{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:D47DJEDJTYXFGPRX3FDYU6IYSB","short_pith_number":"pith:D47DJEDJ","schema_version":"1.0","canonical_sha256":"1f3e3490699e2e533e37d9478a79189067435ed1381bb49d71eb2e6539d49035","source":{"kind":"arxiv","id":"1507.05754","version":1},"attestation_state":"computed","paper":{"title":"Unimodality of the independence polynomials of some composite graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bao-Xuan Zhu, Qinglin Lu","submitted_at":"2015-07-21T09:11:15Z","abstract_excerpt":"Let $I(G;x)$ denote the independence polynomial of a graph $G$. In this paper we study the unimodality properties of $I(G;x)$ for some composite graphs $G$.\n  Given two graphs $G_1$ and $G_2$, let $G_1[G_2]$ denote the lexicographic product of $G_1$ and $G_2$. Assume $I(G_1;x)=\\sum_{i\\geq0}a_ix^i$ and $I(G_2;x)=\\sum_{i\\geq0}b_ix^i$, where $I(G_2;x)$ is log-concave. Then we prove (i) if $I(G_1;x)$ is log-concave and $(a^2_i-a_{i-1}a_{i+1})b^2_1\\geq a_ia_{i-1}b_2$ for all $1\\leq i \\leq \\alpha(G_1)$, then $I(G_1[G_2];x)$ is log-concave; (ii) if $a_{i-1}\\leq b_1a_i$ for $1\\leq i\\leq \\alpha(G_1)$, "},"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":"1507.05754","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-21T09:11:15Z","cross_cats_sorted":[],"title_canon_sha256":"6f8db99a6019e2201682333eef74bae3fe16301e3728490509ec22e46b2707f4","abstract_canon_sha256":"595d13a5d0f62e7eafb1aefb7fef4169dc4c239dc69e1a914565ceffc06005c4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:32.575599Z","signature_b64":"THRC86yAS6Fdd8NQ4cEMPPpoM+tdDOA5AbMufc98BVrq9dbgDv2S6zJ9PjA1asz8xOkkzJh3MyCc0yHEnIH8Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1f3e3490699e2e533e37d9478a79189067435ed1381bb49d71eb2e6539d49035","last_reissued_at":"2026-05-18T01:36:32.574760Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:32.574760Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Unimodality of the independence polynomials of some composite graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bao-Xuan Zhu, Qinglin Lu","submitted_at":"2015-07-21T09:11:15Z","abstract_excerpt":"Let $I(G;x)$ denote the independence polynomial of a graph $G$. In this paper we study the unimodality properties of $I(G;x)$ for some composite graphs $G$.\n  Given two graphs $G_1$ and $G_2$, let $G_1[G_2]$ denote the lexicographic product of $G_1$ and $G_2$. Assume $I(G_1;x)=\\sum_{i\\geq0}a_ix^i$ and $I(G_2;x)=\\sum_{i\\geq0}b_ix^i$, where $I(G_2;x)$ is log-concave. Then we prove (i) if $I(G_1;x)$ is log-concave and $(a^2_i-a_{i-1}a_{i+1})b^2_1\\geq a_ia_{i-1}b_2$ for all $1\\leq i \\leq \\alpha(G_1)$, then $I(G_1[G_2];x)$ is log-concave; (ii) if $a_{i-1}\\leq b_1a_i$ for $1\\leq i\\leq \\alpha(G_1)$, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.05754","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":"1507.05754","created_at":"2026-05-18T01:36:32.574890+00:00"},{"alias_kind":"arxiv_version","alias_value":"1507.05754v1","created_at":"2026-05-18T01:36:32.574890+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.05754","created_at":"2026-05-18T01:36:32.574890+00:00"},{"alias_kind":"pith_short_12","alias_value":"D47DJEDJTYXF","created_at":"2026-05-18T12:29:17.054201+00:00"},{"alias_kind":"pith_short_16","alias_value":"D47DJEDJTYXFGPRX","created_at":"2026-05-18T12:29:17.054201+00:00"},{"alias_kind":"pith_short_8","alias_value":"D47DJEDJ","created_at":"2026-05-18T12:29:17.054201+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/D47DJEDJTYXFGPRX3FDYU6IYSB","json":"https://pith.science/pith/D47DJEDJTYXFGPRX3FDYU6IYSB.json","graph_json":"https://pith.science/api/pith-number/D47DJEDJTYXFGPRX3FDYU6IYSB/graph.json","events_json":"https://pith.science/api/pith-number/D47DJEDJTYXFGPRX3FDYU6IYSB/events.json","paper":"https://pith.science/paper/D47DJEDJ"},"agent_actions":{"view_html":"https://pith.science/pith/D47DJEDJTYXFGPRX3FDYU6IYSB","download_json":"https://pith.science/pith/D47DJEDJTYXFGPRX3FDYU6IYSB.json","view_paper":"https://pith.science/paper/D47DJEDJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1507.05754&json=true","fetch_graph":"https://pith.science/api/pith-number/D47DJEDJTYXFGPRX3FDYU6IYSB/graph.json","fetch_events":"https://pith.science/api/pith-number/D47DJEDJTYXFGPRX3FDYU6IYSB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/D47DJEDJTYXFGPRX3FDYU6IYSB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/D47DJEDJTYXFGPRX3FDYU6IYSB/action/storage_attestation","attest_author":"https://pith.science/pith/D47DJEDJTYXFGPRX3FDYU6IYSB/action/author_attestation","sign_citation":"https://pith.science/pith/D47DJEDJTYXFGPRX3FDYU6IYSB/action/citation_signature","submit_replication":"https://pith.science/pith/D47DJEDJTYXFGPRX3FDYU6IYSB/action/replication_record"}},"created_at":"2026-05-18T01:36:32.574890+00:00","updated_at":"2026-05-18T01:36:32.574890+00:00"}