents a new spectral version of the graph Zarankiewicz problem: How large can be the maximum eigenvalue of the signless Laplacian of a graph of order $n$ that does not contain a specified complete bipartite subgraph. A conjecture is stated about general complete bipartite graphs, which is proved for infinitely many cases.\n  More precisely, it is shown that if $G$ is a graph of order $n,$ with no subgraph isomorphic to $K_{2,s+1},$ then the largest eigenvalue $q(G)$ of the signless Laplacian of $G$ satisfies \\[ q(G)\\leq\\frac{n+2s}{2}+\\frac{1}{2}\\sqrt{(n-2s)^{2}+8s}, \\] with equalit","authors_text":"Laura Patuzzi, Maria Aguieiras A. de Freitas, Vladimir Nikiforov","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-02T15:23:20Z","title":"Maxima of the Q-index: graphs with no K_s,t"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.00625","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:7f972e2f7767d0d842aea5b308ade3233ecd3acb0eda201f00c925cf778914f9","target":"record","created_at":"2026-05-18T01:37:24Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"ddc28c39ddb1bb383695e72f6e59bb3bcb0860cfbd2d968d5ed4d9484b10a79e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-02T15:23:20Z","title_canon_sha256":"42b4d2cfc278dc118c1ae8712e2f95749f5327b2df03b246e428a22893a8ad0a"},"schema_version":"1.0","source":{"id":"1507.00625","kind":"arxiv","version":1}},"canonical_sha256":"a84c9ed2cd448334460c74da4eba7d278ab00deb5eb8ce8878a2fa4f5c4b5313","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a84c9ed2cd448334460c74da4eba7d278ab00deb5eb8ce8878a2fa4f5c4b5313","first_computed_at":"2026-05-18T01:37:24.068657Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:37:24.068657Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yOU4oQ0rytsqWasoI3pWVD7aSj53pC+AyX2j0SQeLgyxGXKkEDXFMD2E1+1WWPHTF7PUvhIPUhix+gCsxQ0oBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:37:24.069391Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.00625","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7f972e2f7767d0d842aea5b308ade3233ecd3acb0eda201f00c925cf778914f9","sha256:1f00b9263f70dd349301046eeb776c3b555e470da2558bf5611cffb0689f5b5a"],"state_sha256":"b6f8a1c30c1af01102d9fb7e586b6ac74edf35adf37851e463f9d71b47e3cff6"}