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We introduce a new graph operation called the clique cover product, denoted by $G^{\\mathscr{C}}\\star H^U$, as follows: for each clique $C_i\\in \\mathscr{C}$, add a copy of the graph $H$ and join every vertex of $C_i$ to every vertex of $U$. We prove that the independence polynomial of $G^{\\mathscr{C}}\\star H^U$ $$I(G^{\\mathscr{C}}\\star H^U;x)=I^q(H;x)I(G;\\frac{xI(H-U;x)}{I(H;x)}),$$ which generalizes some known results on independence polynomials of corona and "},"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":"1309.7673","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-09-29T23:00:43Z","cross_cats_sorted":[],"title_canon_sha256":"b82cc0b015461f18c9852f31908808d14d224dbacaeadd3a1c46fbd9346a8110","abstract_canon_sha256":"d20cd4cde8324b6686c29a4f8b2745d49b6df9c17a7f31a8e93b8b946e8bc02b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:11:48.763718Z","signature_b64":"LOIIcn1gf1537fwpA4J9/E13aNyU0FBYpybnnKwWli3ODfWwPCgCoYEG4sKDzfmDUtkAoY7fhtajuW2xFhuXBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0929e4fd2ef0117149f949c36fbeafebb3b36cf658fb9beef869a2e0017b5456","last_reissued_at":"2026-05-18T03:11:48.763239Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:11:48.763239Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Operations of graphs and unimodality of independence polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bao-Xuan Zhu","submitted_at":"2013-09-29T23:00:43Z","abstract_excerpt":"Given two graphs $G$ and $H$, assume that $\\mathscr{C}=\\{C_1,C_2,\\ldots, C_q\\}$ is a clique cover of $G$ and $U$ is a subset of $V(H)$. We introduce a new graph operation called the clique cover product, denoted by $G^{\\mathscr{C}}\\star H^U$, as follows: for each clique $C_i\\in \\mathscr{C}$, add a copy of the graph $H$ and join every vertex of $C_i$ to every vertex of $U$. We prove that the independence polynomial of $G^{\\mathscr{C}}\\star H^U$ $$I(G^{\\mathscr{C}}\\star H^U;x)=I^q(H;x)I(G;\\frac{xI(H-U;x)}{I(H;x)}),$$ which generalizes some known results on independence polynomials of corona and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.7673","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":"1309.7673","created_at":"2026-05-18T03:11:48.763323+00:00"},{"alias_kind":"arxiv_version","alias_value":"1309.7673v1","created_at":"2026-05-18T03:11:48.763323+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.7673","created_at":"2026-05-18T03:11:48.763323+00:00"},{"alias_kind":"pith_short_12","alias_value":"BEU6J7JO6AIX","created_at":"2026-05-18T12:27:38.830355+00:00"},{"alias_kind":"pith_short_16","alias_value":"BEU6J7JO6AIXCSPZ","created_at":"2026-05-18T12:27:38.830355+00:00"},{"alias_kind":"pith_short_8","alias_value":"BEU6J7JO","created_at":"2026-05-18T12:27:38.830355+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/BEU6J7JO6AIXCSPZJHBW7PVP5O","json":"https://pith.science/pith/BEU6J7JO6AIXCSPZJHBW7PVP5O.json","graph_json":"https://pith.science/api/pith-number/BEU6J7JO6AIXCSPZJHBW7PVP5O/graph.json","events_json":"https://pith.science/api/pith-number/BEU6J7JO6AIXCSPZJHBW7PVP5O/events.json","paper":"https://pith.science/paper/BEU6J7JO"},"agent_actions":{"view_html":"https://pith.science/pith/BEU6J7JO6AIXCSPZJHBW7PVP5O","download_json":"https://pith.science/pith/BEU6J7JO6AIXCSPZJHBW7PVP5O.json","view_paper":"https://pith.science/paper/BEU6J7JO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1309.7673&json=true","fetch_graph":"https://pith.science/api/pith-number/BEU6J7JO6AIXCSPZJHBW7PVP5O/graph.json","fetch_events":"https://pith.science/api/pith-number/BEU6J7JO6AIXCSPZJHBW7PVP5O/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BEU6J7JO6AIXCSPZJHBW7PVP5O/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BEU6J7JO6AIXCSPZJHBW7PVP5O/action/storage_attestation","attest_author":"https://pith.science/pith/BEU6J7JO6AIXCSPZJHBW7PVP5O/action/author_attestation","sign_citation":"https://pith.science/pith/BEU6J7JO6AIXCSPZJHBW7PVP5O/action/citation_signature","submit_replication":"https://pith.science/pith/BEU6J7JO6AIXCSPZJHBW7PVP5O/action/replication_record"}},"created_at":"2026-05-18T03:11:48.763323+00:00","updated_at":"2026-05-18T03:11:48.763323+00:00"}