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Combin. 25(1) (2018) \\#P1.46) proved that the complete bipartite graph $K_{1,n}$ is stable and posed the question: \\textbf{Are all complete bipartite graphs stable?}\n  We answer this question by establishing the following results:\n  \\begin{itemize}\n  \\item The complete bipartite graphs $K_{2,n}$ and $K_{3,n}$ are stable.\n  \\item For any integer $k\\geq0"},"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":"2505.24381","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-05-30T09:10:28Z","cross_cats_sorted":[],"title_canon_sha256":"c9b611ee50d79629cc5758a14d62257e7e6c0f573b70b060f0a443978382a216","abstract_canon_sha256":"31750074cfdb090b75e58a0a765001a4c1cbc5d546035f0d86930d853c2bca5d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:12:48.034681Z","signature_b64":"t6yBGmQx/PLvqtYxQmXwJ1l8CZo7PCw0VdJEP6HpMbIHoC0iI3Z4Q/Wn5frR8ntpTVYOQWinuEjlZhD3OjMhAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4d9f1306f3ebace7fed821b4a1e9665f212946f361a85ff4194dadefa6b2202a","last_reissued_at":"2026-07-05T11:12:48.034158Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:12:48.034158Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The stability of independence polynomials of complete bipartite graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bo Ning, Guo Chen, Jianhua Tu","submitted_at":"2025-05-30T09:10:28Z","abstract_excerpt":"The independence polynomial of a graph is termed {\\it stable} if all its roots are located in the left half-plane $\\{z \\in \\mathbb{C} : \\mathrm{Re}(z) \\leq 0\\}$, and the graph itself is also referred to as stable. Brown and Cameron (Electron. J. Combin. 25(1) (2018) \\#P1.46) proved that the complete bipartite graph $K_{1,n}$ is stable and posed the question: \\textbf{Are all complete bipartite graphs stable?}\n  We answer this question by establishing the following results:\n  \\begin{itemize}\n  \\item The complete bipartite graphs $K_{2,n}$ and $K_{3,n}$ are stable.\n  \\item For any integer $k\\geq0"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.24381","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/2505.24381/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":"2505.24381","created_at":"2026-07-05T11:12:48.034212+00:00"},{"alias_kind":"arxiv_version","alias_value":"2505.24381v1","created_at":"2026-07-05T11:12:48.034212+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2505.24381","created_at":"2026-07-05T11:12:48.034212+00:00"},{"alias_kind":"pith_short_12","alias_value":"JWPRGBXT5OWO","created_at":"2026-07-05T11:12:48.034212+00:00"},{"alias_kind":"pith_short_16","alias_value":"JWPRGBXT5OWOP7WY","created_at":"2026-07-05T11:12:48.034212+00:00"},{"alias_kind":"pith_short_8","alias_value":"JWPRGBXT","created_at":"2026-07-05T11:12:48.034212+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/JWPRGBXT5OWOP7WYEG2KD2LGL4","json":"https://pith.science/pith/JWPRGBXT5OWOP7WYEG2KD2LGL4.json","graph_json":"https://pith.science/api/pith-number/JWPRGBXT5OWOP7WYEG2KD2LGL4/graph.json","events_json":"https://pith.science/api/pith-number/JWPRGBXT5OWOP7WYEG2KD2LGL4/events.json","paper":"https://pith.science/paper/JWPRGBXT"},"agent_actions":{"view_html":"https://pith.science/pith/JWPRGBXT5OWOP7WYEG2KD2LGL4","download_json":"https://pith.science/pith/JWPRGBXT5OWOP7WYEG2KD2LGL4.json","view_paper":"https://pith.science/paper/JWPRGBXT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2505.24381&json=true","fetch_graph":"https://pith.science/api/pith-number/JWPRGBXT5OWOP7WYEG2KD2LGL4/graph.json","fetch_events":"https://pith.science/api/pith-number/JWPRGBXT5OWOP7WYEG2KD2LGL4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JWPRGBXT5OWOP7WYEG2KD2LGL4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JWPRGBXT5OWOP7WYEG2KD2LGL4/action/storage_attestation","attest_author":"https://pith.science/pith/JWPRGBXT5OWOP7WYEG2KD2LGL4/action/author_attestation","sign_citation":"https://pith.science/pith/JWPRGBXT5OWOP7WYEG2KD2LGL4/action/citation_signature","submit_replication":"https://pith.science/pith/JWPRGBXT5OWOP7WYEG2KD2LGL4/action/replication_record"}},"created_at":"2026-07-05T11:12:48.034212+00:00","updated_at":"2026-07-05T11:12:48.034212+00:00"}