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We prove that $\\xi_k$ can be determined by considering a finite set of graphs and show that $\\xi_k=2$ for $k\\le 2$, $\\xi_3=1.430\\cdots$, $\\xi_4=1.361\\cdots$ and $\\xi_5=1.317\\cdots$. We also prove that for any bridgeless graph $G=(V,E)$, if all roots of $F(G,\\lambda)$ ar"},"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":"1403.1916","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-03-08T01:46:59Z","cross_cats_sorted":[],"title_canon_sha256":"9aec45dc16386c66d130d31eb7d1c95b0681e6910bab2222c1976313f025fbfd","abstract_canon_sha256":"d807d72da54a98ec57533cea069425081e6ff45466bb7f5735a6b778325f7cf9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:56:51.289819Z","signature_b64":"VMzMy7Y5fMI3W3x6AVTLF/ggobvDcehUcEUaMadEdGn0fUj/3UR3cnZAZtJ+SQzHtqOYU3NLvsF1ZAQyMvmeDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"33dffd158b36fa92f1b73ea51bbd76ef40511aa2ed0bc8d25b3cd1d6fd83e53a","last_reissued_at":"2026-05-18T02:56:51.289303Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:56:51.289303Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Zero-free Intervals of Flow Polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Fengming Dong","submitted_at":"2014-03-08T01:46:59Z","abstract_excerpt":"This article studies real roots of the flow polynomial $F(G,\\lambda)$ of a bridgeless graph $G$. For any integer $k\\ge 0$, let $\\xi_k$ be the supremum in $(1,2]$ such that $F(G,\\lambda)$ has no real roots in $(1,\\xi_k)$ for all graphs $G$ with $|W(G)|\\le k$, where $W(G)$ is the set of vertices in $G$ of degrees larger than $3$. We prove that $\\xi_k$ can be determined by considering a finite set of graphs and show that $\\xi_k=2$ for $k\\le 2$, $\\xi_3=1.430\\cdots$, $\\xi_4=1.361\\cdots$ and $\\xi_5=1.317\\cdots$. We also prove that for any bridgeless graph $G=(V,E)$, if all roots of $F(G,\\lambda)$ ar"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.1916","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":"1403.1916","created_at":"2026-05-18T02:56:51.289387+00:00"},{"alias_kind":"arxiv_version","alias_value":"1403.1916v1","created_at":"2026-05-18T02:56:51.289387+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1403.1916","created_at":"2026-05-18T02:56:51.289387+00:00"},{"alias_kind":"pith_short_12","alias_value":"GPP72FMLG35J","created_at":"2026-05-18T12:28:30.664211+00:00"},{"alias_kind":"pith_short_16","alias_value":"GPP72FMLG35JF4NX","created_at":"2026-05-18T12:28:30.664211+00:00"},{"alias_kind":"pith_short_8","alias_value":"GPP72FML","created_at":"2026-05-18T12:28:30.664211+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/GPP72FMLG35JF4NXH2SRXPLW55","json":"https://pith.science/pith/GPP72FMLG35JF4NXH2SRXPLW55.json","graph_json":"https://pith.science/api/pith-number/GPP72FMLG35JF4NXH2SRXPLW55/graph.json","events_json":"https://pith.science/api/pith-number/GPP72FMLG35JF4NXH2SRXPLW55/events.json","paper":"https://pith.science/paper/GPP72FML"},"agent_actions":{"view_html":"https://pith.science/pith/GPP72FMLG35JF4NXH2SRXPLW55","download_json":"https://pith.science/pith/GPP72FMLG35JF4NXH2SRXPLW55.json","view_paper":"https://pith.science/paper/GPP72FML","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1403.1916&json=true","fetch_graph":"https://pith.science/api/pith-number/GPP72FMLG35JF4NXH2SRXPLW55/graph.json","fetch_events":"https://pith.science/api/pith-number/GPP72FMLG35JF4NXH2SRXPLW55/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GPP72FMLG35JF4NXH2SRXPLW55/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GPP72FMLG35JF4NXH2SRXPLW55/action/storage_attestation","attest_author":"https://pith.science/pith/GPP72FMLG35JF4NXH2SRXPLW55/action/author_attestation","sign_citation":"https://pith.science/pith/GPP72FMLG35JF4NXH2SRXPLW55/action/citation_signature","submit_replication":"https://pith.science/pith/GPP72FMLG35JF4NXH2SRXPLW55/action/replication_record"}},"created_at":"2026-05-18T02:56:51.289387+00:00","updated_at":"2026-05-18T02:56:51.289387+00:00"}